INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 40, 1—13 (1997) APPLICATION OF WAVELETS ON THE INTERVAL TO NUMERICAL ANALYSIS OF INTEGRAL EQUATIONS IN ELECTROMAGNETIC SCATTERING PROBLEMS GAOFENG WANG Synopsys, Inc., 700 East Middlefield Road, Mountain »iew, CA 94043, º.S.A. SUMMARY The wavelet expansions on the interval are employed for solving the problems of the electromagnetic (EM) scattering from two-dimensional (2-D) conducting objects. The arbitrary configurations of scatterers are modeled using the boundary element method (BEM). By using the wavelets on the interval as basis and test functions, a sparse matrix equation is generated from the integral equation under study. The resulted sparse matrix equation allows the use of sparse matrix solvers or multi-level iterations for rapid solution. The utilization of wavelets on the interval circumvents the difficulties in the application of the wavelets on the real line to finite interval problems, and has no periodicity constraint to the unknown function that is usually imposed by periodic wavelets. Numerical examples are provided and compared with the previously published data or other methods. KEY WORDS: electromagnetic fields and scattering; wavelets on the interval; numerical analysis; integral equation; boundary element method I. INTRODUCTION The applications of wavelets in computational electromagnetics have recently attracted a great deal of attention. Traditionally, the conventional numerical analysis of integral equations leads to a dense matrix due to the global nature of integral equations. For a large-scale problem, the solution procedure of a dense matrix equation is often prohibitively slow. A salient feature of the wavelets is that using wavelets as basis functions in the numerical solution of integral equations can lead to sparse matrix equations (e.g. see References 1—5). The wavelets have localization properties in both space and frequency. Using the wavelets as the basis functions generates strong decorrelation among the expansion coefficients and weakens the global coupling effects in integral equations. Thus, it is very likely to attain a sparse matrix equation from an integral equation after discretization. Moreover, the multiresolution analysis inherent in a wavelet expansion naturally provides a multi-level scheme, which has proven to be a very stable and fast convergent iterative method (e.g. see Reference 6). Originally, wavelets are bases on the whole straight real line. Some difficulties arise when applying such wavelets on the whole real line to problems over finite bounded domains, in particular, to problems over curved domains (e.g. see References 2, 5 and 7). To overcome these difficulties, a hybrid wavelet expansion and boundary element method (HWBM) was introduced in References 3 and 5. However, the periodic wavelets initially used in the HWBM are only good for the unknown functions that possess a periodic nature. In Reference 8, semi-orthogonal spline wavelets on [0, 1] were introduced to solve first-kind integral equations. The semi-orthogonal CCC 0029—5981/97/010001—13 ( 1997 by John Wiley & Sons, Ltd. Received 8 September 1995 Revised 18 May 1996 2 G. WANG spline wavelets on [0, 1] are applicable to non-periodic unknown functions on finite bounded domains, but no longer lead to zero residual methods since the basis functions are not completely orthogonal. Recently, the intervallic orthogonal wavelets, constructed from Coiflets,9 were applied to solve surface integral equations in Reference 10. But only the scaling functions were used in the wavelet expansions of Reference 10, which results in an analysis at only the finest level instead of a multiresolution analysis and does not take advantage of the wavelet functions. Here, the orthogonal wavelets on the interval, constructed from Daubechies compactly supported wavelets,7 are employed in the HWBM to accelerate the numerical solution of integral equations on bounded domains. This is illustrated by solving the EM scattering problems of 2-D conducting objects. By applying the BEM geometrical representation, arbitrary curved contours of scatterers are mapped into the definition domain [0, 1] of the wavelets on the interval. Both the scaling functions and the wavelet functions on [0, 1] are employed to form multi-level wavelet expansions. The unknown current is then expressed using such a multi-level wavelet expansion, and thus a multiresolution analysis is generated. The theoretical content is presented in the next section, which consists of three subsections. The first subsection is dedicated to the orthogonal wavelets on the interval, their expansion and the accompanying multiresolution analysis. After a brief review of the integral equation for 2-D EM scattering problems given in the second subsection, the third subsection presents an HWBM analysis using the orthogonal wavelet on the interval. Numerical results are included in Section 3, followed by a brief conclusion in Section 4. 2. THEORY 2.1. Orthogonal wavelets on the interval and multiresolution analysis Given the scaling function /(x) and the mother wavelet t(x) of the Daubechies N vanishing moment family on the real line, an orthogonal wavelet on [0, 1] can be constructed by preserving the interior wavelet (and scaling) functions and adding adapted edge wavelet (and scaling) functions such that their union generates all polynomials on [0, 1] up to degree N!1.7 More precisely, at resolution level 2m*2N, the scaling function /M (x) and the wavelet function m,n tM (x) on [0, 1] can be written as m,n 2m@2/-%&5 (2mx) if 0)n(N n (1) /M (x)" 2m@2/(2mx!n) if N)n(2m!N m,n [2m(x!1)] if 2m!N)n(2m 2m@2/3*')5 n~2m and 2m@2t-%&5 (2mx) if 0)n(N n tM (x)" 2m@2t(2mx!n) if N)n(2m!N m,n [2m(x!1)] if 2m!N)n(2m 2m@2t3*')5 n~2m (2) where /-%&5 (x), t-%&5 (x), /3*')5 (x), and t3*')5 (x) are, respectively, the (n#1)th left edge scaling n n n n function, the (n#1)th left edge wavelet function, the (!n)th right edge scaling function, and the (!n)th right edge wavelet function, which are completely characterized by two sets of two-scale relations.7 Figure 1 shows the edge wavelet and scaling functions constructed from the Daubechies wavelet with vanishing moment N"6. APPLICATION OF WAVELETS ON THE INTERVAL TO NUMERICAL ANALYSIS Figure 1. (a, b) (continued) 3 4 G. WANG Figure 1. The wavelet on [0, 1] constructed from the Daubechies orthogonal wavelet with vanishing moment N"6. Note that the abbreviations /-%&5 , / , /3*')5 , t-%&5 , t , and t3*')5 denote the scaling and wavelet functions 2m@2/-%&5 (2mx), m,n m,n m,n m,n m,n m,n n 2m@2/(2mx!n), 2m@2/3*')5 [2m(x!1)], 2m@2t-%&5 (2mx), 2m@2t(2mx!n), and 2m@2t3*')5 [2m(x!1)], respectively n n n APPLICATION OF WAVELETS ON THE INTERVAL TO NUMERICAL ANALYSIS 5 An orthogonal multiresolution analysis on [0, 1] can be formed from the above orthogonal wavelet on [0, 1] as follows: = V1 LV1 LV1 L· · · with Z V1 "¸2([0, 1]) (3) mÒ mÒ`1 mÒ`2 m m/mÒ where V1 "SpanM/M D n"0, 1, . . . , 2m!1N and m is any positive integer satisfying 2mÒ*2N. m m,n 0 Furthermore, an orthogonal wavelet decomposition can be obtained as V1 "V1 = W1 , m*m m`1 m m 0 (4) where W 1 "Span MtM D n"0, 1, 2 , 2m!1N. The scaling functions M/M N and the m m,n m,n 0)n(2m wavelet functions MtM N0)n(2m constitute an orthonormal basis in V1 and W1 , respectively. m,n m m By repeatedly applying the decomposition relation (4), one can readily prove that the following set of wavelet and scaling functions forms an orthonormal basis in V1 : m g (x)"/M (x), 0)n(2mÒ n mÒ ,n (5) g2.{#n"tM (x), 0)n(2m{, m )m@(m m{,n 0 Note that this basis contains the wavelet functions with various length scales ranging from 2~mÒ to 2~m`1 besides the scaling functions with length scale 2~mÒ. An orthogonal multiresolution analysis (i.e. (m!m #1)-level analysis) must be generated by this basis. One merit of this basis is 0 that the total number of the wavelet and scaling functions is exactly 2m at resolution level 2m. Any function f (x)3¸2([0, 1]) can be approximated, up to length scale 2~m, by its projection at resolution level 2m as 2m~1 h(x)+P1 h(x)" + h g (x) (6) m n n n/0 where P1 is the projection operator onto the subspace V1 and h is the inner product of h(x) and m m n g (x). When m approaches the positive infinity, the projection P1 h(x) converges to the original n m function h(x) as implied by (3). 2.2. Integral equation The integral equation for EM scattering from a 2-D conducting object can be formulated as11 (time variation e+ut is suppressed) PC G(r, r@) J(r@) dr@"!S(r), r3C (7) where J(r@) is the unknown surface current over the object, C is the 2-D contour of the object in a Cartesian xy plane and the definition domain of J(r@), S(r) is the known excitation and G(r, r@) is the 2-D Green’s function. More precisely, the excitation S(r) is the z-component of the incident electric field for transverse magnetic (TM) wave case and the incident magnetic field for transverse electric (TE) wave case. The Green’s function is given by !kgH(2) (kDr!r@D) for TM case 0 1 (2) G(r, r@)" 2d(r!r@)#jkH (kDr!r@D) for TE case 1 4 ]cos [n L (r@), r!r@] (8) 6 G. WANG where H (2) (·) and H (2) (·) denote, respectively, the second kind, zero order and first-order Hankel 1 0 functions, k is the wave number, g is the intrinsic (wave) impedance, d(·) is the Dirac delta function and n̂ is the unit normal vector directed out of the object. 2.3. HWBM analysis and matrix equation In order to apply expansion (6) in integral equation (7), a map between the contour C and the interval [0, 1] must be established, namely, one must seek a one-to-one onto map r")(f) (9) where r3C and f3[0, 1]. This can be done readily using the BEM geometrical representation described in References 3 and 5, which leads to a piecewise parametric representation of the contour C. Instead of a reiteration of the procedure described in References 3 and 5, only a brief outline is given here. For further details, the readers are referred to References 3 and 5. Roughly speaking, the establishment of the map (9) can be completed in three steps. First, through the conventional boundary element method,12 the contour C is discretized into a series of boundary elements, and then each of the boundary elements is mapped into a one-dimensional standard element via the shape functions or interpolation functions. Second, the standard elements are mapped into corresponding portions of the interval [0, 1] through a linear map. Finally, a map between the contour C and the interval [0, 1] is obtained by combining the two maps generated from the two steps described above. By using the map between C and [0, 1], integral equation (7) becomes 1 P0 G[r, )(f@)]h(f@) K K dr@ df@"!S(r), r3C df@ (10) where h(f)"J[)(f)]. Using (6) and the Galerkin method, (10) leads to a matrix equation [A ][h ]"[B ] jn n~1 j (11) where j, n"1, 2, 3, . . . , 2m, and P0 C P0 G[)(f), )(f@)] gn~1 (f@) K df@ K df@ D gj~1 (f) K df K df 1 dr df B "! S[)(f)]g (f) j~1 K df K j P0 A " jn 1 1 dr@ dr These integrals for computing the entry A of the system matrix A and the entry B of the jn j excitation vector B can be evaluated through either the fast wavelet transform7 or the conventional numerical integration such as the Guassian quadrature.13 3. NUMERICAL RESULTS A computer program has been coded in C language for the technique described in preceding section. In this section, numerical examples are presented to illustrate validity and the merits of this technique. The wavelet on the interval employed in the following numerical computations is constructed from the Daubechies orthogonal wavelet with vanishing moment N"6 (see Figure 1). The lowest resolution level is chosen as 2mÒ"24. A parameter of interest for 2-D EM scattering is the radar cross-section (RCS), defined as the width for which the incident wave APPLICATION OF WAVELETS ON THE INTERVAL TO NUMERICAL ANALYSIS 7 carries sufficient power to produce, by omnidirectional radiation, the same scattered power density in a given direction as the scattered field from the induced surface current J(r@). The formulas in Reference 11 are adopted for the RCS computations in the following examples. All the examples are performed on a SUN-SPARC-10 workstation, and roughly factors of 1—3·3 in the CPU time savings of this technique against the standard moment method have been recorded. It has been found that this technique has no CPU time savings for the small problems under 32 unknowns. The advantage of this technique comes to the surface when the unknown number increases. It runs roughly over three times faster than the standard moment method when the unknown number reaches 256. It can be expected to gain more CPU time saving for even larger problems. In the first example, the problem of plane wave scattering from an elliptic conducting cylinder is presented. The cross-section of this cylinder is an ellipse with a major axis of two wavelengths and an axial ratio of 4. The incident wave is assumed to propagate along the minor axis of the ellipse. The scattering field patterns (i.e. bistatic radar cross-section), as computed by Andreasen,14 the conventional moment method with 64 pulse basis functions and this approach with resolution level 25 are depicted in Figure 2 versus the observation angle. The observation angle is measured in relative to the direction of incidence. Namely, the observation angle takes zero at the incident wave propagation direction. The agreements among these results are excellent. Figure 3 plots the results for the backscattering radar cross-section of a rectangular conducting cylinder, as computed by using the conventional moment method with 64 pulse basis functions and this technique with resolution level 25, versus the azimuth angle h (i.e. the angle between the observation direction and the positive x-axis direction). The corners of the rectangular crosssection of this cylinder locate at (!j/4, j/2), (j/4, j/2), (j/4, !j/2) and (!j/4, !j/2) in the xy plane, respectively, where j denotes the wavelength. The agreements between two sets of results are again excellent. In the above two examples, the definition domains of the unknown currents are closed contours and thus the unknown currents have a periodic nature. The third example is dedicated to the plane wave scattering of a pair of thin hemi-circular metallic tubes. Here, the definition domain of the unknown current consists of two open contours—two hemi-circular arcs centred at (0, 3·5j) and (0, !3·5j) of the xy plane, respectively. Figure 4 illustrates the scattering field patterns obtained by this technique and the conventional moment method. The wavelet expansion on [0, 1] at resolution level 26 is employed to expand the unknown current distribution over each of the two hemi-circular arcs. Namely, 128 wavelet (scaling) functions are adopted in the computation using this technique. In the computation applying the conventional moment method, 256 pulse basis functions are totally used over the two hemi-circular arcs. To achieve the same accuracy, it is observed that less basis functions are needed in this wavelet approach than the conventional moment method. This supports a theoretical result of the wavelet theory that the wavelet expansions converge faster, i.e. fewer coefficients are needed to represent a given function than other expansions.9 The major merit of a wavelet-based method is the generation of a sparse matrix from an integral operator. To illustrate the matrix sparseness produced by this technique, let us consider a scatterer with relatively larger electrical size—a thin conducting right-angled corner reflector of eight-wavelength arms. The 2-D contour of the reflector consists of two straight lines, namely, the line segment from (0, 8j) to (0, 0) and the line segment from (0, 0) to (8j, 0) in the xy plane. The wavelet expansion on [0, 1] at resolution level 28 is employed to expand the unknown current over the reflector. Since 256 wavelet functions are involved in this expansion, a system matrix with size 256]256 is generated. Figure 5 shows the sparseness structures of the system matrix 8 G. WANG Figure 2. Scattering field patterns of an elliptic conducting cylinder computed by Andreasen, this wavelet approach and the conventional moment method: (a) TM case; (b) TE case . APPLICATION OF WAVELETS ON THE INTERVAL TO NUMERICAL ANALYSIS 9 Figure 3. Backscattering radar cross-sections of a rectangular conducting cylinder computed by using the wavelets on the interval and the conventional moment method: (a) TM case; (b) TE case . 10 G. WANG Figure 4. Scattering field patterns of a pair of thin hemi-circular metallic tubes computed by using the wavelets on the interval and the conventional moment method: (a) TM case; (b) TE case . APPLICATION OF WAVELETS ON THE INTERVAL TO NUMERICAL ANALYSIS 11 Figure 5. Typical sparseness structures of the system matrix obtained by using the wavelets on the interval: (a) threshold 10~3, TM case; (b) threshold 10~2, TM case; (c) threshold 10~3, TE case; (d) threshold 10~2, TE case after applying thresholds 10~3 and 10~2. The black ink indicates the remaining non-zero elements after discarding each element whose magnitude relative to that of the largest one is smaller than a selected threshold. The ratio of the number of the remaining non-zero elements to the total number of elements in the system matrix is obtained as 17·88 and 6·4 per cent in TM case and 8 and 3.26 per cent in TE case for the respective thresholds selected above. Since the wavelet expansion at resolution level 28 is used, a multi-level (five-level) analysis is realized. From Figure 5, the multi-level structures can be observed in the system matrix as well. A multi-level iteration can be utilized to solve the sparse matrix equation. The multi-level iterative methods fully take advantages of the sparseness and multi-level structures of the system matrix.6 Another approach to accelerate the solution of the matrix equation by making use of the sparseness is to employ a sparse matrix solver such as the conjugate gradient method.13 Figure 6 depicts the magnitudes of current distributions, computed by using the sparse matrices shown in Figure 5, in comparison with the exact solution obtained by using the original full matrix. The normalized length of the 2-D contour takes values 0, 0·5 and 1 at points (0, 8j), (0, 0) and (8j, 0), respectively. The incident plane wave propagates along the positive x-axis. In this example, the currents distribute over a bounded domain and have non-periodic nature for both TM and TE cases. No artificial jumps or oscillations near the edges are caused by using the wavelet basis functions on the interval as shown in Figure 6 for the current distributions. Artificial jumps or oscillations are commonly observed near the edges for the solution function, if the wavelets on the whole real line are deployed as basis functions in the problems with finite bounded solution domains (e.g., see Reference 2). 12 G. WANG Figure 6. Magnitudes of current distributions computed using the sparse matrices with various thresholds: (a) TM case; (b) TE case . APPLICATION OF WAVELETS ON THE INTERVAL TO NUMERICAL ANALYSIS 13 4. CONCLUSION An HWBM analysis of EM scattering problems using the wavelets on the interval has been presented. The BEM geometrical representation is employed to establish maps between the arbitrary curved contours of the conducting objects and the definition domain [0, 1] of the wavelets on the interval. Through these maps, the unknown currents over the conducting objects are expanded in terms of the wavelet and scaling functions on the interval. A sparse matrix equation is obtained from the integral equation under study. The use of the wavelets on the interval overcomes the difficulties in the application of the wavelets on the real line to finite domain problems, and has no periodicity constraint to the unknown function which is usually imposed by the periodic wavelets. This approach allows accurate modelling of curved solution domains, conversion of integral equations to sparse matrix equations, and orthogonal multiresolution analysis on finite domains. REFERENCES 1. H. Kim and H. Ling, ‘On the application of fast wavelet transform to the integral equation of electromagnetic scattering problems’, Microwave Opt. ¹echnol. ¸ett., 6, 168—173 (1993). 2. B. Z. Steinberg and Y. Leviatan, ‘On the use of wavelet expansions in the method of moments’, IEEE ¹rans. Antennas Propagat., AP-41, 610—619 (1993). 3. G. Wang, ‘Numerical techniques for electromagnetic modeling of high speed circuits’, Ph.D. Dissertation, University of Wisconsin-Milwaukee, October 1993. 4. K. Sabetfakhri and L. P. B. 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