On the Use of High-Frequency Asymptotic Concepts for the Development of Efficient Adaptive

Numerical Integration Algorithms Giorgio Carluccio 1 and Matteo Albani 2

Department of Information Engineering, University of Siena Via Roma 56, 53100 Siena, Italy

1 giorgio.carluccio@dii.unisi.it 2 matteo.albani@ dii.unisi.it

Abstract—The possibility of reducing the sampling point

density in the numerical evaluation of radiation integrals is discussed by resorting to asymptotic high-frequency technique concepts. It is shown that the numerical evaluation of the radiation integrals becomes computationally more efficient by introducing an adaptive sampling. By this approach the number of sampling points results drastically smaller than the standard Nyquist sampling rate.

I. INTRODUCTION The field scattered by a scattering body or by an aperture in

the free space (or in an unbounded homogenous medium) can be described in terms of a double integral of the form

( ) ( )−= ∫∫ jkf

SI G e dSpp , (1)

in which ( )G p is a slowly varying function, and ( )f p is a phase function, both of which depend on the 2D vector

( ),≡ u vp that parameterizes the integration variables on the domain S , and k is the wavenumber of the medium. It is also well known that, in the asymptotic high-frequency (HF) regime, when k is large, the dominant contributions to the integral I come from the neighbourhood of some “critical” points located in the interior of S or on its boundary ∂S ; i.e., stationary phase points in the interior of S at which the gradient of the phase function vanishes, points of the boundary on which the tangential derivative of the phase function vanishes (partial stationary points), and the corner points of the integration domain, respectively, [1], [2]. By applying these concepts to the PO radiation integral, it was shown in [3], [4] that the use of the standard regular Nyquist sampling rate leads to an over-sampling density of points on the surface of integration. This results in redundant and non-efficient numerical integration algorithms. This aspect becomes more and more crucial when the dimensions of the scattering object are electrically large; i.e., large in terms of wavelength. This situation is encountered frequently in various electromagnetic application, such as the prediction of the radar cross section (RCS) of complex targets, or for the computation of radiation characteristics of antennas in their operating environments; i.e., on board of aircraft, ships, satellites, etc., [5], [6]. For these reasons in [3] the authors

proposed an adaptive sampling rule based on the spatial local variation of the phase function ( )f p .

In this paper we present two adaptive sampling rules, which permit to significantly reduce the sampling point number. The proposed formulations are compared to integration algorithm based on standard sampling Nyquist rate.

II. FORMULATION Let us consider, without loss of generality, the PO radiation

integral that describes the scattering phenomena of a general surface illuminated by an arbitrary source. The same treatment can be analogously applied either to any kind of spherical source or to a generic incident field whose wavefront is known analytically. The numerical integration of the PO field is carried out after splitting the original integration domain into M triangular surfaces MS , which are the most useful shapes when dealing with mesh of arbitrary shapes. Hence, the PO field is computed by adding the contributions from each of the M triangles; i.e.,

( ) ( )

1

m

m

Mjkf

m mSm

I G e dS−

=

=∑∫∫ pp . (2)

We propose two adaptive meshing techniques: the first is based on linear amplitude and phase interpolation of the integrand over each triangular sub-domain; while the second is based on a constant amplitude and a quadratic phase expansion of the integrand over each triangular sub-domain. By this assumptions it is possible to analytically evaluate in a closed-form expression the integral over the triangular sub-domains.

A. Linear Amplitude – Linear Phase Solution We subdivide the integration surface into triangles small

enough to approximate the integration function as linear both in amplitude and phase. Namely, we approximate the phase as linear,

( ) ( ) ( )0 0m m m m mf f≈ + ⋅ −p p u p p . (3)

where ( )0m mf= ∇u p , with 0mp barycenter of the m triangle. We multiply the integrand in the thm term of (2) by

2010 URSI International Symposium on Electromagnetic Theory

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( ) ( )0 0m m m mjk fe ⎡ ⎤+ ⋅ −⎣ ⎦p u p p . By this operation, the new integrand is a slowly varying function. The residual slow variation is interpolated with a linear function, thus obtaining,

( ) ( ) ( ) ( ) ( ) ( ), ,3

, ,1

i m m m i mm jkf jkjkfm i m i m m

iG e G e b e− − ⋅ −−

=

≈∑ v u p vpp v p ,

(4)

where ( ),i m mb p denotes a linear basis function which equals 1 at the ith vertex ,i mv and vanishes on the opposite side. The integral I can be finally expressed as

( ) ( ) ( ) ( )

( ) ( )

,.

.

3

, ,1 1

3

, ,1 1

.

m m i mi m

m

i m

Mjkjkf

i m i m m mSm iM

jkfi m i m

m i

I G e b e dS

G e B

− ⋅ −−

= =

−

= =

≈

=

∑∑ ∫∫

∑∑

u p vv

v

v p

v(5)

The residual integral ,i mB is calculated analytically as a Fourier transform, obtaining the final closed form expression

( ) ( )( ) ( )

( ) ( )( )

( ) ( )( )

2, ,

1, ,

, 22, , 1, ,

1, , 2, ,

2 231, , 2, ,

232, 1, 2, ,

3

1

m i m i m

m i m i m

i m

m i m i m m i m i m

m i m i m i m i m

m i m i m m i m i m

jk

m i m i m m i m i m

jk

m

Bk

jk

e

jk

e

jk

+

+

+ +

+ +

+ +

− ⋅ −

+ + +

− ⋅ −

= −⎡ ⎤ ⎡ ⎤⋅ − ⋅ −⎣ ⎦ ⎣ ⎦

⎡ ⎤⋅ − + −⎣ ⎦+⎡ ⎤ ⎡ ⎤⋅ − ⋅ −⎣ ⎦ ⎣ ⎦

+⎡ ⎤⋅ − ⋅ −⎣ ⎦

+⋅

u v v

u v v

u v v u v v

u v v v v

u v v u v v

u v v u v v

u v( ) ( ) 2

1, 2, 1, ,

,i m i m m i m i m+ + +

⎡ ⎤− ⋅ −⎣ ⎦v u v v

(6)

where the sums 1, 2i i+ + are intended “mod 3”; i. e.:, the index 4 corresponds to 1 and the index 5 corresponds to 2. The final expression (6) is a regular function at any incidence/observation aspects except for removable singularities; indeed when each one of the denominators in (6) vanishes, ,i mB admits a limit which has an analytic closed form expression, here omitted for the sake of brevity. Such expressions will shown during the presentation. An analogous result is present in the literature [7]. The meshing rule adopted here is based on the accuracy of the linear approximation (4). If (4) is a good approximation, the mth triangle contribution is calculated by (6), otherwise the mth triangle is split into two sub-triangle and the same procedure is iterated. It is important to note that the integral ,i mB present a linear phase. From an asymptotic point of view, this means that the main contributions of the integral come from the vertexes of the integration domain. Thus, it is very important to match the integrand of I at these critical points in order to correctly reconstruct the integration process

performed numerically. This correct match is realized by the introduction of the sampling function ,i mb .

B. Constant Amplitude – Quadratic Phase Solution The second proposed approach is based on a constant

amplitude approximation and on a quadratic phase approximation. For each triangle mS , the procedure consists in computing the phase error between the actual phase at the

0mp barycenter of the triangle and the approximated one, at the triangle vertexes. This approximation is performed by using the Taylor expansion of the phase function ( )f p in the neighbourhood of 0mp by evaluating analytically the gradient

mu and the Hessian matrix ( )0mfΗ p of the phase function at

0mp . If this phase approximation is not valid, the triangle is split and the procedures is iterated. By using this approach the integral I can be expressed as

( ) ( ) ( )0 01 1

m

m

M Mjkf

m m m mSm m

I G e dS G I−

= =

≈ =∑ ∑∫∫ pp p , (7)

where

( ) ( ) ( )

( ) ( ) ( )0 0

0 0 012

m m m m m

m m m m mf

f f= + ⋅ −

+ − ⋅ ⋅ −

p p u p p

p p Η p p p. (8)

In [8] it was shown that integrals of the form mI can be analytically expressed in terms of UTD transition functions associated to each critical points of the phase function; namely, stationary phase point in the interior of the mth triangle, partial stationary points on the triangle boundary, and vertex points, thus obtaining

3 3

, , ,1 1

s s e e vm m m i m i m i m

i iI I U I U I

= =

= + +∑ ∑ , (9)

where smI is the contribution associated to the stationary phase

point; smU is a step function that is equal to 1 if the stationary

phase point belongs to the mth triangle, otherwise its value is 0; ,

ei mI is the contribution of the partial stationary phase point

on the i side of m triangle; ,ei mU is a step function that is equal

to 1 if the partial stationary phase point belongs to the boundary of the mth triangle and 0 otherwise; ,

vi mI are the

contributions associated to the vertex critical points. We omit the analytical expression of mI for the sake of brevity. These expressions will be deeply analyzed during the presentation. The reader can find the details of expression (9) in [8].

An important aspect of using (9) in (7) is that the value of the integrand at the critical point on each triangle is not the actual value of the integrand in (1). Indeed it was assumed that the amplitude on each triangle is approximated by the amplitude value at the barycenter. This means that, although the quadratic phase approximation permits to consider very

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large triangles on the integration domain, it is necessary to introduce an additive meshing criterion, in order to check the accuracy of the amplitude representation of the integrand. If this criterion is not accounted for, the use of expression (7) can lead to non accurate results. This is another aspect in which the interpretation of the numerical integration algorithm in terms of asymptotic high-frequency techniques has a clear physical interpretation.

It is worth nothing that expressions (6) and (9) can be considered extensions of Gordon’s formula [9] and they reduce to it as a special case for constant amplitude and linear phase.

III. NUMERICAL RESULTS In order to show a practical application of the presented

formulations, we use them to numerically evaluate the PO radiation integral, that describes the scattering of a perfectly conducting Non-Uniform Rational B-Spline (NURBS) surface [10]. We perform the numerical integration in the uv NURBS parametric domain. The considered surface can be enclosed in a box, whose dimensions are about 6.67 6.67 3λ λ λ× × (Fig. 1). We consider an electric dipole located at ( )0,0,12 λ′ =r with

a unit momentum oriented along ( )1ˆ 0,1,12

=q . The

coordinates of the observation point scan a 15λ radius circle, with center at the origin of the reference system and an azimuth angle 0ϕ = ° , while the elevation angle ϑ ranges from 180− ° to 180° . Fig. 2 compares the magnitude of the PO scattered magnetic field computed by using three numerical algorithms. The black continuous line refers to a standard brute force numerical integration algorithm, in which the maximum side length of the meshing triangles is set equal to / 20λ in order to obtain an accurate result. In this case, the number of triangles in which the surface is divided is 60578 at any observation aspects. The blue dashed line refers to the solution presented in Section II-A. By using the proposed adaptive integration technique the mean number of triangles, that are necessary to obtain the same accuracy of the brute force method, is 5189. The maximum number of triangles encountered during the ϑ scan is 8310. Finally, the red dash-dotted line refers to the solution presented in Section II-B. In this case the mean number of triangles is 66, while the maximum number is 104. In the regions close to

150 , 110 , 60ϑ = − ° − ° − ° and, symmetrically, the regions close to 60 ,110 ,150ϑ = ° ° ° , the pattern obtained by the solution presented in Section II-B slightly differs from those obtained by the other two solutions. It can by improved by increasing the number of the meshing triangles, that anyway will be smaller than those obtained by using standard techniques or the solution proposed in Section II-A.

IV. CONCLUSION The use of asymptotic high-frequency concepts was

fruitfully adopted to derive novel quadrature rules for the efficient numerical evaluation of radiation integrals.

Fig. 1. Geometry relevant to the example of the scattering of a smooth convex NURBS surface illuminated by an electric dipole.

Fig. 2. Magnitude of the scattered magnetic field obtained by using three numerical algorithms. The black continuous line (Standard Tech.) refers to a standard brute force numerical integration algorithm. The blue dashed line (Solution A) refers to the solution presented in Section II-A, while the red dash-dotted line (Solution B) refers to the solution presented in Section II-B.

Two different schemes where proposed demonstrating the capability of drastically reducing the sampling rate with respect to the standard Nyquist criterion. Authors are now working to a further step, where the integration rule may consider linear amplitude and quadratic phase function to combine the benefits of the two proposed strategies.

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