Exact Evaluation of Time-Domain Physical Optics Integral on Quadratic Triangular Surfaces

On the closed-form evaluation of the PO integral using the Radon transform interpretation for linear triangles

“…Accurate and fast evaluation of the physical optics (PO) integral is very important on the application of PO approximation to the analysis of electrically large scatterers [1][2][3][4][5][6][7]. In the literature, it is possible to find plenty of works that search for an approximate or exact expression to the PO integral both in the time and frequency domains for different geometric formations [1][2][3][4][5][6][7][8][9]. An elegant way to evaluate the closed-form expression of the PO integral for linear triangular patches in time domain (and also Fourier transformed to frequency domain) is presented in [2] using the Radon transform (RT) interpretation.…”

“…In [2], the PO integral over a linear triangular surface is reduced to a line integral over the intersection of the linear triangle and the plane formed by the propagation direction of the incident plane wave and the observation direction, called k r -plane, then it is concluded that the closed-form expression to the line integral should be a triangle function since the domain of the PO integral is a linear triangle. This approach is also extended to NURBS surfaces [8], quadratic triangular patches [9], different source and observation configurations [3,4], even applied to time domain integral equations [10]. However, the closed-form expression presented in [2] can be obtained mathematically following the steps applied in [9] for exact evaluation of the PO integral for quadratic triangles.…”

“…This approach is also extended to NURBS surfaces [8], quadratic triangular patches [9], different source and observation configurations [3,4], even applied to time domain integral equations [10]. However, the closed-form expression presented in [2] can be obtained mathematically following the steps applied in [9] for exact evaluation of the PO integral for quadratic triangles. In addition, for a time domain simulation, it is possible to encounter with a case such that all scatterer can be completely fit into two consecutive time samples, resulting in no intersection.…”

“…The incident field is assumed to be an impulsively excited plane wave and far scattered fields are observed as in [2]. Note that the formulation presented in this letter is based on the exact evaluation scheme in [9] and its simplification to linear triangles, which yields a less complex formulation and closed-form evaluation. (ii) For the USR case, a formula, which has a rectangular shape in time, is proposed.…”

Transient analysis of scattering from impedance surfaces using physical optics approximation

“…The PO approximation can be applied in both frequency [3] and time [4, 5] domains, also a different variety of surface modelling can be used to discretise the scatterer's surface, e.g. planar (linear) triangles [3, 4], quadratic triangles [6–8], NURBS surfaces [9–13] etc. In all cases mentioned above, the research on the PO approximation focuses on the accurate and efficient evaluation of highly oscillatory radiation integral, which is called PO integral [3–17].…”