Contents
1 Derivation of CC Equations For an Infinite One-Dimensional Perfectly
Reflecting Rough Surface
2
2 Incident and Total Fields
6
3 Dirichlet Problem
8
4 Periodic Surface
9
5 Derivation of SC Equations For an Infinite One-Dimensional Perfectly
Reflecting Rough Surface
11
6 SC Equations For a Periodic Surface
13
7 SS Equations For a Periodic Surface
15
8 Energy
16
9 Wavelet Analysis
17
10 Computational Results
10.1 Example 1, λ/L 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.1 Case A, No Grazing . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.2 Case B, Near-Grazing Incidence/Reflection . . . . . . . . . . . . .
10.2 Example 2, λ/L ≈ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.1 Case A, No Grazing . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.2 Case B, Near-Grazing Incidence/Reflection . . . . . . . . . . . . .
10.3 Example 3, λ/L 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.1 Case A, No Grazing . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.2 Case B, Near-Grazing Incidence/Reflection . . . . . . . . . . . . .
10.4 Example 4, Very Rough Surface . . . . . . . . . . . . . . . . . . . . . . .
10.5 Example 5, Highly Oscillatory Surface with Continuous Derivative . . . .
10.6 Wavelet Transforms Applications . . . . . . . . . . . . . . . . . . . . . .
10.6.1 Example 1, Case A: λ/L 1, No Grazing, CC1 matrices, Daub6
filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6.2 Example 2, Case A: λ/L ≈ 1, No Grazing, CC1 matrices, Daub6
filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6.3 Example 3, Case A: λ/L 1, No Grazing, CC1 matrices, Daub 6
filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6.4 Example 1, Case A: λ/L 1, No Grazing, CC2 matrices, Daub 6
filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6.5 Example 2, Case A: λ/L ≈ 1, No Grazing, CC2 matrices, Daub 6
filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6.6 Example 3, Case A: λ/L 1, No Grazing, CC2 matrices, Daub 6
filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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36
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. 42
10.7 Fourier Transforms of Periodic Green’s Functions . . . . . . . . . . . . .
10.7.1 Example 1, Case A, λ/L 1, No Grazing . . . . . . . . . . . . .
10.7.2 Example 1, Case B, λ/L 1, Near-Grazing Incidence/Reflection
10.7.3 Example 2, Case A, λ/L ≈ 1, No Grazing . . . . . . . . . . . . .
10.7.4 Example 2, Case B, λ/L ≈ 1, Near-Grazing Incidence/Reflection .
10.7.5 Example 3, Case A, λ/L 1, No Grazing . . . . . . . . . . . . .
10.7.6 Example 3, Case B, λ/L 1, Near-Grazing Incidence/Reflection
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43
44
44
44
44
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44
11 Conclusions
51
A Periodic Green’s Function
52
B Numerical Solution of the CC Equations
55
C Numerical Solution of the SC Equations
57
D Numerical Solution of the CC Equations. Discrete Galerkin Method
58
ii
Abstract
We discuss the scattering of acoustic or electromagnetic waves from one-dimensional rough
surfaces. We restrict the discussion in this report to perfectly reflecting Dirichlet surfaces
(TE-polarization). The theoretical development is for both infinite surfaces and periodic
surfaces, the latter equations derived from the former. Several theoretical developments
are presented. They are characterized by integral equation solutions for the surface current
or normal derivative of the total field. All the equations are discretized to a matrix system
and further characterized by the sampling of the rows and columns of the matrix which is
accomplished in either coordinate space (C) or spectral space (S). The standard equations
are referred to here as CC equations of either first kind (CC1) or second kind (CC2). Mixed
representation equations or SC type are solved as well as SS equations fully in spectral
space. Remarks are also included about wavelet techniques used to solve the integral
equations. These have generated some recent interest for electromagnetic scattering from
simple shapes. Our conclusions for rough surface scattering differ somewhat from the
literature conclusions.
Computational results are presented for scattering from various periodic surfaces.
The results include examples with grazing incidence, a very rough surface and a highly
oscillatory surface. The examples vary over a parameter set which includes the geometrical
optics regime, physical optics or resonance regime, and a renormalization regime. Results
of wavelet applications are also included as well as the Fourier transform of the periodic
Green’s function.
The objective of this study was to determine the best computational method for
these problems. Briefly the SC method was the fastest but did not converge for large
slopes or very rough surfaces. The SS method was slower and had the same convergence
difficulties as SC. The CC methods were extremely slow but always converged. The
simplest approach is to try the SC method first. Convergence, when the method works,
is very fast. If convergence doesn’t occur then try SS and finally CC.
1
1
Derivation of CC Equations For an Infinite OneDimensional Perfectly Reflecting Rough Surface
We consider the scattering from an infinite one-dimensional rough surface specified by
z = s(x) (see Figure 1). For the examples we consider in this report the surface is
perfectly reflecting and periodic. In this section we first consider the surface to be infinite
and specify it to be periodic later in Section 4. Our computational results are restricted to
periodic surface cases. Notationally we have a spatial 2-vector x = (x, z) = (x1 , x2 ) and
its restriction to the surface xs = (x, s(x)). The gradient operator is ∂i = ∂x∂ i (i = 1, 2)
and the normal derivative ∂n = ni ∂i where ni is the normal to the surface and repeated
subscripts are summed (here from 1 to 2). Fields are represented by ψ and correspond to
a velocity potential (acoustics) [9] or the y−component of the electric (Dirichlet boundary
value problem) or magnetic (Neumann boundary value problem) fields. Since the surface
is one-dimensional its generator is parallel to the y−axis and no polarization change occurs
during scattering from such a surface. The electromagnetic problem thus reduces to a
scalar one, which is what we treat. All fields are time-harmonic so that a factor exp(−iωt)
is suppressed throughout (ω is circular frequency and t is time).
The scattered field satisfied the scalar Helmholtz equation (k1 = 2π
is the wavenumber
λ
and λ is wavelength)
+
.
(1.1)
(∂i ∂i + k12 )ψ sc (x) = 0, x ∈ DR
The free-space two-dimensional Green’s function G(2) for this problem satisfies the nonhomogeneous Helmholtz equation
(∂i ∂i + k12 )G(2) (x, x ) = −δ(x − x ),
(1.2)
where the right hand side is the Dirac delta function in two-dimensions. G(2) is explicitly
given by [9, p. 54]
i (1)
G(2) (x, x ) = H0 (k1 |x − x |),
(1.3)
4
the Hankel function of zeroth-order, first kind. Its Fourier transform relation is
G(2) (x, x ) =
1 iα . (x−x )
G̃(α)d
α,
e
(2π)2
where
G̃(α) =
α2
1
2
− k1+
(1.4)
(1.5)
and we have chosen k1+ = lim→0+ (k1 + i
) to indicate that we have an outgoing wave.
+
is specified by the characteristic function
The region DR
θ+ (x) = θ(z − s(x))θ(R − r),
(1.6)
in the limit as R → ∞. Here θ is the usual Heaviside function
θ(x) =
1,
0,
2
x > 0,
x < 0,
(1.7)
sc
+
HR
sc
in
sc
+
DR
R
s(x)
-
DR
-
HR
Figure 1: Infinite perfectly reflecting one-dimensional rough surface z = s(x). Incident
+
(in) and scattered (sc) wave are indicated. The region DR
is specified by z > s(x) and
r = |x| < R and is bounded by the rough surface and the semicircle HR+ at radius R. The
−
) and semi-circle (HR− ) below the surface are also illustrated.
complementary region (DR
and r = |x|. The function θ+ thus represents the region bounded by the rough surface
s(x) truncated at R and the upper semicircle at radius R denoted by HR+ (see Figure 1).
Vertical segments joining the surface and the semicircle can also be included [11] but are
omitted in the interests of brevity.
To form equations for the scattered field we use Green’s theorem. Multiply Eq. (1.1)
by G(2) and Eq. (1.2) by ψ sc and subtract the resulting equations. We get
∂i ∂i G(2) (x, x )ψ sc (x ) − G(2) (x, x )∂i ψ sc (x ) = −δ(x − x )ψ sc (x ).
(1.8)
Next, multiply Eq. (1.8) by θ+ (x ), integrate over all space (in x ) and then integrate by
parts using the vector derivative of the characteristic function
where
∂j θ+ (x ) = nj (x )δ(z − s(x ))θ(R − r ) − nj (R)δ(r − R)θ(z − s(x )),
(1.9)
nj (x ) = δj2 − δj1 s(x ),
(1.10)
3
is the non-unit normal to the surface s and
nj (R) = ∂j r |r=R ,
(1.11)
the radial normal to the semicircle HR+ . Two surface integrals result. The integral over
the semicircle is
+
HR
∂r G(2) (x, x )ψ sc (x ) − G(2) (x, x )∂r ψ sc (x )
r =R
Rdθ ,
(1.12)
where θ is the integration angle in [− π2 , π2 ]. If ψ sc satisfies a Sommerfeld radiation condition
this integral vanishes as R → ∞. More generally, so long as ψ sc does not contain any
horizontally propagating plane waves, the integral vanishes as R → ∞ [11]. We include
the latter restriction since we treat the case of plane wave incidence in our periodic surface
examples later. If there is an incident plane wave we must admit scattered plane waves
in order to balance the total energy on this far away semicircle [10]. For horizontal plane
wave incidence and scattering other equations result than the ones we quote below [11].
We thus assume that Eq. (1.12) vanishes as R → ∞. The result using Eq. (1.9) is
a single integral over the infinite surface s∞ (x). To write this result in convenient form,
define single (S) and double (D) layer acoustic potentials with respective densities u and
v as
(Su)(x) =
G(2) (x, xs )u(xs )dx ,
(1.13)
s∞
and
(Dv)(x) =
s∞
∂n G(2) (x, xs )v(xs )dx ,
(1.14)
as well as the normal derivative of ψ sc
N sc (x) = ∂n ψ sc (x).
(1.15)
The resulting equations can then be written as
(Dψ sc )(x) − (SN sc )(x) = θ+ (x)ψ sc (x),
(1.16)
+
−
which for x ∈ D∞
gives the representation of the scattered field, and for x ∈ D∞
(the
lower region below the surface) the equation is referred to as an extinction theorem [19].
To form surface integral equations use the limiting properties of single and double
layer potentials [4]. Define the limits from above (+) and below (−) as
lim± (Su)(x) = (Su)± (xs ),
(1.17)
lim± (Dv)(x) = (Dv)± (xs ).
(1.18)
(Su)+ (xs ) = (Su)− (xs ),
(1.19)
x→xs
and
x→xs
The single layer is continuous
4
and the double layer has a jump discontinuity
(Dv)+ (xs ) − (Dv)− (xs ) = v(xs ),
(1.20)
1
(Dv)± (xs ) = PV(Dv)(xs ) ± v(xs ),
2
(1.21)
with each limit defined as
where PV stands for Cauchy Principal Value. Using these limiting values the surface
integral equation which follows from Eq. (1.16) is
1
PV(Dψ sc )(xs ) − (SN sc )(xs ) = ψ sc (xs ).
2
(1.22)
The kernel terms in this integral equation have both arguments in coordinate space (on
the surface) so that a discretized version of them will yield (kernel) matrices whose rows
and columns result from coordinate-space sampling. We thus refer to this equation as a
coordinate-coordinate (CC) equation.
A second equation can be formed by taking the normal derivative of Eq. (1.16) for
+
x ∈ D∞
. The normal derivative of the single layer potential is discontinuous with limits
1
∂n (Su)± (xs ) = PV∂n (Su)(xs ) − u(xs ),
2
(1.23)
and the normal derivative of the double layer has the same limit from above and below
but is singular and we take its Hadamard Finite Part (FP) [14]
lim ∂n (Dv)(x) = FP∂n (Dv)(xs ).
x→x±
s
(1.24)
[Note: For now the use of PV and FP notation is purely formal. We discuss the details
later]. The result is another CC integral equation
1
FP∂n (Dψ sc )(xs ) − PV∂n (SN sc )(xs ) = N sc (xs ).
2
(1.25)
The usual boundary value problems we wish to discuss involve total field quantities. We
treat the incident field next in Section 2 and combine the results into integral equations
on the total field.
5
2
Incident and Total Fields
We form integral equations of the total field (ψ T ) and normal derivative (N T ). These are
defined by
(2.1)
ψ T (x) = ψ i (x) + ψ sc (x),
and
N T (x) = N i (x) + N sc (x),
(2.2)
in terms of the incident (i) field and its normal derivative. ψ i (x) satisfies the homogeneous
Helmholtz equation
(∂j ∂j + k12 )ψ i (x) = 0.
(2.3)
Our examples later are for a single plane wave
ψ i (x) = φp (x) = Deik1 (α0 x−β0 z) ,
(2.4)
where α0 = sin(θi ), β0 = cos(θi ), and θi is the angle of incidence defined from the
z−direction. We generally choose D = 1. We also omit horizontally incident waves,
so β0 > 0. More generally, we could have a continuous superposition (or spectral decomposition) of plane waves
i
c
ψ (x) = φ (x) =
where
⎧
⎨
m(μ) = ⎩
∞
−∞
I(μ)eik1 [μx−m(μ)z] dμ,
1
(1 − μ2 ) 2 ,
μ ≤ 1,
i(μ2 −
μ > 1.
1
1) 2 ,
(2.5)
(2.6)
The density I(μ) is a continuous function. Eq. (2.4) is a special case if we let I(μ) be the
distribution Dδ(μ − α0 ).
1
Asymptotically for r = (x2 + z 2 ) 2 large, Eqs. (2.4) and (2.5) behave quite differently. Eq. (2.4) has no limit and Eq. (2.5) (for continuous I(μ)) behaves like a cylindrical
+
wave and satisfies the incoming cylindrical radiation condition in D∞
and an outgoing
−
cylindrical wave radiation condition in D∞ , the domain below the surface.
The results of Green’s theorem can be summarized by defining the bracket integral of
Green’s theorem on a surface P for any field φ
(2)
[G , φ; x, P ] =
The result is [11]
P
G(2) (x, xp )∂n φ(xp ) − ∂n G(2) (x, xp )φ(xp ) dx .
lim [G(2) , φp ; x, HR+ ] = φp (x),
(2.8)
lim [G(2) , φp ; x, HR− ] = 0.
(2.9)
R→∞
and
(2.7)
R→∞
For a single plane wave the equation corresponding to Eq. (1.16) is
(Dφp )(x) − (SN p )(x) = θ+ (x)φp (x) − φp (x),
6
(2.10)
where the last term follows from Eq. (2.8) and N p is the normal derivative of the plane
wave. Combining Eqs. (2.10) and (1.16) we have a representation for the total field at x
(off the surface)
θ+ (x)ψ T (x) = φp (x) + (Dψ T )(x) − (SN T )(x),
(2.11)
and in the limit as x approaches the surface s(x) the surface integral equation
1 T
ψ (xs ) = φp (xs ) + PV (Dψ T )(xs ) − (SN T )(xs ).
2
(2.12)
+
A second equation can be derived using the normal derivative of Eq. (2.11) for x ∈ D∞
and subsequently taking the limit as x approaches the surface. It is
1 T
N (xs ) = N p (xs ) + FP ∂n (Dψ T )(xs ) − PV ∂n (SN T )(xs ).
2
(2.13)
−
Further, using Green’s theorem in D∞
on G(2) and φc and combining the result with the
scattered field results in Section 1, an analogous set of equations to Eqs. (2.12) and (2.13)
results (with the modification that the plane wave term is replaced by the continuous
superposition).
Thus, for an incident field of either form (excluding horizontal plane waves) we can
write Eqs. (2.12) and (2.13) as
1 T
ψ (xs ) = ψ i (xs ) + PV (Dψ T )(xs ) − (SN T )(xs ),
2
(2.14)
and
1 T
N (xs ) = N i (xs ) + FP ∂n (Dψ T )(xs ) − PV ∂n (SN T )(xs ).
(2.15)
2
These are both coordinate-coordinate (CC) integral equations on the boundary unknowns
ψ T and N T . Both are valid for an infinite surface with the restriction that no horizontal
plane waves occur. For completeness and later use we include the field representation
which follows from Eq. (2.11) for an incident field satisfying the above restrictions. It is
θ+ (x)ψ T (x) = ψ i (x) + (Dψ T )(x) − (SN T )(x).
7
(2.16)
3
Dirichlet Problem
For the Dirichlet (D) boundary value problem
ψ T (xs ) = 0.
(3.1)
Acoustically this describes a soft surface and electromagnetically it is the case of TEpolarization. With this condition Eq. (2.14) becomes
ψ i (xs ) = (SN T )(xs ),
(3.2)
which is referred to as CC1, a first-kind integral equation [21] for the remaining boundary
unknown N T . Eq. (2.15) becomes
1 T
N (xs ) = N i (xs ) − PV ∂n (SN T )(xs ),
2
(3.3)
which is an integral equation of the second kind, and we refer to it as the CC2 equation.
Often the two equations are linearly combined as follows. Choose real constants α and
β, multiply Eq. (3.2) by α and Eq. (3.1) by β, and add the resulting equations. Since all
the functions are evaluated on the surface they are functions of a single variable. Define
the incident field function by
F i (x) = βψ i (xs ) + αN i (xs ).
(3.4)
The resulting added equations can be put in the impedance form
i
F (x) =
∞
−∞
Z D (x, x )N T (x )dx ,
(3.5)
where the “impedance” kernel is defined symbolically as
1
Z D (x, x ) = αδ(x − x ) + α PV ∂n G(2) (xs , xs ) + βG(2) (xs , xs ).
2
(3.6)
For α = 0 we have CC1. For β = 0, CC2, and for β = 1 and α arbitrary we have what is
referred to as the Combined Field Integral Equation (CFIE) (see [17]). It becomes particularly important for scattering from bounded bodies where the CC1 and CC2 solutions
contain different resonances but the CFIE solutions do not.
8
4
Periodic Surface
For a periodic surface with period L, s(x + L) = s(x). We can then reduce Eq. (3.5) to an
integration over a single period cell, say from − L2 to L2 . The single layer potential term in
Eq. (3.2) can be written as
(SN T )(xs ) =
∞
−∞
G(2) (xs , xs )N T (x )dx
(4.1)
In (x),
(4.2)
∞
=
n=−∞
where
In (x) =
(2n+1) L
2
(2n−1) L
2
G(2) (xs , xs )N T (x )dx .
(4.3)
In Eq. (4.3) use the Weyl representation for the Green’s function [9, p. 63]
G(2) (xs , xs ) =
πi
(2π)2
∞
1 ik1 [μ(x−x )+m(μ)|s(x)−s(x )|]
e
dμ,
m(μ)
−∞
(4.4)
(with m(μ) defined by Eq. (2.6)), the Floquet (pseudo-)periodicity of the boundary unknown
N T (x + nL) = eik1 α0 nL N T (x),
(4.5)
and the change of variables x = x − nL.
We can then write Eq. (4.1) on the domain [− L2 , L2 ] as
T
(SN )(xs ) =
L
2
−L
2
Gp1 (x, x )N T (x )dx ,
(4.6)
where Gp1 is the periodic Green’s function with wavenumber k1 given by
Gp1 (x, x ) =
πi
(2π)2
∞
−∞
∞
1 ik1 [μ(x−x )+m(μ)|s(x)−s(x )|] e
.
eink1 L(α0 −μ) dμ.
m(μ)
n=−∞
(4.7)
For scalar arguments x and x , Gp1 is confined to the surface.
Next, use the Poisson sum [23]
∞
n=−∞
eint = 2π
∞
δ(t + 2πj),
(4.8)
j=−∞
where t = k1 L(α0 − μ). The delta function in Eq. (4.8) reduces to
δ(t + 2πj) =
where
1
δ(μ − αj ),
k1 L
λ
αj = α0 + j .
L
9
(4.9)
(4.10)
This is the Bragg equation with αj = sin(θj ) and θj the angle of the j th outgoing Bragg
wave. Using Eqs. (4.8) to (4.10) to evaluate Eq. (4.7) we get
Gp1 (x, x ) =
with
∞
1 ik1 [αj (x−x )+βj |s(x)−s(x )|]
i λ e
,
4π L j=−∞ βj
⎧
⎨
βj = ⎩
1
(1 − αj ) 2 ,
|αj | ≤ 1,
i(αj 2 −
|αj | > 1.
1
1) 2 ,
(4.11)
(4.12)
Other representations for this periodic Green’s function, useful for its evaluation in computations, are discussed in Appendix A. Further, it is also convenient to have a representation
for this function off the surface, and it is obviously given by
Gp1 (x, x ) =
∞
1 ik1 [αj (x−x )+βj |z−z |]
i λ e
.
4π L j=−∞ βj
(4.13)
Similarly, the normal derivative of the single layer potential term in Eq. (3.3) can be
reduced to a single period cell. The result is
PV
where
∞
−∞
L
2
∂n G(2) (xs , xs )N T (x )dx = − L Gp1 (x, x )N T (x )dx ,
−2
Gp1 (x, x ) = [nj ∂j Gp1 (x, x )]|z=s(x)
|z =s(x )
(4.14)
(4.15)
follows from Eq. (4.13). The slash on the integral in Eq. (4.14) represents Cauchy principal
value (if appropriate). Using Eqs. (4.6) and (4.14), Eq. (3.4) reduces to
i
F (x) =
L
2
−L
2
ZpD1 (x, x )N T (x )dx ,
(4.16)
where the impedance kernel is given by
1
ZpD1 (x, x ) = αδ(x − x ) + α PV Gp1 (x, x ) + βGp1 (x, x ).
2
(4.17)
Again, for α = 0 we have the CC1 equation, for β = 0 the CC2 equation, and for
β = 1 and α arbitrary the CFIE equation. These are the equations which are solved for
our examples. The numerical solution is discussed in Appendix B. Discretization of Eq.
(4.16) yields an impedance matrix whose rows and columns result from sampling both in
coordinate space, thus the acronym coordinate-coordinate or CC.
10
5
Derivation of SC Equations For an Infinite OneDimensional Perfectly Reflecting Rough Surface
In the previous four sections we confined our attention to problems where both rows and
columns of the matrix to be inverted were sampled in coordinate space. Here we derive
a mixed representation where the rows are sampled in the conjugate spectral (S) space
and the columns still sampled in the coordinate space. These are the SC equations. A
straightforward derivation without using any of the results in the first four sections can
be found in the literature [7]. A different derivation which however yields the same results
proceeds as follows.
Use the representation for the total field given by Eq. (2.16) for the Dirichlet problem
T
(ψ (xs ) = 0). We have
ψ sc (x) = −(SN T )(x),
and
ψ i (x) = (SN T )(x),
+
x ∈ D∞
,
(5.1)
−
x ∈ D∞
.
(5.2)
The Weyl representation Eq. (4.4) with kx = k1 μ and Kx = k1 m(μ) (see Eq. (2.6)) and
written off the surface in the x−variable is
G(2) (x, xs ) =
πi
(2π)2
∞
−∞
1 ik1 [μ(x−x )+m(μ)|z−s(x )|]
e
dμ.
m(μ)
(5.3)
For z > max[s(x)] we can remove the absolute value in the phase of Eq. (5.3) and for
these values of z write Eq. (5.1) using Eqs. (1.13) and (5.3) as
ψ sc (x) =
where
−i
A(μ) =
4πm(μ)
∞
−∞
∞
−∞
A(μ)eik1 [μx+m(μ)z] dμ,
(5.4)
e−ik1 [μx +m(μ)s(x )] N T (x )dx .
(5.5)
Once we know N T we can thus evaluate A(μ) and the scattered field. Eq. (5.4) is a
1
spectral representation of the scattered field. For large r = (x2 + z 2 ) 2 (where x = r sin θ
and z = r cos θ), a stationary phase evaluation of Eq. (5.4) can be written in terms of
scattering amplitude T (θ) as
eik1 r
ψ sc (x) ∼ T (θ) √ ,
(5.6)
r
which is an outgoing cylindrical wave where
1
T (θ) = (−2πi) 2 A(μsp ),
(5.7)
and the stationary phase point is μsp = sin θ. The amplitude A(μ) is thus directly related
to the scattering amplitude.
To solve for N T use Eq. (5.2) and the representation Eq. (5.3) now for z < min[s(x)].
The result is
∞
i
ψ (x) =
I(μ)eik1 [μx−m(μ)z] dμ,
(5.8)
−∞
11
which is just Eq. (2.5) and where
I(μ) =
i
4πm(μ)
∞
−∞
e−ik1 [μx−m(μ)s(x)] N T (x)dx.
(5.9)
Given the properties of the incident field, i.e. I(μ), we solve the first kind integral equation
Eq. (5.9) for N T and use this to evaluate the scattered field. The kernel of the integral
equation is now a function of μ (spectral, S) and x (coordinate, C) and the method is
referred to as spectral-coordinate (SC).
We can write Eqs. (5.5) and (5.9) in the symmetric representation
I ± (μ) =
∞
−∞
e−ik1 [μx∓m(μ)s(x)] N T (x)dx,
where
I ± (μ) = 4πi m(μ)
12
−I(μ)
A(μ) .
(5.10)
(5.11)
6
SC Equations For a Periodic Surface
For a periodic surface s(x + L) = s(x), and we can write Eqs. (5.10) as
∞
n=−∞
where
Q±
n (μ) =
±
Q±
n (μ) = I (μ),
(2n+1) L
2
(2n−1) L
2
e−ik1 [μx∓m(μ)s(x)] N T (x)dx.
(6.1)
(6.2)
Again, change variables to x = x − nL, and use the Floquet periodicity of N T given by
Eq. (4.5). The result is
ink1 L(α0 −μ) ±
Q0 (μ).
(6.3)
Q±
n (μ) = e
Use of the Poisson sum, Eq. (4.8), and Eq. (4.9) yield for Eq. (6.1)
∞
λ ±
Q0 (μ)
δ(μ − αj ) = I ± (μ).
L
j=−∞
(6.4)
Integration of both sides of Eq. (6.4) over the μ−domain [αn −
Lλ , αn +
Lλ ] where 0 < < 1
yields
αn + λ
λ ±
L ±
Q0 (αn ) =
(6.5)
λ I (μ)dμ.
L
αn −
L
For a single incident plane wave (see Eq. (5.8))
I(μ) = Dδ(μ − α0 ),
(6.6)
and, for a periodic surface, the scattered field spectra are discrete
A(μ) =
∞
n=−∞
An δ(μ − αn ).
(6.7)
(This can be seen by reducing Eq. (5.5) to the integration over a single periodic cell).
Using these results in Eq. (6.5) and the definitions Eq. (5.11) we get
λ ±
Q0 (αn ) = I ± (αn ),
L
where
I ± (αn ) = 4πi βn
−Dδn0
An .
(6.8)
(6.9)
T
T
has
The integrals Q±
0 have dimensions of length times the dimensions of N . Also, N
dimensions of inverse length times the dimensions of the field. It is convenient to scale
out this inverse length by defining the function N(x) as
N T (x) = ik1 N(x),
13
(6.10)
so that N(x) has the same dimensions as the field ψ T . (The scaling Eq. (6.10) obviously
relates to the fact that differentiation of a wave-like field quantity produces a factor ik1 .)
The result can be written as
P0± (αn ) = F ± (αn ),
(6.11)
where
P0± (αn )
1
=
L
−L
2
−L
2
e−ik1 [αn x∓βns(x)] N(x)dx,
and
±
F (αn ) = 2βn
−Dδn0
An .
(6.12)
(6.13)
The method of solution is to solve the first kind equation for N(x) (the “ + ” equation)
then evaluate the “ − ” equation for An . The scattered field from Eqs. (5.4) and (6.7) is
then
∞
ψ sc (x, z) =
n=−∞
An eik1 (αn x+βn z) .
(6.14)
The kernels in Eq. (6.12) are functions of μ = αn and x, i.e. a discrete spectral parameter n and a coordinate variable, thus again the spectral-coordinate (SC) acronym. An
alternative derivation of these results can be found in the literature [7].
14
7
SS Equations For a Periodic Surface
We derive the spectral-spectral (SS) equations from the SC equations in Section 6. The
method is to expand the boundary unknown in the topological or surface wave basis [15]
in Eq. (6.12)
∞
N(x) =
j =−∞
Ñj eik1 (αj x−βj s(x)) .
(7.1)
Note that this “basis” is not a complete set. The result is first a system of linear equations
for the vector of expansion coefficients in the discrete spectral domain Ñ = {Ñj }
with the components
K̃Ñ = F+ ,
(7.2)
Fj+ = −2β0 Dδj0 ,
(7.3)
and the matrix whose entries are
K̃jj 1
=
2π
π
−π
L
e−i(j−j )y eik1 (βj −βj )s( 2π y) dy,
(7.4)
where the integrals have been scaled to [π, π], and second, the set of equations to evaluate
for the Aj coefficients is
M̃jj Ñj = 2βj Aj ,
(7.5)
j
where the matrix elements are
M̃
jj 1
=
2π
π
−π
L
e−i(j−j )y e−ik1 (βj +βj )s( 2π )y dy.
(7.6)
The scattered field is then given by Eq. (6.14). Rows and columns of both K̃ and M̃
are indexed in the (discrete) spectral integer j and the method is referred to as spectralspectral (SS).
15
8
Energy
We know that the sum of the scattered energy must equal the energy in the incident wave
j
|Aj |2 Re(βj ) = β0 D 2 .
(8.1)
Only real (Re) orders carry energy away from the surface. We set D = 1 for our trials
and quote the condition as
Re(βj )
|Aj |2
= 1.
(8.2)
β0
j
The left hand side of Eq. (8.2) is referred to as the normalized energy. The Aj are
computed and then we determine how well the energy check Eq. (8.2) is satisfied. It is a
necessary but not sufficient condition of accuracy.
16
9
Wavelet Analysis
The matrix system for the CC equations reduces (cf. Eq. (B.10)) to
ZNT = Fi ,
(9.1)
KN = F+ ,
(9.2)
for the SC equations (cf. Eq. (C.4)) to
and for the SS equations (cf. Eq. (7.2)) to
K̃Ñ = F+ .
(9.3)
Ax = b,
(9.4)
Each system thus has the matrix form
for the vector of unknowns x where A and b are given.
We apply wavelet transform methods to this system of equations in an attempt to
speed up the process of solving for the vector x. The goal is to sparsify the matrix which
makes the matrix solution faster. Several different types of wavelets can be used. Among
those we used are the Daubechies class of orthogonal wavelets [6]. The transform is
performed using matrix multiplication and is a similarity transform in the first stage. If
W is the orthogonal matrix which performs the wavelet transform on the columns of a
matrix, then the wavelet transform à of the matrix A is given by
à = WAWT ,
(9.5)
since W−1 = WT (similarity transform). Applying this transformation to Eq. (9.4) we
get
WAWT Wx = Wb,
(9.6)
and the system is just
Ãx̃ = b̃,
(9.7)
where x̃ = Wx and b̃ = Wb.
Pick a threshold , and replace à by Ã(t) as follows
(t)
Ãij
Solve the new system
=
0,
Ãij
|Ãij | ≤ ,
otherwise.
Ã(t) x̃ = b̃,
(9.8)
(9.9)
for x̃. Then compute x = WT x̃, which is the thresholded solution of Eq. (9.4).
Several researchers have recently proposed the use of wavelets in the solution of matrix
equations similar to (9.1) [12, 13, 18, 25, 26].
A comparison of our results with wavelet and thresholded wavelet methods to these
other references is difficult since the latter are applied to canonical geometric shapes such
17
as corner reflectors [13, 25], circular cylinders [12, 25], a thin wire and thin flat plate [18],
and a semicircular array of parallel thin cylinders [26]. Nominally the equations used in
these references are similar to Eq. (4.16) with α = 0, i.e. to our CC1 equations. But,
because of the simple geometric shapes, the Green’s functions have a much simpler form
(only a single Hankel function) than ours. Ours involve a rough surface whose height and
slope vary over a parametric set and contain other parameters such as angle of incidence
and surface period. Our Green’s functions are thus much more complicated and take
a much longer time to evaluate. See the discussion in Appendix A. Nevertheless some
points of agreement and disagreement are possible. First, the wavelet transform can be
used to sparsify the impedance matrix and to make the resulting inversion more efficient
in the sense that the inversion of just this matrix is faster. But this is not the full
story. The times necessary to set up and undo the wavelet transform, and to do a matrix
search to threshold can be many times the inversion time of the untransformed impedance
matrix. This is particularly true for the SC and SS methods where the straight inversion
is much faster. We discussed this in a previous report [2]. Second, these other references
quite often reach a sparsity of 10 % or so (only 10 % of the matrix elements remain
after thresholding). We could not reach these levels of sparsity without our matrices
becoming singular unless we dramatically oversampled to begin with. See Figs. 11-15 and
the discussion in Section 10.6. In addition it is claimed in all the references that very
sparse matrices did not significantly affect the accuracy. No references had the energy
check levels we obtained. (The method of moments solution was assumed to be the exact
result). We found (see Figs. 11-15) that the accuracy decreased dramatically as sparsity
increased. In fact, sparsification is not recommended for these rough surface scattering
problems. Note that this conclusion is based on the fact that we computed a matrix
inverse whereas the references used some iterative technique. Sparsification may help
iterative techniques.
It should be remembered that we have control over sampling. Proponents of wavelets
go immediately to large matrices which are often the result of oversampling. Significant
sparsification may not appreciably alter the accuracy. We on the other hand often achieved
better results with small dense matrices and no wavelet techniques.
18
10
Computational Results
This section presents timing and reliability results for several formalisms on several surfaces. λ/Δx is a measure of pulse width, with higher numbers corresponding to faster
sampling. λ/Δx = 10 is often quoted, but it can be either oversampling or undersampling.
The surface graphs use equal scales on the horizontal and vertical axes, so apparent tangency is true tangency. The best CC method is used for the surface current and scattered
amplitude plots, since spectral methods often have incorrect currents. An additional result is referred to as CG, a CC Galerkin approach with Fourier basis functions for a first
kind equation. This is very similar to CC1, but the self-cell integral is performed exactly,
not using the first few terms in an expansion. A discussion of the numerical methods for
CC, SC, and CG methods can be found in Appendices B through D respectively. The SS
numerical technique is treated in Section 7. For our calculations with CG, we set α = 0
and β = 1 (see Appendix D).
In Sections 10.1 to 10.5 we treat a variety of rough surface examples. In Sections 10.1
to 10.3 respectively we consider the cases when λ/L 1 (geometrical optics regime),
λ/L ≈ 1 (resonance or physical optics regime), and λ/L 1 (sometimes referred to as a
renormalization regime). In Section 10.4 we treat a very rough surface with a maximum
slope of about 25. In Section 10.5 we present results of a case with a highly oscillatory
surface with the oscillations increasing as the end points of the period are approached.
The results of these computations can be summarized as follows. The CC methods
always worked well in the sense that the error was small for a sufficiently large matrix.
This is illustrated in Figures 2-9 and the accompanying tables contained in Sections 10.1
to 10.5 (Examples 1-5). For the CC methods, fill time refers to the time necessary to
compute the matrix elements, here the time to compute the periodic Green’s function
and its normal derivative. This took a great deal of time (see the discussion in Appendix
B), and the result was that the CC methods were extremely slow.
The fill time for the SC method was several orders of magnitude faster than CC
(this is because we were only evaluating a function, as shown in Appendix C). SC was
clearly the solution method of preference when it worked. It failed to work for very rough
(Example 4) and highly oscillatory (Example 5) surfaces. The fill time for the SS method
was between that for CC and SC and consisted of the evaluation of matrix elements of
the form given in Eq. 7.4. The SS method is based on the same type of topological
basis expansion as the SC method and it had the same convergence difficulties as the SC
method.
In Section 10.6 we use Examples 1-3 in conjunction with the wavelet transform
methodology explained in Section 9 where we discussed some conclusions. We note a
significant decrease in accuracy with sparsity. We gain no significant advantage by using
the wavelet transform, and find that it can often be disadvantageous.
In Section 10.7 we present magnitudes of the periodic Green’s function (parts (a) in
Figs. 16-21) and the magnitudes of the Fourier transform (parts (b) in Figs. 16-21). The
former generally reproduce results noted by others, while the latter are new. In particular,
we note the similarity of Figs 18(b) and 19(b) to crystal diffraction patterns [5] and Figs
20(b) and 21(b) to the diffraction patterns of a rectangular slit [3].
19
10.1
Example 1, λ/L 1
10.1.1
Case A, No Grazing
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
0.075
0.01563553622559
20◦
Error = log10 |1 − Normalized Energy|
Matrix
Formalism
Size
λ/Δx Fill Time
SS
128 by 128
4788
SS
138 by 138
6272
SS
148 by 148
7930
SC
128 by 128
2.0
0.64
SC
138 by 138
2.2
0.74
SC
148 by 148
2.3
0.82
CC1
64 by 64
1.0
501
CC1
128 by 128
2.0
2039
CC1
256 by 256
4.0
8165
CC2
64 by 64
1.0
808
CC2
128 by 128
2.0
3283
CC2
256 by 256
4.0
13201
CG
65 by 65
1.0
515
CG
129 by 129
2.0
2133
CG
257 by 257
4.0
8401
20
Linear Solution
Time
Error
0.24
-14.8
0.74
-15.3
0.88
-15.4
0.62
-15.7
0.72
-15.7
0.91
-15.1
0.11
-0.2
0.61
-1.6
4.18
-2.7
0.10
0.1
0.62
-5.7
4.04
-8.0
0.10
-0.2
0.61
-1.3
4.15
-3.4
Surface and Incoming Waves
Scattered Energy Distribution
0.1
0.4
0.3
0.08
0.2
0.06
0.1
0
0.04
−0.1
−0.2
0.02
−0.3
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−0.06
0.5
−0.04
−0.02
0
(a)
0.02
0.04
0.06
(b)
|N(x)| vs x
Real(N(x)) vs x
2.5
2.05
2
2
1.5
1.95
1
0.5
1.9
0
1.85
−0.5
−1
1.8
−1.5
1.75
−2
−2.5
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
1.7
−0.5
0.5
(c)
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d)
Figure 2: Example 1, Case A: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC2 formalism,
matrix size 256 by 256.
21
10.1.2
Case B, Near-Grazing Incidence/Reflection
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
0.075
0.01566499626662
75◦
Error = log10 |1 − Normalized Energy|
Matrix
Formalism
Size
λ/Δx Fill Time
SS
128 by 128
1305
SS
138 by 138
2058
SS
148 by 148
2901
SC
128 by 128
2.0
0.76
SC
138 by 138
2.2
0.74
SC
148 by 148
2.3
0.83
CC1
64 by 64
1.0
541
CC1
128 by 128
2.0
2194
CC1
256 by 256
4.0
8781
CC2
64 by 64
1.0
848
CC2
128 by 128
2.0
3450
CC2
256 by 256
4.0
13851
CG
65 by 65
1.0
574
CG
129 by 129
2.0
2296
CG
257 by 257
4.0
9070
22
Linear Solution
Time
Error
0.24
-0.9
0.76
0.1
0.86
-0.6
0.59
-0.5
0.73
-0.9
0.88
-3.1
0.10
-0.5
0.60
-0.4
4.11
-4.7
0.11
-0.3
0.59
-0.8
3.98
-4.9
0.11
-0.5
0.62
-0.5
4.12
-3.9
Surface and Incoming Waves
Scattered Energy Distribution
0.4
0.3
0.1
0.2
0.05
0.1
0
0
−0.1
−0.2
−0.05
−0.3
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
−0.15
0.5
−0.1
−0.05
(a)
1.2
1
1
0.5
0.8
0
0.6
−0.5
0.4
−1
0.2
−0.3
−0.2
−0.1
0
0.1
|N(x)| vs x
1.5
−0.4
0.05
(b)
Real(N(x)) vs x
−1.5
−0.5
0
0.1
0.2
0.3
0.4
0
−0.5
0.5
(c)
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d)
Figure 3: Example 1, Case B: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC1 formalism,
matrix size 256 by 256.
23
10.2
Example 2, λ/L ≈ 1
10.2.1
Case A, No Grazing
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
0.25
0.95
20◦
Error = log10 |1 − Normalized Energy|
Matrix
Formalism
Size
λ/Δx Fill Time
SS
2 by 2
0.29
SS
6 by 6
1.63
SS
10 by 10
4.7
SS
14 by 14
10.1
SS
18 by 18
18.2
SS
22 by 22
29
SC
2 by 2
1.9
0.11
SC
6 by 6
5.7
0.02
SC
10 by 10
9.5
0.02
SC
14 by 14
13.3
0.05
SC
18 by 18
17.1
0.06
CC1
64 by 64
60.8
298
CC1
128 by 128 122
1224
CC1
256 by 256 243
4993
CC2
64 by 64
60.8
478
CC2
128 by 128 122
1959
CC2
256 by 256 243
8126
CG
65 by 65
61.8
310
CG
129 by 129 123
1231
CG
257 by 257 244
4898
A−1
A0
=
=
Linear Solution
Time
Error
0.01
-0.4
0
-1.7
0
-2.2
0
-2.3
0
-2.4
0.01
-2.3
0
−∞
0
-2.5
0.01
-2.7
0.01
-2.7
0.01
-2.7
0.11
-4.7
0.63
-5.3
4.2
-5.9
0.10
-5.5
0.60
-6.4
4.1
-7.3
0.11
-8.3
0.61
-9.5
4.05
-10.7
−0.17963135247142 − 0.65697150178152i
−0.75962914626508 + 0.17615097897977i
24
Surface and Incoming Waves
Scattered Energy Distribution
0.5
0.6
0.4
0.5
0.3
0.4
0.2
0.3
0.1
0
0.2
−0.1
0.1
−0.2
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−0.4
0.5
−0.3
−0.2
−0.1
0
0.1
(a)
(b)
Real(N(x)) vs x
|N(x)| vs x
0.2
0.3
3.5
0.5
0
3
−0.5
2.5
−1
−1.5
2
−2
1.5
−2.5
−3
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
1
−0.5
0.5
(c)
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d)
Figure 4: Example 2, Case A: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC1 formalism,
matrix size 256 by 256.
25
10.2.2
Case B, Near-Grazing Incidence/Reflection
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
0.25
0.95
75◦
Error = log10 |1 − Normalized Energy|
Matrix
Formalism
Size
λ/Δx Fill Time
SS
3 by 3
0.38
SS
7 by 7
2.1
SS
11 by 11
5.7
SS
15 by 15
11.9
SS
19 by 19
20.2
SS
23 by 23
31.4
SC
3 by 3
2.9
0.01
SC
7 by 7
6.7
0.03
SC
11 by 11
10.5
0.04
SC
15 by 15
14.3
0.05
SC
19 by 19
18.1
0.07
SC
23 by 23
21.9
0.07
SC
27 by 27
25.7
0.09
CC1
64 by 64
60.8
303
CC1
128 by 128 122
1182
CC1
256 by 256 243
4799
CC2
64 by 64
60.8
469
CC2
128 by 128 122
1928
CC2
256 by 256 243
7787
CG
65 by 65
61.8
305
CG
129 by 129 123
1216
CG
257 by 257 244
4824
A−2
A−1
A0
=
=
=
Linear Solution
Time
Error
0
-1.1
0
-1.9
0
-2.0
0
-2.0
0.01
-1.9
0.01
-1.8
0
-1.4
0
-2.8
0.01
-3.4
0
-3.6
0.01
-3.7
0.02
-3.7
0.02
-3.6
0.11
-5.7
0.58
-6.3
4.27
-6.9
0.12
-6.1
0.61
-7.0
4.06
-7.9
0.11
-9.7
0.62
-9.8
4.14
-9.7
0.07250263029849 + 0.00497683319193i
−0.01402209947690 − 0.18205512555385i
−0.91996242949398 + 0.13259130973127i
26
Surface and Incoming Waves
Scattered Energy Distribution
0.5
0.6
0.4
0.3
0.4
0.2
0.2
0.1
0
0
−0.2
−0.1
−0.4
−0.2
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
−0.8
0.5
−0.6
−0.4
−0.2
(a)
0
0.2
0.4
0.6
0.8
(b)
|N(x)| vs x
Real(N(x)) vs x
1.1
1
1
0.8
0.9
0.6
0.8
0.4
0.7
0.6
0.2
0.5
0
0.4
−0.2
−0.4
−0.5
0.3
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.2
−0.5
0.5
(c)
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d)
Figure 5: Example 2, Case B: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC2 formalism,
matrix size 256 by 256.
27
10.3
Example 3, λ/L 1
10.3.1
Case A, No Grazing
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
2.5
100
20◦
Error = log10 |1 − Normalized Energy|
Matrix
Formalism
Size
SS
1 by 1
SS
5 by 5
SS
9 by 9
SS
13 by 13
SS
17 by 17
SS
21 by 21
SS
29 by 29
SC
1 by 1
SC
5 by 5
SC
9 by 9
SC
13 by 13
SC
17 by 17
SC
21 by 21
CC1
64 by 64
CC1
128 by 128
CC1
256 by 256
CC2
64 by 64
CC2
128 by 128
CC2
256 by 256
CG
65 by 65
CG
129 by 129
CG
257 by 257
A0 =
λ/Δx
100
500
900
1300
1700
2100
6400
12800
25600
6400
12800
25600
6500
12900
25700
Fill Time
0.36
1.27
3.9
8.5
15.6
25.5
55.4
0
0.02
0.02
0.04
0.05
0.07
149
526
2042
267
893
3317
159
542
2091
Linear Solution
Time
Error
0
-2.0
0
-4.7
0.01
-7.5
0.01
-6.2
0.01
-7.7
0.01
-8.2
0.01
-5.3
0
-15.7
0.01
−∞
0.01
-15.4
0.01
−∞
0.01
-13.1
0
-10.3
0.1
-15.4
0.6
-15.4
4.1
-15.1
0.1
-6.7
0.6
-8.7
3.9
-9.0
0.11
-15.1
0.63
−∞
4.16
-14.8
−0.99185964722787 + 0.12733593444511i
28
Surface and Incoming Waves
Scattered Energy Distribution
5
0.9
4
0.8
3
0.7
0.6
2
0.5
1
0.4
0
0.3
0.2
−1
0.1
−2
−4
−3
−2
−1
0
1
2
3
0
−0.6
4
−0.4
−0.2
0
(a)
9
−1
8
−2
7
−3
6
−4
5
−5
4
−6
3
−7
2
−8
1
−0.3
−0.2
−0.1
0
0.6
|N(x)| vs x
0
−0.4
0.4
(b)
Real(N(x)) vs x
−9
−0.5
0.2
0.1
0.2
0.3
0.4
0
−0.5
0.5
(c)
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d)
Figure 6: Example 3, Case A: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC1 formalism,
matrix size 256 by 256.
29
10.3.2
Case B, Near-Grazing Incidence/Reflection
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
2.5
100
75◦
Error = log10 |1 − Normalized Energy|
Matrix
Formalism
Size
SS
1 by 1
SS
5 by 5
SS
9 by 9
SS
13 by 13
SS
17 by 17
SS
21 by 21
SS
29 by 29
SC
1 by 1
SC
5 by 5
SC
9 by 9
SC
13 by 13
SC
17 by 17
SC
21 by 21
CC1
64 by 64
CC1
128 by 128
CC1
256 by 256
CC2
64 by 64
CC2
128 by 128
CC2
256 by 256
CG
65 by 65
CG
129 by 129
CG
257 by 257
λ/Δx
100
500
900
1300
1700
2100
6400
12800
25600
6400
12800
25600
6500
12900
25700
Fill Time
0.24
1.25
3.9
8.5
15.5
25.2
54.7
0
0.01
0.03
0.05
0.05
0.07
154
539
2108
277
917
3442
165
557
2145
Linear Solution
Time
Error
0
-3.1
0
-5.8
0.01
-8.6
0.01
-8.0
0.01
-8.8
0
-8.2
0.02
-5.6
0
-15.2
0.01
-15.7
0.01
-15.2
0.01
-15.7
0
-13.7
0.01
-10.2
0.1
-12.7
0.6
-12.9
4.2
-13.2
0.1
-6.7
0.6
-8.7
3.9
-9.0
0.1
-12.7
0.6
-12.9
4.0
-13.2
A0 = −0.99938168553634 + 0.03516029884120i
30
Surface and Incoming Waves
Scattered Energy Distribution
5
0.8
4
0.6
3
0.4
2
0.2
1
0
0
−0.2
−1
−0.4
−2
−0.6
−1
−4
−3
−2
−1
0
1
2
3
−0.5
0
(a)
(b)
|N(x)| vs x
Real(N(x)) vs x
0
2.5
−0.5
2
−1
1.5
−1.5
1
−2
0.5
−2.5
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.5
4
0.1
0.2
0.3
0.4
0
−0.5
0.5
(c)
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d)
Figure 7: Example 3, Case B: (a) Surface and incoming waves, (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC1 formalism,
matrix size 256 by 256.
31
10.4
Example 4, Very Rough Surface
Times are in seconds of CPU time required by a SPARC 20 workstation.
This section presents results for a surface with extremely large slopes. We show that
our code still converges for such cases. The maximum slope, πd/L, is about 25. λ is
relatively small here. It is expected that better results should be attainable with larger
values of λ.
Interface
S(x)
d/L
λ/L
θi
Perfectly Reflecting
−(d/2) cos(2πx/L)
8
0.05
20◦
Error = log10 |1 − Normalized Energy|
Formalism
CC2
CC2
CG
CG
Matrix
Size
λ/Δx Fill Time
512
25.6
9870
1024
51.2
39488
513
25.7
7288
1025
51.3
28872
32
Linear Solution
Time
Error
28
1.9
250
-3.9
29
-0.2
250
-0.2
Surface and Incoming Waves
Scattered Energy Distribution
300
0.25
250
200
0.2
150
0.15
100
50
0.1
0
−50
0.05
−100
−150
−10
−8
−6
−4
−2
0
2
4
6
8
0
−0.2
10
−0.15
−0.1
−0.05
0
(a)
(b)
Real(N(x)) vs x
|N(x)| vs x
0.05
0.1
250
100
80
200
60
40
150
20
0
100
−20
−40
50
−60
−80
−100
−10
−8
−6
−4
−2
0
2
4
6
8
0
−10
10
(c)
−8
−6
−4
−2
0
2
4
6
8
10
(d)
Figure 8: (a) Surface and incoming waves (carefully note the scale), (b) Scattered energy
distribution, and surface current ((c) real part (d) modulus) for the CC2 formalism, matrix
size 1024 by 1024.
33
10.5
Example 5, Highly Oscillatory Surface with Continuous
Derivative
Due to the roughness of the surface, it is sampled uniformly in arc length (instead of x).
S(x)
d/L
λ/L
θi
−(d/2) cos (2πx/L + 10π(2x/L)3 ) , |x| ≤ L/2
S(x + nL) = S(x),
elsewhere
0.15
0.05
20◦
Error = log10 |1 − Normalized Energy|
Matrix
Formalism
Size
λ/Δx Fill Time
SS
40 by 40
1316
SS
48 by 48
1887
SS
56 by 56
2482
SC
40 by 40
2.0
0.26
SC
48 by 48
2.4
0.16
SC
56 by 56
2.8
0.23
CC1
512 by 512
25.6
22406
CC1
1024 by 1024 51.2
89674
CC2
512 by 512
25.6
36635
CC2
1024 by 1024 51.2
146970
CG
513 by 513
25.7
22111
CG
1025 by 1025 51.3
88367
34
Linear Solution
Time
Error
0.01
-0.2
0.05
1.2
0.08
1.7
0.05
1.4
0.05
1.5
0.10
0.7
28.0
-0.6
247
-0.8
28.3
-2.0
246
-3.1
30.2
-2.4
249
-4.4
Surface and Incoming Waves
Scattered Energy Distribution
8
6
0.1
4
2
0
0.05
−2
−4
−6
−10
−8
−6
−4
−2
0
2
4
6
8
10
0
−0.05
0
(a)
0.05
0.1
(b)
|N(x)| vs x
Real(N(x)) vs x
45
30
40
20
35
30
10
25
0
20
15
−10
10
−20
5
−30
−10
−8
−6
−4
−2
0
2
4
6
8
0
−10
10
(c)
−8
−6
−4
−2
0
2
4
6
8
10
(d)
Figure 9: Example 5: (a) Surface and incoming waves, (b) Scattered energy distribution,
and surface current ((c) real part (d) modulus) for the CG formalism, matrix size 1025
by 1025.
35
10.6
Wavelet Transforms Applications
Part I: Sparsity and Accuracy with Wavelets, CC1
This section will analyze sparsity and accuracy of results when wavelet transforms
are used. Its purpose is to test our hypothesis that wavelets lose the accuracy that
a rapidly-sampled system brings, and they do not provide a significant speedup on a
coarsely-sampled system.
The initial results agree with this hypothesis. These results show that a transformed
matrix with 75% sparsity has much worse results than a non-transformed dense matrix
with 1/4 the number of elements. Also, CC2, with its strong diagonal, has fewer problems
in sparse situations than CC1. CC1 often becomes singular at a low sparsity. As our test
cases, we use Examples 1-3 with no grazing (from Sections 10.1 to 10.3).
36
10.6.1
Example 1, Case A: λ/L 1, No Grazing, CC1 matrices, Daub6 filter
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
0.075
0.01563553622559
20◦
Error = log10 |1 − Normalized Energy|
Error vs. Sparsity
2
1
................................
0 2........................2
............
.....2
2
...........................2
...........................2
...........................2
...........................2
...........................2
...........................2
........................................................................................................................................ .................... 2.............
......
...
....... ...........
2 2 2 2
.2
....
........................
.........
...
.
2
.
.
..
.
.
3
.
.
..
...
...
...
...
..
...
−1
..
...
.
.
.
.
.
...
...
..
...
.......
...........................3
...........................3
...........................3
...........................3
...........................
.3
..............................3
...........................3
..
3.......................3
...
..
.
..
.
.
.
...
...
...
...
...
...
...
−2
...
..
..
..
..
..
..
...
...
..
.
......................................................................................................................
.
.......... ....
...........
−3 Error
... ..
... ..
.....
.
3
−4
−5
−6
−7
−8
−9
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fraction Zero Elements
Figure 10: Error vs. Sparsity, CC1 matrices for Example 1, Case A. 2 — 64 by 64
matrices, 3 — 128 by 128 matrices, — 256 by 256 matrices. If a plot stops before 80%
sparsity, it has become singular or had an error worse than the graph can accomodate.
37
10.6.2
Example 2, Case A: λ/L ≈ 1, No Grazing, CC1 matrices, Daub6 filter
−(d/2) cos(2πx/L)
0.25
0.95
20◦
S(x)
d/L
λ/L
θi
Error = log10 |1 − Normalized Energy|
Error vs. Sparsity
2
1
0
−1
−2
−3
Error
−4
−5
−6
2
3
3
2
3
2
2
3
2
3
2
3
2
3
2
3
3
2
2
3
3
2
3
3
2
3
2
3
2
2
−7 2
3
−8 −9
2
...
.... ..
.... ...
.... . ..
.... ...............
.
.
.
.... ... .......
.... ...
......
.........
.... ....
....
..
......... .....
......
...
.. ....
........ ...
..........
...
.... .......... .... .....
...................................................
......
.
.
.
.
.
.
.
.
.
.
.
.
.
..... .......
...
...
..............
..
..... ...........
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.......... ........
...
... ...... ....
....... ...........
..............
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...
..... .......
...... ...................................
...... ... .........
.....
.
.
..............
.
.
.
.
.
.
.
.
.
.
.
.
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.
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.
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.
......
......
...
. .....
.
..
.. ...........
...
.....
.
.
.
.
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.
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.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
....
......
...
.. ... ...
..
......
...
....
...... ......
....
... ...
...
.....
.....................
........
....
...
...........
.....
..
..
.............
........... .................
....
.. .................
..
.......
.......................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
......... .. ..
......
....
.
...
...
............
.........
..
....
.....
...
....
.......
..
....
...
.....
....
.. ................
...
..
.....
... ....
...
..
.......................
....
............
... ...
....
...
.
.....
.
.
.
.
.
.... ........................................
.
.....
....
....
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...
.
.
.
.
.
.
.
......
...
..
.
......
.
.
...
.
...
..
......
..
...
....
... ..
...
...
......
.. ..
...
.....
...
... ..
... ..........
.
.... ..... ....
.........
.. ... ..
... ... ...
... .... ...
.... ... ....
.. .. ..
... ... ....
... ... ...
.... .... ...
.. .. ..
... .... ....
... ... ...
.... ........
.. ....
... ......
... .......
.... ..... ...
.. ... ..
... ... ...
...... ...
.. ... ...
.
... ... ....
... ...
..
... ...
...
.. ...
.
...
..
.. ...
.
.
.. ..
...
.. ..
...
... ..
...
... ....
...
....
..
..
..
...
...
..
..............................
..
..
...
...
..
0.0
3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fraction Zero Elements
Figure 11: Error vs. Sparsity, CC1 matrices for Example 2, Case A. 2 — 64 by 64
matrices, 3 — 128 by 128 matrices, — 256 by 256 matrices. If a plot stops before 80%
sparsity, it has become singular or had an error worse than the graph can accomodate.
38
10.6.3
Example 3, Case A: λ/L 1, No Grazing, CC1 matrices, Daub 6 filter
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
2.5
100
20◦
Error = log10 |1 − Normalized Energy|
Error vs. Sparsity
−3
−4
−5
−6
−7
−8
Error
−9
−10
−11
−12
−13
−14
−15
...
2
... .....
... ....
2
..
......
. .....
......
....
......
...
....
......
.
.
.
.
....
.
....
...............................
..
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....
..
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....
....
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..
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....
.
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..
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..
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2
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3
2
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2
3
3
2
3
3
3
3
3
2
3
2
3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fraction Zero Elements
Figure 12: Error vs. Sparsity, CC1 matrices for Example 3, Case A. 2 — 64 by 64
matrices, 3 — 128 by 128 matrices, — 256 by 256 matrices. If a plot stops before 80%
sparsity, it has become singular or had an error worse than the graph can accomodate.
39
Part II: Sparsity and Accuracy with Wavelets, CC2
10.6.4
Example 1, Case A: λ/L 1, No Grazing, CC2 matrices, Daub 6 filter
−(d/2) cos(2πx/L)
0.075
0.01563553622559
20◦
S(x)
d/L
λ/L
θi
Error = log10 |1 − Normalized Energy|
Error vs. Sparsity
2
1
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0 2
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Error
−4
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2
3
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2
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3
3
−8 −9
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fraction Zero Elements
Figure 13: Error vs. Sparsity, CC2 matrices for Example 1, Case A. 2 — 64 by 64
matrices, 3 — 128 by 128 matrices, — 256 by 256 matrices. If a plot stops before 80%
sparsity, it has become singular or had an error worse than the graph can accomodate.
40
10.6.5
Example 2, Case A: λ/L ≈ 1, No Grazing, CC2 matrices, Daub 6 filter
−(d/2) cos(2πx/L)
0.25
0.95
20◦
S(x)
d/L
λ/L
θi
Error = log10 |1 − Normalized Energy|
Error vs. Sparsity
2
1
0
−1
−2
−3
Error
−4
−5
−6
−7
2
3
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−8
−9
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fraction Zero Elements
Figure 14: Error vs. Sparsity, CC2 matrices for Example 2, Case A. 2 — 64 by 64
matrices, 3 — 128 by 128 matrices, — 256 by 256 matrices. If a plot stops before 80%
sparsity, it has become singular or had an error worse than the graph can accomodate.
41
10.6.6
Example 3, Case A: λ/L 1, No Grazing, CC2 matrices, Daub 6 filter
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
2.5
100
20◦
Error = log10 |1 − Normalized Energy|
Error vs. Sparsity
3
2
1
0
−1
−2
−3
Error
−4
−5
−6
−7
−8
.
3
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−9 −10
2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fraction Zero Elements
Figure 15: Error vs. Sparsity, CC2 matrices for Example 3, Case A. 2 — 64 by 64
matrices, 3 — 128 by 128 matrices, — 256 by 256 matrices. If a plot stops before 80%
sparsity, it has become singular or had an error worse than the graph can accomodate.
42
10.7
Fourier Transforms of Periodic Green’s Functions
In all cases, the Fourier transform used is
Ĝp1 (kx , kx ) =
≈
L/2 L/2
−L/2
j
m
−L/2
e−i(kx x+kx x ) Gp1 (x, x ) dx dx
e−i(kx xj +kx xm ) Gp1 (xj , xm )Δxj Δxm ,
where the Green’s function values are obtained from the CC1 matrix. We use as our
cases Examples 1-3 for both non-grazing and near-grazing incidence and reflection. Note
that the function Gp1 (x, x ) is the periodic Green’s function evaluated on the surface. See
Appendix A for the explicit representations used.
43
10.7.1
Example 1, Case A, λ/L 1, No Grazing
S(x)
d/L
λ/L
θi
10.7.2
Example 1, Case B, λ/L 1, Near-Grazing Incidence/Reflection
S(x)
d/L
λ/L
θi
10.7.3
−(d/2) cos(2πx/L)
0.25
0.95
75◦
Example 3, Case A, λ/L 1, No Grazing
S(x)
d/L
λ/L
θi
10.7.6
−(d/2) cos(2πx/L)
0.25
0.95
20◦
Example 2, Case B, λ/L ≈ 1, Near-Grazing Incidence/Reflection
S(x)
d/L
λ/L
θi
10.7.5
−(d/2) cos(2πx/L)
0.075
0.01566499626662
75◦
Example 2, Case A, λ/L ≈ 1, No Grazing
S(x)
d/L
λ/L
θi
10.7.4
−(d/2) cos(2πx/L)
0.075
0.01563553622559
20◦
−(d/2) cos(2πx/L)
2.5
100
20◦
Example 3, Case B, λ/L 1, Near-Grazing Incidence/Reflection
S(x)
d/L
λ/L
θi
−(d/2) cos(2πx/L)
2.5
100
75◦
44
|Gp1 (x, x )|
−0.4
−0.3
−0.2
x
−0.1
0
0.1
0.2
0.3
0.4
−0.4
−0.3
−0.2
−0.1
0
xprime
0.1
0.2
0.3
0.4
(a)
|Ĝp1 (kx , kx )|
−40
−30
−20
kx
−10
0
10
20
30
40
−40
−30
−20
−10
0
kxprime
10
20
30
40
(b)
Figure 16: Example 1, Case A, modulus of (a) periodic Green’s function and (b) Fourier
transform of periodic Green’s function.
45
|Gp1 (x, x )|
−0.4
−0.3
−0.2
x
−0.1
0
0.1
0.2
0.3
0.4
−0.4
−0.3
−0.2
−0.1
0
xprime
0.1
0.2
0.3
0.4
(a)
|Ĝp1 (kx , kx )|
−40
−30
−20
kx
−10
0
10
20
30
40
−40
−30
−20
−10
0
kxprime
10
20
30
40
(b)
Figure 17: Example 1, Case B, modulus of (a) periodic Green’s function and (b) Fourier
transform of periodic Green’s function.
46
|Gp1 (x, x )|
−0.4
−0.3
−0.2
x
−0.1
0
0.1
0.2
0.3
0.4
−0.4
−0.3
−0.2
−0.1
0
xprime
0.1
0.2
0.3
0.4
(a)
|Ĝp1 (kx , kx )|
−40
−30
−20
kx
−10
0
10
20
30
40
−40
−30
−20
−10
0
kxprime
10
20
30
40
(b)
Figure 18: Example 2, Case A, modulus of (a) periodic Green’s function and (b) Fourier
transform of periodic Green’s function.
47
|Gp1 (x, x )|
−0.4
−0.3
−0.2
x
−0.1
0
0.1
0.2
0.3
0.4
−0.4
−0.3
−0.2
−0.1
0
xprime
0.1
0.2
0.3
0.4
(a)
|Ĝp1 (kx , kx )|
−40
−30
−20
kx
−10
0
10
20
30
40
−40
−30
−20
−10
0
kxprime
10
20
30
40
(b)
Figure 19: Example 2, Case B, modulus of (a) periodic Green’s function and (b) Fourier
transform of periodic Green’s function.
48
|Gp1 (x, x )|
−0.4
−0.3
−0.2
x
−0.1
0
0.1
0.2
0.3
0.4
−0.4
−0.3
−0.2
−0.1
0
xprime
0.1
0.2
0.3
0.4
(a)
|Ĝp1 (kx , kx )|
−40
−30
−20
kx
−10
0
10
20
30
40
−40
−30
−20
−10
0
kxprime
10
20
30
40
(b)
Figure 20: Example 3, Case A, modulus of (a) periodic Green’s function and (b) Fourier
transform of periodic Green’s function.
49
|Gp1 (x, x )|
−0.4
−0.3
−0.2
x
−0.1
0
0.1
0.2
0.3
0.4
−0.4
−0.3
−0.2
−0.1
0
xprime
0.1
0.2
0.3
0.4
(a)
|Ĝp1 (kx , kx )|
−40
−30
−20
kx
−10
0
10
20
30
40
−40
−30
−20
−10
0
kxprime
10
20
30
40
(b)
Figure 21: Example 3, Case B, modulus of (a) periodic Green’s function and (b) Fourier
transform of periodic Green’s function.
50
11
Conclusions
Computational results have been presented for scattering from various periodic surfaces.
The results include examples with grazing incidence, a very rough surface and a highly
oscillatory surface. The examples vary over a parameter set which includes the geometrical optics regime, physical optics and resonance regime, and a renormalization regime.
Results of wavelet applications are also included as well as the Fourier transform of the
periodic Green’s function.
The main objective of this study was to determine the best computational method
for these problems. Briefly the SC method was the fastest but did not converge for large
slopes or very rough surfaces. The SS method was slower and had the same convergence
difficulties as SC. The CC methods were extremely slow but always converged. The
simplest approach is to try the SC method first. Convergence, when the method works,
is very fast. If convergence doesn’t occur then try SS and finally CC. Results for the
remaining mixed representation (CS) can be found in the literature [22]. The use of
wavelet transform methods for these rough surface scattering problems is generally not
recommended since they take so long to implement and sparsification for our problems
often dramatically reduced accuracy.
51
A
Periodic Green’s Function
In Section 4 we derived the first (or spectral) representation of Gp1 given by
∞
1 ik1 [αj (x−x )+βj |s(x)−s(x )|]
i λ e
.
4π L j=−∞ βj
Gp1 (x, x ) =
(A.1)
A second representation follows from using Eqs. (4.1) to (4.3). We have
∞
−∞
G
(2)
(xs , xs )N T (x )dx
=
∞
n=−∞
In ,
(A.2)
where now we use the Hankel function representation for G(2) (from Eq. (1.3) )
In =
(2n+1) L
2
(2n−1) L
2
1
i (1)
H0 (k1 [(x − x )2 + (s(x) − s(x ))2 ] 2 )N T (x )dx .
4
(A.3)
Shift the integration variable (x = x − nL), and use the periodicity of s(x) and the
Floquet periodicity of the boundary unknown Eq. (4.5). The result is
∞
−∞
G
(2)
(xs , xs )N T (x )dx
=
L
2
−L
2
Gp1 (x, x )N T (x )dx ,
(A.4)
where now
Gp1 (x, x ) =
∞
1
i 2
2
(1)
eik1 α0 nL H0 (k1 {[(x − (x + nL))] + [s(x) − s(x )] } 2 ),
4 n=−∞
(A.5)
which is the representation of Gp1 in terms of a phased periodic array of Hankel functions.
A third representation can also be derived using Eq. (A.5) [24]. From tables [20] we
have the Laplace transform representation
(1)
H0 (
√
s2
+
a2 )
2i
= − eis
π
∞
0
1
2
2
−st cos[a(t − 2it) ]
e
dt.
1
(t2 − 2it) 2
(A.6)
Transforming this equation using t = u2 we have
(1)
H0 (
√
s2
+
a2 )
4i
= − eis
π
where
∞
0
2
e−su D(a, u)du,
(A.7)
1
D(a, u) =
cos[au(u2 − 2i) 2 ]
1
(u2 − 2i) 2
.
(A.8)
Rewrite the sum in Eq. (A.5) in three parts, the n = 0 term, a sum from 1 to ∞, and a
sum from −1 to −∞. Let n → −n in the latter sum. If we define as the s−variable in
Eq. (A.7)
s± = k1 (±(x − x) + nL),
(A.9)
52
and use
a = k1 [s(x) − s(x )],
(A.10)
then Eq. (A.5) can be written using Eq. (A.7) as
1
i (1)
H0 (k1 [(x − x )2 + (s(x) − s(x ))2 ] 2 )
4
∞
∞
1
2
ik1 α0 nL is+
+
e
e
e−s+ u D(a, u)du
π n=1
0
∞
1 −ik1 α0 nL is− ∞ −s− u2
e
e
e
D(a, u)du.
+
π n=1
0
Gp1 (x, x ) =
(A.11)
The summations can be performed. Using Eq. (A.9) define the coefficients of n in Eq.
(A.11)
b± = k1 L(u2 − i[1 ± α0 ]),
(A.12)
and then the sums are
∞
e−nb± =
n=1
If we further define
e−b±
.
1 − e−b±
p± = eik1 L(1±α0 ) ,
(A.13)
(A.14)
then Eq. (A.11) can be written as
1
i (1)
H0 (k1 [(x − x )2 + (s(x) − s(x ))2 ] 2 )
4
2
1 + ik1 (x−x ) ∞ e−u k1 (x −x+L)
D(a, u)du
+
p e
π
1 − p+ e−k1 Lu2
0
2
1 − ik1 (x −x) ∞ e−u k1 (x−x +L)
D(a, u)du,
+
p e
π
1 − p− e−k1 Lu2
0
Gp1 (x, x ) =
(A.15)
which is the third representation for Gp1 on the surface.
The same analysis follows for the function off the surface. Extend a to b where
b = k1 (z − z ),
(A.16)
and we have the general off-the-surface representation
1
i (1)
H0 (k1 [(x − x )2 + (z − z )2 ] 2 )
4
2
1 + ik1 (x−x ) ∞ e−u k1 (x −x+L)
p e
D(b, u)du
+
π
1 − p+ e−k1 Lu2
0
2
1 − ik1 (x −x) ∞ e−u k1 (x−x +L)
p e
D(b, u)du,
+
π
1 − p− e−k1 Lu2
0
Gp1 (x, x ) =
(A.17)
which is used to compute the normal derivative of Gp1 as in Eq. (4.14).
To compute the impedance kernel in Section 4 we use the Green’s function representation in various ways. To compute Gp1 (x, x ) for example consider the following cases:
53
(a) For x = x and s(x), s(x ) far apart we use the spectral sum Eq. (A.1).
(b) For x = x but s(x) close to s(x ) use Eq. (A.15) and directly evaluate it. We
approximate the integrals using piecewise Gaussian quadrature. The integrands
decay rapidly and we determine where the integrand is negligible and approximate
the number of integration intervals to achieve good accuracy.
(c) For x = x we again use the representation Eq. (A.15) as follows. In the two integral
terms set x = x and evaluate as in case (b). The Hankel function term in Eq. (A.15)
must be treated as described in Appendix B by evaluating the self-cell integral.
Although Eq. (A.5) is the canonical representation for the periodic Green’s function in
two-dimensions we generally do not use it for evaluation purposes since the convergence
is slow and evaluation time per term is long.
To evaluate Gp1 we have corresponding cases:
(a) For x = x and s(x), s(x ) far apart take the normal derivative of Eq. (A.1).
(b) For x = x but s(x) close to s(x ) use a direct evaluation of Eq. (A.15).
(1)
(c) For x = x the normal derivative of the Hankel function H0
(1)
np ∂p H0 (k1 |xs
is
(1)
−
xs |)
np |xs − xs |p H1 (k1 |xs − xs |)
,
=−
|x − x |
(A.18)
which is finite in the limit as xs → xs
(1)
lim np ∂p H0 (k1 |xs − xs |) = −
xs →x
s
s (x )
i
,
π 1 + [s (x )]2
(A.19)
so the self-cell evaluation is not a problem.
Although we do not use it for our calculations, we can also define the Fourier transform
of the periodic Green’s function. It is given by
Ĝp (kx , kx ) =
≈
L
2
−L
2
−L
2
j
L
2
m
e−i(kx x + kx x ) Gp (xs , xs )
e−i(kx xj + kx xm ) Gp (xj , x m )Δxj Δxm
Several examples are presented in Section 10.
54
(A.20)
B
Numerical Solution of the CC Equations
In Section 4 we derived the CFIE integral equation given by
F i (x) =
L
2
−L
2
ZpD1 (x, x )N T (x )dx ,
(B.1)
where the impedance kernel is given by
1
ZpD1 (x, x ) = αδ(x − x ) + α PV Gp1 (x, x ) + βGp1 (x, x ),
2
(B.2)
F i (x) = αN i (xs ) + βψ i (xs ).
(B.3)
and
A standard discretization of Eq. (B.1) is the method-of-moments approach, which utilizes
a pulse basis and collocation in x. There are three steps in using this approach. First,
partition the interval [− L2 , L2 ] into a certain number (say M) subintervals, with the pth
interval called Δp . Then Eq. (B.1) can be written as
i
F (x) =
M p=1 Δp
ZpD1 (x, x )N T (x )dx .
(B.4)
Second, choose a representative point (such as the midpoint) in each subinterval, with the
pth given by xp , and approximate N T (x ) for x ∈ Δp as N T (xp ). Consequently,
i
F (x) ≈
M
p=1
T
N (xp )
Δp
ZpD1 (x, x )dx .
(B.5)
Third, collocate this equation for x equal to each of the chosen points, yielding
F i (xq ) ≈
M
p=1
N T (xp )
Δp
ZpD1 (xq , x )dx .
(B.6)
When the kernel is not singular (i.e. when p = q), approximate the integrals as
Δp
ZpD1 (xq , x )dx ≈ Δp ZpD1 (xq , xp ).
(B.7)
When the kernel is singular, we must analytically integrate any singular terms. As seen
in representation Eq. (A.11) for Gp1 , the only singular term is the Hankel function. For
x ≈ x ,
(1)
H0
2
k1 [(x − x ) + (s(x) − s(x
1
2 2
)) ]
1
k1
2i
[(x − x )2 + (s(x) − s(x ))2 ] 2 ,
≈ 1+
γ +ln
π
2
(B.8)
55
with Euler’s constant γ ≈ 0.5772. For small Δq , we can therefore approximate the integral
as follows
Δq
(1)
H0
2
k1 [(xq − x ) + (s(xq ) − s(x
≈
Δq
2
− Δq
2
1
2 2
)) ] dx
k1 |
| 1 + (s (xq ))2 2i
) d
1+
γ + ln(
π
2
k1 1 + (s (xq ))2 4i Δq
2i
2
) +
γ + ln(
ln d
= Δq 1 +
π
2
π
0
k1 1 + (s (xq ))2 eγ Δq 2i
) .
= Δq 1 + ln(
π
4e
(B.9)
This is the only nonrepairable singularity present in ZpD1 , so we can now form the matrix
equation we wish to solve
Fi = ZNT ,
(B.10)
where Fni = F i (xn ), NnT = N T (xn ), and
⎧
α
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎨
Zmn = Δn
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
+ α PV Gp1 (xm , xn ) + β U(xm , xn )
+ iβ4
1+
2i
π
ln
k1
√
1+(s (xn ))2 eγ Δn
4e
α PV Gp1 (xm , xn ) + βGp1 (xm , xn ),
,
m = n,
(B.11)
m = n,
where U(xm , xn ) is the nonsingular part of Gp1 (x, x ), namely, the two integral terms in
Eq. (A.15) which are evaluated at x = x . The latter term in the m = n equation in Eq.
(B.11) follows from Eq. (B.9)
The scattered amplitudes are given by an approximation to the “−” version of Eqs.
(6.11)-(6.13)
M
1
2βn An =
e−ik1 [αn xp +βn s(xp )] N(xp )Δp .
(B.12)
L p=1
56
C
Numerical Solution of the SC Equations
In Section 6 we derived the SC integral representation:
1 − 2 −ik1 [αn x−βn s(x)]
e
N(x)dx.
L − L2
L
−2Dβn δn0 =
(C.1)
We approximate the integral with the discrete quadrature rule given by
−L
2
−L
2
f (x)dx ≈
M
p=1
wp f (xp ),
(C.2)
where {wp } and {xp } are the weights and sampled points, respectively. Therefore, Eq.
(C.1) becomes
M
1
−2Dβn δn0 =
wp e−ik1 [αn xp −βn s(xp )] N(xp ).
(C.3)
L p=1
This can be written as the matrix equation
KN = F+ ,
(C.4)
where Fm+ = −2Dβ0 δm0 (see Eq. (7.3), Nn = N(xn ), and
Kmn =
wn −ik1 [αm xn −βm s(xn )]
e
.
L
(C.5)
The scattering amplitudes are obtained via the following approximation of Eq. (6.12)
2βn An =
M
1
wp e−ik1 [αn xp +βn s(xp )] N(xp ).
L p=1
(C.6)
More details on a particular numerical method for solving this system of equations can
be found in [16].
57
D
Numerical Solution of the CC Equations. Discrete
Galerkin Method
An alternative to the method-of-moments approach in Appendix B is set forth in [1]. This
method allows one to integrate the logarithmic singularity in the CC1 equation exactly,
avoiding the series approximation used in the method-of-moments approach.
A rigorous derivation of this method, with error analysis, can be found in [1]. We
present a more intuitive derivation here. First, we parameterize the coordinate variable,
Lt
using x(t) = 2π
− L2 . Then Eq. (4.16) becomes
2
2π
0
ZpD1 (t, t )N(x(t ))dt =
4π i
F (x(t)),
L
(D.1)
where
1
ZpD1 (t, t ) = αδ(x(t) − x(t )) + α PV Gp1 (x(t), x(t )) + βGp1 (x(t), x(t )).
2
(D.2)
Now we treat the singularity in Gp1 (x, x ) at x = x . We can write
1
1
Gp1 (x, x ) = Gp1 (x, x ) +
ln [(x − x )2 + (s(x) − s(x ))2 ] 2
2π
1
1
2
2 2
−
ln [(x − x ) + (s(x) − s(x )) ] ,
2π
(D.3)
where the singularity in the bracketed expression at x = x is repairable. In addition, the
logarithmic term can be written as
1
ln [(x(t) − x(t ))2 + (s(x(t)) − s(x(t )))2 ] 2
where
B̃(t, t ) =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
1
= ln |2e− 2 sin(
1
e− 12 [(x(t)−x(t ))2 +(s(x(t))−s(x(t )))2 ] 2 ln
,
|2 sin( t−t
)|
2
1
L
ln e− 2 2π
1 + [s (x(t))]2 ,
t − t
)| + B̃(t, t ),
2
(D.4)
t − t = 2mπ,
(D.5)
t − t = 2mπ.
Since B̃(t, t ) is not singular, we will group it with the other nonsingular or repairable
terms of the kernel:
B(t, t ) = αδ(x(t) − x(t )) + 2α PV Gp1 (x(t), x(t ))
1 1
1
2
2 2
+β 2Gp1 (x(t), x(t ))+ ln [(x(t)−x(t )) +(s(x(t))−s(x(t ))) ] − B̃(t, t ) (,D.6)
π
π
Then Eq. (D.1) becomes
2π
0
β
B(t, t )N(x(t ))dt −
π
2π
0
1
ln |2e− 2 sin(
58
t − t
4π i
)|N(x(t ))dt =
F (x(t)).
2
L
(D.7)
Now we expand N(x) in a truncated Fourier series
N(x(t )) ≈
n
m=−n
Nm eimt ,
(D.8)
and substitute this expression into Eq. (D.7). The second integral can then be evaluated
exactly:
2π
1
t − t imt
βNm eimt
β
)|e dt =
.
(D.9)
ln |2e− 2 sin(
− Nm
π
2
max{1, |m|}
0
Therefore, we can write Eq. (D.7) as
n
m=−n
Nm
2π
0
B(t, t )eimt dt +
βeimt
4π i
=
F (x(t)).
max{1, |m|}
L
(D.10)
We now approximate the remaining integral in a manner which avoids redundant computation of the complicated function B(t, t ). We divide the interval [0, 2π] into 2n + 1 equal
2π
(j + 12 ), j = 0, ..., 2n, is the midpoint of the jth subinterval.
intervals, so that tj = 2n+1
Use the integral approximation
2π
0
f (t )dt ≈
2n
2π f (tj )
2n + 1 j=0
(D.11)
in Eq. (D.10) and collocate the resulting equation at the tj points, yielding
n
m=−n
Nm
2n
βeimtj
4π i
2π F (x(tj )),
B(tj , tk )eimtk +
=
2n + 1 k=0
max{1, |m|}
L
j = 0, ..., 2n. (D.12)
This linear system is used to determine the expansion coefficients of N(x(t)). Convergence
of this method is guaranteed [1]. The scattering amplitudes are computed using the
approximation:
2βm Am =
2n
1 e−ik1 [αm x(tj )+βm s(x(tj ))] N(x(tj )).
2n + 1 j=0
59
(D.13)
Acknowledgements and Disclaimer
Effort sponsored by the Air Force Office of Scientific Research, Air Force Materials Command, USAF, under the Multi-University Research Initiative (MURI) program Grant #
F49620-96-1-0039.
The US Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The views and conclusions
contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied of the AFOSR
or the US Government.
Erdmann’s research was supported in part by an Undergraduate Research Grant from
the Colorado Advanced Software Institute (CASI) and a Grant-in-Aid of Research from
Sigma Xi, The Scientific Research Society.
Discussions with Dr. Gary Brown and Capt. Jeff Boleng were very helpful.
60
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