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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. Katehi, ‘Analysis of integrated millimeter-wave and submillimeter-wave waveguides using
orthonormal wavelet expansions’, IEEE ¹rans. Microwave ¹heory ¹ech., MTT-42, 2412—2422 (1994).
5. G. Wang, ‘A hybrid wavelet expansion and boundary element analysis of electromagnetic scattering from conducting
objects’, IEEE ¹rans. Antennas Propagat., AP-43, 170—178 (1995).
6. G. Wang, ‘A multilevel formulation of the wavelet expansion methods in electromagnetic field computations’, IEEE
¹rans. Antennas Propagat., submitted.
7. A. Cohen, I. Daubechies and P. Vial, ‘Wavelets on the interval and fast wavelet transforms’, Appl. Comput. Harmonic
Analy., 1, 54—81 (1993).
8. J. C. Goswami, A. K. Chan and C. K. Chui, ‘On solving first-kind integral equations using wavelets on a bounded
interval’, IEEE ¹rans. Antennas Propagat., AP-43, 614—622 (1995).
9. I. Daubechies, ¹en ¸ectures on ¼avelets, SIAM Press, Philadelphia, 1992.
10. G. W. Pan and J. Y. Du, ‘On solving surface integral equations using intervallic wavelets’, in Proc. 1995 Progress in
Electromagnetics Res. Symp. (PIERS’95), Seattle, Washington, July 1995, p. 132.
11. R. F. Harrington, Field Computation by Moment Methods, IEEE Press, New York, 1993.
12. C. A. Brebbia, J. C. F. Telles and L. C. Wrobel, Boundary Element ¹echniques, Springer, Berlin, 1984.
13. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C: ¹he Art of Scientific
Computing, 2nd edn, Cambridge University Press, Cambridge, 1992.
14. M. G. Andreasen, ‘Scattering from parallel metallic cylinders with arbitrary cross sections’, IEEE ¹rans. Antennas
Propagat., AP-12, 746—754 (1964).
.
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