## 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

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.