Efficient adaptive numerical integration algorithms for the evaluation of surface radiation integrals in the high-frequency regime



[1] 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. Using this approach, the number of sampling points is found to be drastically smaller than that resulting from a standard Nyquist sampling rate.

1. Introduction

[2] The field scattered by an object or radiated 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

equation image

where G(p) is a slowly varying function and f(p) is a phase function, both of which depend on the 2-D vector p ≡ (u, v) that parameterizes the integration variables on the domain D and k is the wave number of the medium. Under typical approximations, like physical optics (PO) for scattering surfaces or Kirchhoff approximation for apertures, both G(p) and f(p) are estimated in an analytical closed form so that the scattered field can be explicitly calculated by integrating (1) numerically. On the other hand, it is well known that, in the asymptotic high-frequency regime, when k is large, the dominant contributions to the integral I come from the neighborhood of some “critical” points located in the interior of D or on its boundary ∂D; i.e., stationary phase points in the interior of D 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 [Jones and Kline, 1958; Chako, 1965]. By applying these concepts to the Physical Optics (PO) radiation integral, which is in the form (1), Burkholder and Lee [2005] and Gordon [1994] showed that the use of the standard regular Nyquist sampling rate leads to an oversampling density of points on the surface of integration, thus resulting in a redundant and nonefficient numerical integration. This aspect becomes crucial when the dimensions of the scattering object are electrically large; i.e., large in terms of the wavelength. This situation is frequently encountered in various electromagnetic applications, 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., [Kim and Burnside, 1986; Marhefka and Burnside, 1992].

[3] In the literature, various approaches have been developed in order to improve the efficiency of the numerical calculation of the radiation integral I. Ludwig [1968] proposed to divide the integration domain into rectangular subregions where both a linear amplitude and linear phase approximation of the integrand is considered. The integration in these subregions is obtained in a closed form. Generalizations of this approach, which allow quadratic phase and amplitude variation, were proposed by Crabtree [1991] and Delgado et al. [2007]. For these approaches it is shown that a closed form solution for the evaluation of the radiation integral does not exist. Crabtree [1991] suggested evaluation of this integral by using Fresnel integrals in one parametric direction while integrating the other numerically (using techniques such as Gaussian quadrature) or by using approximate phase expansions which allow a double Fresnel approximation of the integral. Delgado et al. [2007] exploited some polynomial approximations of the error function to describe a quadratic variation of the phase of the integrand. More recently, Vico-Bondia et al. [2010] presented an algorithm for the computation of the PO integral based on a quadratic approximation of the integrand both for the amplitude and the phase over small triangles, in which the integration domain has been divided. The contribution of each triangle is computed by using a path deformation technique. A different approach was proposed by Boag and Letrou [2003, 2005]. This technique is based on a subdomain decomposition of the scatterer and a subsequent aggregation of the scattering patterns associated to the subdomains. The reconstruction of the radiation pattern is realized by using a multilevel hierarchical algorithm for achieving an improvement in efficiency. Finally, Burkholder and Lee [2005] presented an adaptive sampling rule based on the spatial local variation of the phase function f(p), which permits the reduction of the sampling point number.

[4] In this paper we present an adaptive integration algorithm, inspired to high-frequency asymptotic concepts, which allows a significant reduction of the sampling point number. The integration domain is divided into triangles where the integral is evaluated in an efficient way by resorting either to closed form expressions if the integrand is approximated linearly both in amplitude and in phase or to canonical uniform theory of diffraction (UTD) transition functions [Kouyoumjian and Pathak, 1974; Hill, 1990; Carluccio et al., 2010] if the integrand is approximated linearly in amplitude and quadratically in phase. The latter integration approach allows the use of triangles that are large in terms of the wavelength, thus obtaining a significant reduction of the sampling point number without loss of accuracy.

2. Integration Algorithm

[5] Let us consider the radiation integral I that describes the radiation phenomena from a general regular surface S, with S:Dequation image2equation image3, in which D is the polygonal domain where the surface S is defined (Figure 1a). The approach presented here can be applied to the numerical evaluation of integrals with the general form of I, where the integrand is known analytically.

Figure 1.

(a) Polygonal integration domain D. (b) First subdivision in triangles of the integration domain D. (c) Final mesh where the original integration domain D has been divided into M triangles Am.

2.1. Adaptive Mesh

[6] The numerical integration is carried out after splitting the original integration domain into triangular patches, which are the most useful shapes when dealing with mesh of arbitrary shapes. We propose the following adaptive meshing criterion, which is based on the accuracy of the integral evaluation on a single triangle. The original integration domain is first divided into a minimum number of triangles (Figure 1b). On each triangle an integrand approximation is performed which allows the integral evaluation in an efficient way. Then, each triangle is divided into two subtriangles on each of which the integrand is approximated and the integral is evaluated by the same approach used in the original triangle. If the difference between the integral on the entire triangle and the sum of the integration results on the two subtriangles is lower than a fixed threshold, the meshing refinement on this part of the integration domain stops and the integration results are stored; otherwise the triangle subdivision is iterated until the stop criterion is satisfied. Let M be the number of triangles Am obtained at the end of the meshing process (Figure 1c). The radiation integral I is computed by adding the M integrals evaluated on the meshed domain; i.e.,

equation image

As briefly described above, in this work we consider two different kind of approximation for the integrand on each triangle: the first is based on a linear amplitude and phase approximation of the integrand (rule 1); while the second is based on a linear amplitude and a quadratic phase expansion of the integrand (rule 2). The integration rules based on these approximations are presented in sections 2.2 and 2.3. For the sake of simplicity, we will consider a single canonical triangle A, and the subscript m will be omitted.

2.2. Rule 1: Linear Amplitude—Linear Phase

[7] Let us consider the integral

equation image

We approximate the integration function as linear both in amplitude and phase. Namely, we consider the linear approximation of the phase

equation image

where u0 = ∇f(p0), with p0 denoting the barycenter of the triangle. We multiply the integrand F(p) by eequation image. By this “demodulation”, the residual integrand becomes a slowly varying function which can be linearly interpolated, thus obtaining,

equation image

where bi(p) denotes a linear basis function which equals 1 at the ith vertex vi and vanishes on the opposite side (Figure 2). The integral J can be finally expressed as

equation image


equation image

The integral Bi(u0) is calculated analytically as a Fourier transform. By introducing the vectors li = vi+1vi, for i = 1, 2, 3, where the subscript i + 1, is intended “mod 3” (i. e., v4v1), one obtains the final closed form expression

equation image

where SA is the area of the triangle and again the subscripts i + 1, i + 2 are intended “mod 3” (i. e., l4l1, l5l2). The Fourier transform (8) is a regular function at any incidence/observation aspects except for removable singularities; indeed when any of the denominators in (8) vanishes, Bi admits a limit which has an analytic closed form expression. An analogous result was derived by Ludwig [1968] by approximating the integrand as linear both in amplitude and phase on rectangular subdomains.

Figure 2.

Sampling function bi on a generic triangle of the mesh.

[8] Expression (6) can be thought of as a surface quadrature rule with sampling points at triangle vertexes vi and weights Bi(u0) depending on the phase progression rate u0. The use of the vertexes as sampling points is convenient because function samples can be reused among contiguous triangles and while nesting the rule in the adaptive process. Furthermore, since the integral in (7) presents a linear phase, from an asymptotic point of view the main contributions to the integral J come from the vertexes. Thus, for the accurate evaluation of the integral I, it is very important to match the integrand of J at these critical points.

2.3. Rule 2: Linear Amplitude—Quadratic Phase

[9] The second proposed integration rule is based on a linear amplitude approximation and on a quadratic phase approximation. We consider the quadratic Taylor expansion for the phase function

equation image

where h0 is the Hessian matrix of the phase function evaluated at the barycenter p0 of the triangle. Similarly to the previous case, after “demodulating” the integrand F(p) by the above quadratic phase expansion, the remaining slowly varying integrand is linearly interpolated through the basis functions bi which samples the function at the vertexes, thus obtaining

equation image


equation image


equation image

in which ui = u0 + h0 · (vip0). Again formula (10) can be interpreted as a quadrature rule with sampling points at triangle vertexes vi and weights Ci(u0, h0) now depending on both the phase progression rate u0 and the phase curvature h0. Since the integrand in (11) presents the quadratic phase (12), from an asymptotic point of view, the main contributions to each integral Ci are due to three different kind of critical points of different asymptotic order (Figure 3): (1) one stationary phase point ps = p0h0−1 · u0, if present; (2) three partial stationary phase points pse,ℓ = vl[(vps) · h0 · l]/(l · h0 · l), with ℓ = 1, 2, 3, if present; and (3) three vertexes vq, with q = 1, 2, 3. It is worth noting that, as the three ϕi(12) associated to the three respective terms in (10) all differ only by a constant, the three integrals Ci share the same critical points which are therefore independent of i. Consequently, each integral Ci can be exactly decomposed in [Carluccio et al., 2010]

equation image

By defining the quantities

equation image

the stationary phase point contribution of Ci is given by

equation image

where δ = 1 if the eigenvalues of the Hessian matrix h0 are both positive, δ = −1 if they are both negative, and δ = j if they have opposite sign. In (13), Us is the Heaviside step function, that is equal to 1 if psA, or Us = 0 otherwise; in other words, this contribution has to be accounted for only if the stationary phase point ps lies inside the triangle A.

Figure 3.

Critical points associated to the quadratic phase expansion.

[10] The partial stationary phase point contributions in (13) are

equation image

where ηi,ℓ = γi, −γiβi, βi, for ℓ = i, i + 1, i + 2 (again intended “mod 3”), respectively. Also, in (16)equation image and equation image denote the unit vectors along and normal (outgoing) to the ℓth side of the triangle, respectively (see Figure 4); use,ℓ = u0 + h0 · (pse,ℓp0) is the gradient of the phase function at pse,ℓ, while equation image is the UTD wedge transition function [Kouyoumjian and Pathak, 1974],

equation image

with arg equation image ∈ (−3/4π, π/4), whose argument

equation image

is the measure of the phase distance between the stationary phase point ps and the partial stationary phase point pse,ℓ on the ℓth edge. In (13), Use is the Heaviside step function, that is equal to 1 if pse,ℓequation imageA, or Use = 0 otherwise; accordingly, this type of contribution has to be considered only if a the partial stationary phase point is present on the ℓth side of the triangle. The vertex contributions Ci,qv are given by

equation image

and contain the UTD vertex transition function [Hill, 1990; Carluccio et al., 2010],

equation image


equation image

is the generalized Fresnel integral [Capolino and Maci, 1995]. The argument of the equation image and equation image transition functions in (19) are defined by

equation image

where the upper (lower) sign applies when ℓ = q (ℓ = q − 1), and q and ℓ indexes identify the qth vertex and the ℓth side of the triangle, respectively. The xq,ℓ parameters are a measure of the phase distance between the partial stationary phase point on the ℓth side, pse,ℓ, and the qth vertex vq. The wq parameter is defined as

equation image

In (22) and (23) the branch of the square root is chosen so that −3/4π < arg(equation image) < π/4.

Figure 4.

Vectors for the geometrical description of a generic triangle of the mesh.

[11] As described by Carluccio et al. [2010], the UTD transition function equation image allows both the uniform description of the coalescence of a single partial stationary phase point and a vertex, and the uniform description of the simultaneous coalescence of the stationary phase point, the vertex, and the two partial stationary phase points, which lie on the two edges joining at the vertex.

3. Numerical Results

[12] In order to show a practical application of the presented algorithms, we adopt them for the numerical evaluation of the PO radiation integral describing the scattering by a perfectly conducting parabolic surface (Figure 5). Note that, since the integrand for the calculation of the PO scattered electric field is a vector function F = Gejkf, the algorithm is extended to the vector case simply by using the same quadrature rules to calculate a vector integral J from vector quantities F, and by applying the convergence test on the norm of the vector integral ∣J∣, when splitting any triangle. In this application, the phase function f is the sum of the distances from the source to the integration point on the surface, and from the integration point to the observation point; i.e., the total ray path length. Therefore, f is known analytically as well as its gradient and Hessian matrix. Namely, let us consider the parabolic surface defined by the equation z = (x2 + y2)/(4fp) − fp, where fp = 15λ is the focal length and the point (x, y) ranges within an octagonal domain D circumscribed by a circle of radius ρ = 15λ. The parabola is illuminated by a unit strength (IΔℓ = 1A · λ) electric Hertzian dipole located at the focus, which lies at the origin of the reference system, and oriented along the x axis (Figure 5). The observation point scans a circle centered at the origin of the reference system, in the plane ϕ = 30°, while ϑ ranges from −180° to 180° in 361 steps, with ϑ and ϕ denoting the inclination and azimuth angles, respectively, in standard spherical coordinates. The field is evaluated either at a radial distance r = 60λ, in the Fresnel region, or at r = 6000λ, in the Fraunhofer region. The meshing refinement is performed by setting an accuracy threshold ɛ = 2.5 × 10−3 on the relative error for the norm of the electric field evaluated by integrating on a triangle and on its relevant two subtriangles. Table 1 resumes the performances of the proposed algorithms. In particular, in Table 1, rule 1 identifies the algorithm based on the quadrature formula (6) with a linear approximation for both phase and amplitude of the integrand, while rule 2 identifies the algorithm based on (10) with a linear approximation for the amplitude and a quadratic approximation for the phase of the integrand. Furthermore, we also consider for comparison a zero order trapezoidal integration rule (rule 0), in which each integral on a triangle is evaluated by using the quadrature formula

equation image

Table 1 gives the time and the mean number μ of triangles necessary for the evaluation of the electric field on the entire observation scan, respectively. Table 1 also gives the p = 5% and p = 95% percentiles of the number of triangles necessary for the field evaluation, respectively. Finally, Table 1 gives the minimum and the maximum number of triangles encountered in the scan. Table 1 also reports the performance of rule 2 with a larger threshold ɛ = 10−2.

Figure 5.

Parabolic surface illuminated by a focal electric Hertzian dipole. The surface is shaded accordingly to the PO-induced current strength.

Table 1. Algorithm Performances
RuleTime [s]μp = 5%p = 95%MinMax
r = 60λ (Fresnel Zone)
0(ɛ = 2.5 × 10−3)7394711553197978560017631805998
1(ɛ = 2.5 × 10−3)11720246614448946614449154
2(ɛ = 2.5 × 10−3)91305226379208401
2(ɛ = 10−2)70774810842115
r = 6000λ (Fraunhofer Zone)
0(ɛ = 2.5 × 10−3)68941931115150705223289720530
1(ɛ = 2.5 × 10−3)106142453762457619324578
2(ɛ = 2.5 × 10−3)811469421059270
2(ɛ = 10−2)6521853864

[13] It is clear that increasing the order of the integration rule from 0 to 2 leads to an improvement of the efficiency of the numerical integration. Indeed, the number of triangles, and in turn the calculation time, necessary for the field evaluation decreases considerably. Figure 6, in which the magnitude of the PO scattered electric field is reported for the r = 60λ scan, shows that the accuracy of the field evaluated by using different algorithms is the same. Indeed, all the lines associated with different approaches are superimposed. The use of a larger threshold (ɛ = 10−2) in connection with rule 2 results in a very similar accuracy for the field evaluation, while with rule 1 and 0 such a larger threshold results in a less accurate field evaluation in low field parts of the observation scan, and are therefore not considered. The field predicted in the r = 6000λ scan is reported in Figure 7, where for the sake of readability ϑ ranges ±10° around the boresight. It is noticeable that rule 2, for certain observation aspects, only requires 8 triangles for the field evaluation. On the contrary, when using rule 0, the final mesh satisfying the convergence criterion comprises triangles whose side length is on the order of λ/15 ∼ λ/20, which is consistent with a typical nonadaptive accurate mesh for a PO integration.

Figure 6.

Magnitude of the scattered electric field along r = 60λ scan obtained by using rule 0 (dash-dotted line), rule 1 (crosses), rule 2 (solid line), and rule 2 with larger threshold (circles).

Figure 7.

Magnitude of the scattered electric field along r = 6000λ scan obtained by using rule 0 (dash-dotted line), rule 1 (crosses), rule 2 (solid line), and rule 2 with larger threshold (circles).

[14] To show how the adaptive integration algorithms mesh the domain, we consider as an example the observation point at ϑ = 45° in the r = 60λ scan. In Figure 8 the contour plots of the amplitude ∣G∣ (gray lines) and the phase function f (black lines) of the integrand are presented; while Figure 9 shows both the final meshes obtained by using the integration algorithm based on rule 1 (gray mesh) and rule 2 (black mesh). At this squinted observation aspect the stationary phase point do not belong to the integration domain and the meshes are not uniformly distributed on the integration domain. Accordingly to the amplitude and phase variation of the integrand there are regions in which the meshes are denser. However, rule 2 requires a much coarser meshing with many triangles larger than the wavelength; in particular, the larger triangle in the upper right side of the domain has two sides as large as 7.5λ.

Figure 8.

Contour plot of the magnitude of the integrand ∣G∣ (gray lines) and of the phase function f (black lines) in the integration domain for r = 60λ and ϑ = 45°.

Figure 9.

Integration domain meshed by using rule 1 (gray) and rule 2 (black) for r = 60λ and ϑ = 45° with accuracy threshold ɛ = 2.5 × 10−3.

[15] Finally, we studied the behavior of the presented algorithms when varying the dimension of the parabolic scattering surface. We evaluated the scattered electric field on the r = 60λ scan as before, while changing the radius ρ of the circle circumscribing to the parabola definition domain, from ρ = 5λ to ρ = 15λ, with steps of λ. Figure 10 shows in a logarithmic scale the time necessary for the field evaluation by using the various rules: rule 0 (dash-dotted line), rule 1 (dashed line), and rule 2 (solid line) with accuracy threshold ɛ = 2.5 × 10−3. This picture highlights the effectiveness of high-frequency inspired algorithms with respect to the standard trapezoidal rule (rule 0). For the sake of completeness, Figure 11 shows in a logarithmic scale the p = 5% percentile (dash-dotted lines), the p = 95% percentile (dashed lines), and the mean number μ of triangles (solid lines) necessary for the field evaluation by using rule 0 (rhombic markers), rule 1 (square markers), and rule 2 (circle markers) with accuracy threshold ɛ = 2.5 × 10−3. It is apparent that, except for few observation points close to the parabola axis, for which the phase variation is very slow corresponding to the p = 5% percentile, rule 0 requires a huge number of triangles. By adopting rule 1 the mean number of triangles decreases more than one order of magnitude and becomes similar to that required with rule 0 for slow phase variation. Rule 2 allows a further reduction in the number of triangles larger than another order of magnitude. However, by comparing Figure 10 to Figure 11, it becomes apparent that the great reduction in the number of triangles which is allowed by rule 2 with respect to rule 1, does not result in a time reduction of the same order. This is due to the fact that the numerical effort required for applying (10) to each triangle is considerably larger than that required by (6), which does not involve the calculation of UTD transition functions and of the Hessian matrix of the phase function. Authors hope to increase in the future the time saving associated to the use of rule 2 by developing better optimized routines for the calculation of the special functions involved.

Figure 10.

Time (in logarithmic scale) for the field evaluation on the entire r = 60λ scan by using rule 0 (dash-dotted line), rule 1 (dashed line), and rule 2 (solid line) with accuracy threshold ɛ = 2.5 × 10−3.

Figure 11.

The p = 5% percentile (dash-dotted lines), p = 95% percentile (dashed lines), and mean number μ of triangles (solid lines) resulting in the final meshes for the field evaluation on the entire r = 60λ scan by using rule 0 (rhombuses), rule 1 (squares), and rule 2 (circles) with accuracy threshold ɛ = 2.5 × 10−3.

4. Conclusions

[16] The use of asymptotic high-frequency concepts was fruitfully adopted to derive novel quadrature rules for the efficient numerical evaluation of radiation integrals. Two different adaptive algorithms were proposed and discussed demonstrating the capability of drastically reducing the sampling rate with respect to the standard Nyquist criterion, and the time necessary for the field evaluation with respect to standard integration algorithms. The effectiveness of the proposed rule 1 (linear amplitude − linear phase) and rule 2 (linear amplitude − quadratic phase) based algorithms depends on the feature of integrand that is approximated on the triangles of the mesh. A representative example is shown illustrating how rule 2 is capable to drastically reduce the number of required triangles in the final mesh. However, the resulting time saving for the integral calculation is not of the same order because of the numerical effort associated with the quadrature rule to be applied on each triangle, which involves the calculation of special functions and of the phase function Hessian matrix. Authors plan to develop in the next future faster routines for such special functions calculation. This should result in a further improved time reduction.