Plane Wave Basis in Integral Equation

for 3D Scattering

Emmanuel Perrey-Debain, Jon Trevelyan, and Peter Bettess

School of Engineering, University of Durham, Science Laboratories,South Road, Durham, DH1 3LE, Great Britain

Abstract. The classical boundary element formulation for the Helmholtz equation isrehearsed, and its limitations with respect to the number of variables needed to modela wavelength are explained. A new type of interpolation for the potential is describedin which the usual boundary element shape functions are modified by the inclusion of aset of plane waves, propagating in a range of directions. For a given number of degreesof freedom, the frequency for which accurate results can be obtained, using the newtechnique, can be up to ten or fifteen times higher than that of the classical method.

1 Introduction

The Boundary Element Method is a powerful technique for scattering problems.Its most important feature is that it only requires discretrization of the scatteringsurface. However, if conventional boundary finite element spaces are used, thewell-known ‘10 degrees of freedom per wavelength’ rule of thumb is still requiredto get reasonable results, which is a serious handicap for high frequency calcu-lation. Two distinct approachs have recently emerged to address this issue. Thefirst concerns techniques designed to enable rapid solution of very large systemsby using iterative algorithms [1–3]. The second attempts to reduce the complex-ity by considering more elaborate bases to approximate the unknown field. Inthis regard, a number of developments in the Finite Element Method (FEM )community under the generic grouping of Partition of Unity Finite Element

Methods (PUFEM ) [4] showed an outstanding improvement of the approxima-tion properties of the standard FEM when applied to the Helmholtz equation[5,6]. Essentially, instead of using the conventional finite element approximationwithin each element, a set of plane waves travelling in multiple directions is alsoincluded. This procedure permits inclusion of a priori information about the lo-cal behaviour of the solution and the number of variables can be greatly reducedin many cases. In previous years, the use of the plane wave basis as introducedin the PUFEM [4] has been theoretically investigated by de La Bourdonnaye [7]under the title of Microlocal Discretization(MD) for solving scattering problemswith integral equations. The method was developed further by Perrey-Debainet al. [10,11] for bidimensional problems. In practical terms, the results of theirwork show that the plane wave basis enables the supported frequency range to beextended by a factor of 3 to 4 over conventional boundary elements even for non-convex complex geometries. A particular use of the MD where only the incidentplane wave is included in the basis can be found in [8,9]. However, this method

2 Emmanuel Perrey-Debain et al.

only works for convex obstacles and for sufficiently high frequency (which couldbe well above the medium frequency range of interest in some cases). In thisshort paper, we apply the plane wave basis for 3D scattering problems.

2 Mathematical Formulation

Consider a perfectly rigid three dimensional obstacle Ω of regular surface ∂Ωimpinged by an incident time-harmonic wave field ΦI . By using the direct for-mulation via Green’s second identity, the problem can be formulated into aboundary integral equation on the boundary ∂Ω as follows (e−iωt convention):

Φ(x) + 2

∫

∂Ω

∂G(x, y)

∂n(y)Φ(y)dΓy = 2ΦI(x) (1)

where n is the normal unit vector directed into the obstacle, G is the free-spaceGreen’s function G(x, y) = eiκr/(4πr) where r = |x − y| and κ denotes thewave number. The unknown Φ is the physical variable of interest. The scatteringsurface ∂Ω is described with a number K of non-overlapping patches ∂Ωk, k =1, . . . , K such that ∂Ωk is the image of the coordinate set

T = η1 ≥ 0 , η2 ≥ 0 , η1 + η2 ≤ 1 (2)

via a smooth invertible parametrization x = T k(η1, η2) for (η1, η2) ∈ T . Us-ing a piecewise linear boundary finite element space, the quantity Φ on ∂Ωk isapproximated as

Φ =3

∑

p=1

Np(η1, η2)Φp (3)

where functions Np stands for the (P1) Lagrangian polynomial on T and Φp arethe nodal values corresponding to the three vertices of T . The linear approxima-tion (3) leads to a 10 degrees of freedom per wavelength requirement to obtain‘engineering accuracy’ results (say about 1% of error). This requirement can berelaxed if a set of plane waves travelling in multiple directions is also included.Following the PUFEM [4] or MD [7], the new approximation reads as follows

Φ =3

∑

p=1

Np(η1, η2)

Qp∑

q=1

exp(iκξqp · x) Φq

p (4)

where the point x describes the surface ∂Ωk. Directions ξqp are chosen so that

they are regularly distributed on the unit sphere. Coefficients Φqp no longer rep-

resent the nodal values, but are now the amplitudes of the set of Qp plane wavesassociated with the vertex p of ∂Ωk. If it happens that two patches share a com-mon vertex then the associated set of directions and amplitudes are identical.Thus, Φ is piecewise C∞ and globally C0 on ∂Ω.

Plane Wave Basis in Integral Equation for 3D Scattering 3

3 Numerical integration

In recent papers [10,11], the authors found some great success by solving theintegral equation using a direct collocation aproach. The same technique willbe used here. Consider M points xi , i = 1, . . . , M evenly distributed over thescattering surface. Collocating (1) at those points yields the following system

AΦ = (W + K)Φ = b (5)

where the vector Φ contains the plane wave coefficients. The sparse matrix W

can be interpreted as a plane wave interpolation matrix, K is the boundaryelement matrix stemming from the integral of (1) and b is the source vector con-taining the incident field. The non invertibility of the integral operator (1) whenκ is an eigenfrequency of the corresponding interior Dirichlet problem is allevi-ated using Schenck’s method [12]. The computation of the boundary matrix isexpected to be prohibitively long at high frequency since we need to integrate os-cillatory functions over 2D patches of many wavelengths in extent with sufficientaccuracy. Thus, in [8], the authors suggested the use of the stationary phase toaccelerate the calculation. More recently, Darrigrand [9] applied the Fast Multi-

pole Method for the same purpose. In our work, classical quadrature rules havebeen employed. Regular integrals are computed with General Cartesian ProductRules such that the distribution of integration points is homogeneous over T . Todeal with singular integrals, we split the integration domain into m2 trianglesTn of equal size,

T =

m2

⋃

n=1

Tn (6)

The set of triangles Tn located in the close vicinity of the singularity define aplanar polygon over which the integration is performed using Duffy’s coordi-

nates [14]. The regular integration over the other triangles is carried out witha 19-point formula [13]. To avoid recomputing the same geometric informationsindependent of the collocation point, the code has been written in the spirit ofthe Reusable Intrinsic Sample Point algorithm [15]. The rectangular (or possiblysquare) system matrix is finally solved by a QR algorithm.

4 Results

We consider the ellipsoid parametrization:

x =

R1 sin θ cosϕR2 sin θ sinϕ

R3 cos θ

, θ ∈ [0, π] , ϕ ∈ [0, 2π] (7)

8 triangular patches are sufficient to decribe the scatterer. For example, thepatch corresponding to the first octant (θ, ϕ) ∈ [0, π/2]2 is obtained via

θ = (η1 + η2)π

2and ϕ =

(

η1

η1 + η2

)

π

2, (η1, η2) ∈ T (8)

4 Emmanuel Perrey-Debain et al.

Table 1 shows the performance of the method for the particular case of a verticalplane wave impinging upon a unit sphere. In all cases, the number of directionsassociated with each vertex is the same and is therefore referred to by a singlenumber Q. Moreover, the incident plane wave is always included in the set. Nis the total number of variables (here, it is simply given by N = 6Q) and thenumber of collocation points is chosen so that M ∼ 1.5N . ε2 stands for therelative L2(∂Ω)-norm error and τ denotes the number of degrees of freedom perwavelength,

τ = λ

(

N

ar(∂Ω)

)1/2

(9)

where ar(∂Ω) is the area of ∂Ω. The last two rows clearly show the efficiencyof the plane wave basis since only 2.5 degrees of freedom are sufficient to getreasonable results. This behaviour has already been observed in the bidimen-sional case [10]. The major drawback is the CPU time consumed for the matrixevaluation. Indeed, our method requires O(κ6) products instead of O(κ4) in theclassical approach.

Table 1. Scattering by a unit sphere, (calculations performed on a 2 GHz PIV)

Test κ Q N ε2 τ CPU time

# 1 1 98 588 0.0003 % 43 11 min

# 2 10 98 588 0.013 % 4.3 11 min

# 3 30 298 1788 0.9 % 2.5 4 h 40 min

# 4 40 490 2940 1.1 % 2.4 24 h 30 min

Now, we consider the ellipsoid (R1 = 0.2, R2 = 0.5, R3 = 1) illuminated bya spherical source of strength 4π located at (1, 0,−3) for κ = 40. A few testhave been carried out to test the convergence of the method. By taking Q = 218directions (N = 1308), results have been found to be satisfying. This correspondsto τ = 2.9 and the CPU time is less than 2 hours. Figure 1 shows the real partof the total field Φ on the surface of the scatterer.

5 Conclusion

This paper showed the applicability of the plane wave basis in integral equationsfor three dimensional scattering problems. For the particular case of the sphere,2.5 degrees of freedom per wavelength are sufficient to get 1% error resultswhereas conventional finite element basis would require 10 to 15 times morevariables to achieve the same accuracy. At this stage, the major limitation of themethod is the time taken for the calculation of the system matrix. This will besubject to further studies.

Plane Wave Basis in Integral Equation for 3D Scattering 5

References

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(Prentice-Hall, Inc., New Jersey 1994)

Z

X

Y

Real (T)0.9

0.7

0.4

-0.0

-0.4

-0.7

-0.9

Fig. 1. Scattering by an ellipsoid, κ = 40