High Frequency Scattering By Convex Curvilinear Polygons

High Frequency Scattering By Convex Curvilinear Polygons

M. Mokgolele‡,∗, S. Langdon‡, S. N. Chandler-Wilde‡

‡Department of Mathematics,University of Reading, Whitenights, P. O. Box 220, Berkshire RG6 6AX, UK
∗Email: [email protected]

Abstract function of first kind of order zero, n is the normal di-
∂ui(x)
Acoustic and electromagnetic wave scattering prob- rection directed out of Ω, f (x) := ∂n + iηui(x), and
lems can be formulated as the Helmholtz equation with
appropriate boundary conditions. The computational cost η ∈ R\{0} is the coupling parameter. The total field
of standard schemes for these problems grows in direct
proportion to the frequency of the incident wave. Re- throughout D is determined by
cently Chandler-Wilde and Langdon have proposed a
novel Galerkin boundary element method to solve the u(x) = ui(x) − Φ(x, y) ∂u (y)ds(y), x ∈ D.
problem of acoustic scattering by a sound soft convex ∂n
polygon, with computational cost that depends only log- Γ
arithmically on the frequency. They achieved this by in-
corporating into the approximation space the products of In this paper we consider the numerical solution of (1) in
plane wave basis functions with piecewise polynomials the case that Γ is a curvilinear polygon. In §2 we intro-
supported on a graded mesh, with smaller elemets adja- duce our numerical method, in §3 we present some nu-
cent to the corner of the polygon. In this paper we ex- merical results and in §4 we present our conclusion.
tend their scheme to problems of scattering by curvilinear
polygons. 2 Galerkin Boundary Element Method
We begin by parametrising (1) on the boundary of a

curvilinear polygon such as that shown in Fig.1.

1 Introduction
Consider the two-dimensional problem of scattering of

a time-harmonic acoustic incident plane wave

ui(x) = eikx.d, in D := R2\Ω,

where d ∈ R2 is a unit vector representing the direction
of the incident field. The scattered field us := u − ui
(where u denotes the total field) satisfies

∆us + k2us = 0 in D, us = −ui on Γ,

and also the Sommerfeld radiation condition (see

e.g. [3]), where Γ denotes the boundary of the obstacle Figure 1: Scattering by a curvilinear polygon

Ω. Using Green’s theorem and following the usual cou-

pling procedure we obtain a second kind boundary inte- nsd
j=1
gral equation for ∂u We write the boundary as Γ = Γj , where Γj ,
∂n
j = 1, ..., nsd are the nsd sides of the polygon, ordered so

1 ∂u (x) + ∂Φ(x, y) + iηΦ(x, y) ∂u (y) ds(y) that Γj, j = 1, ..., nsh, are in shadow, (so that nj.d > 0
2 ∂n ∂n(x) ∂n
Γ for j = 1, ..., nsh), and Γj, j = nsh + 1, ..., nsd are

illuminated, (so that nj.d < 0 for j = nsh + 1, ..., nsd),

= f (x), x ∈ Γ, (1) with j increasing anticlockwise as shown in Fig 1.

where ∂u ∈ L2(Γ) is the unknown boundary data, Here nj = (nj1, nj2) is the normal derivative to the
∂n
H0(1)(k|x line Γj. For simplicity we assume here that nj.d = 0
Φ(x, y) := i − y|) is the fundamental solu-
4 ∀ j = 1, ..., nsd. If this were not the case, special
tion of the Helmholtz equation in 2D, H0(1) is the Hankel
care would be needed in the “transition zone” around

the shadow boundary nj.d = 0, (see e.g. [4]). We Langdon in [1,2] for the case where Γj is a straight line.
denote the vertices of the polygon by Pj = (pj, qj), For A > λ := 2π/k we define a composite graded mesh
for j = 1, ..., nsd. We set Pn+1 = P1, so that for on [0, A], with a polynomial grading on [0, λ] and a ge-
j = 1, ..., nsd, Γj is the line joining Pj with Pj+1. ometric grading on [λ, A]. For N = 2, 3, ..., the mesh
ΛN,A,qj := {y0, ..., yN+NA} consists of the points yi :=
If we denote γj(s), for j = 1, ..., nsd as the arc length λ(i/N )qj , i = 0, ..., N , where qj := (2ν + 3)/(1 − 2αj)
parameterisation of the curve Γj then x ∈ Γ can be rep- (where αj := 1 − π/(2π − φj)), together with the
points yN+j := λ(A/λ)j/NA, j = 1, ..., NA,, where
resented by NA = ⌈N ∗⌉, is the smallest integer greater than or equal
to N ∗, with N ∗ = − log |A/λ|/(qj log(1−1/N )) chosen
j−1 to ensure a smooth transition between the two parts of the
mesh. We choose our approximate space VΓ,ν to be the
x(s) = Pj + γj s − Lm , union over all sides of polygon of piecewise polynomials
of order ν supported on the mesh Γ+j multiplied by eiks,
m=1 with piecewise polynomials of order ν supported on the
mesh Γj− multiplied by e−iks, where Γ+j := Pj + ΛN,A,qj
for s ∈ (Pj, Pj+1) j = 1, ..., nsd, (2) and Γ−j := Pj+1 − ΛN,A,qj+1 . Then our Galerkin method
approximation ϕN ∈ VΓ,ν is defined by
where Lj is the length of Γj, j = 1, ..., nsd. The interior
angle given by φj ∈ (0, π), j = 1, ..., nsd is the angle (ϕN , ρ) + (KϕN , ρ) = (F, ρ), for all ρ ∈ VΓ,ν. (5)
between the tangents to Γj−1 and Γj at Pj. The angle of
the incident plane wave is given by θ, which is measured 3 Numerical Results

anticlockwise from the downward vertical as shown in

Fig 1.

We rewrite (1) in parametrised form as

L (3)

φ(s) + K(s, t)φ(t)dt = F˜(s),

0

where φ(s) := 1 ∂u (x(s)), nsd
k ∂n
L = Lj,

j=1

K(s, t) := 2 ∂Φ(x(s), x(t)) + iηΦ(x(s), x(t)) |x′ (t)|
∂n(x(s))

and F˜(s) := 2 f (x(s)). We begin by separating the lead-
k
ing order behavior, in the limit as k → ∞. We define

Ψ(s) := 2 ∂ui (x(s)), on illuminated sides,
k ∂n on shadow sides,
0,

then ϕ(s) := φ(s) − Ψ(s) represents the difference be- Figure 2: Scattering by a two sided curvilinear polygon
tween the exact solution of (3) and the solution in the high
frequency limit. Substituting into (3) we get

ϕ(s) + Kϕ(s) = F (s), s ∈ [0, L]. (4) We solve (1) on a two sided curvilinear polygon, which
is parametrised by
L
x(s) = aj + r cos(s/r − bj) (6)
where Kψ(s) := 2 K(s, t)ψ(t)dt, y(s) = r sin(s/r − bj)

0

F (s) := 2 f (s) − Ψ(s) − 2 L
k
K(s, t)Ψ(t)dt. for (x(s), y(s)) ∈ Γj, where
a1 = −a, b1 = cos−1(a/r)
0
a2 = a, b2 = 3 cos−1(a/r) − π
We are going to solve (4) by a Galerkin boundary ele-

ment method. We use the same mesh grading and ap-

proximation space as that used by Chandler-Wilde and

with r = 3 and a = 1.5, as shown in Figure 2. We
take here the incident angle θ = π/2, and ν = 0 in (5).
We fix N = 4 and increase k. The actual and relative L2
errors are shown in Table 1. Here φ is computed using the

same method with a larger number of degree of freedom.
Here MN denotes the total number of degrees of freedom.
It is clear from the results that the relative errors remain
roughly constant as k increases.

Table 1: Relative errors, Scattering by curvilinear poly-
gon

k MN φ − φN φ − φN 2/ φ 2

5 20 2.1058x10−1 3.8363x10−1

10 24 1.2401x10−1 3.3347x10−1

20 24 7.9625x10−2 3.0512x10−1

40 28 5.3473x10−2 3.0133x10−1

4 Conclusions
In this paper we have described a Galerkin scheme

for solving a second kind boundary integral equation on
the boundary of a curvilinear polygon. We have demon-
strated the robustness of the numerical scheme as the
wavenumber increases by numerical experiments. Ideas
regarding the rigorous error analysis of this problem will
appear in [5].

References
[1] S. Arden, S. N. Chandler-Wilde, and S. Langdon, “A

Collocation method for high frequency scattering by
convex polygons”, J. Comp. Appl. Math., 204(2007)
334-343.

[2] S. N. Chandler-Wilde and S. Langdon, “A Galerkin
boundary element method for high frequency scat-
tering by convex polygons”, SIAM J. Numer. Anal.,
45(2007) 610-640.

[3] D. Colton and R. Kress, “Integral Equation Methods
in Scattering Theory”, Wiley, 1983.

[4] V. Dominguez, I. G. Graham, and V. P. Smyshlyaev,
“A hybrid numerical-asymptotic boundary inte-
gral method for high-frequency acoustic scat-
tering”, Numer. Math., DOI:10.1007/s00211-007-
0071-4(2007).

[5] M. Mokgolele, S. Langdon and S. N. Chandler-
Wilde, “High Frequency Scattering By Convex
Curvilinear Polygons”, in preparation.