On the Evaluation of Retarded-Time Potentials for SWG Bases

Abstract

A new approach for obtaining the time samples of the retarded-time scalar and vector potentials due to an impulsively excited Schaubert-Wilton-Glisson (SWG) basis function is presented. The approach is formulated directly in the time domain without any assumptions regarding the temporal behavior of the currents represented by the SWG bases. It is shown that the aforementioned potentials are related to the solid angle formed by the intersection of the tetrahedral supports of the SWG basis and the "hyper-cone" that is centered at the observation point and that has a radius = , where is the speed of light. In particular, the scalar potential is directly proportional to the total solid angle, and the vector potential is a function of the area centroid vector of this solid angle. The validity of the obtained time-domain formulas is demonstrated through comparison of results to those obtained in the frequency domain by using numerical quadrature and transformed into time domain. IndexTerms-Marching-on-in-time (MOT) method, Schaubert-Wilton-Glisson (SWG) basis, time-domain analysis, volume integral equations (VIEs).

A Stable Marching On-In-Time Scheme for Solving the Time-Domain Electric Field Volume Integral Equation on High-Contrast Scatterers

“…The CEE scheme is applied to the complex semidisk S D with radius ω max = 2πf max . The matrix skeletonization required by the CEE scheme is carried out using Ñλ = 1000, Ñs = 1000, and δ = 10 −13 [see (25) and ( 26)], which results in N λ = 13 and N s = 7. Consequently, the coefficients of the CEE scheme are obtained by solving the matrix system in (30) with N λ = 13 and N s = 7.…”

“…Semianalytical techniques listed under group 1) above have been developed for integrals arising from the discretization of the time-domain surface and wire integral equations using piecewise polynomial temporal basis functions. Even though closed-form expressions of retarded-time potentials due to impulsively excited Schaubert-Wilton-Glisson (SWG) spatial basis functions [45] (typically used for discretizing the VIEs) have been derived [25], [26], practical use of these expressions in solving TD-VIEs has not been realized yet.…”

Parallel PWTD-Accelerated Explicit Solution of the Time-Domain Electric Field Volume Integral Equation

“…The implicit MOT scheme require at every time step solution of the linear system [3], which is traditionally constructed upon expanding the flux density with Schaubert-Wilton-Glisson (SWG) spatial basis functions [8] and piecewise polynomial temporal basis functions [9], followed by Galerkin and point testing in space and time, respectively. In addition, modern implicit MOT-based solution of time domain surface and volume integral equations can be made low-and high-frequency stable by using computationally more expensive space-time discretization techniques, such as bandlimited time discretization [7,10], space-time Galerkin testing [11,12], quasi-Helmholtz decomposition [13,14], and highly accurate evaluation of MOT matrix elements [12,[15][16][17][18][19][20]. In contrast, the explicit MOT scheme, usually leverages pulse spatial basis functions and low order temporal basis functions and point testing both in space and time.…”

Marching On-In-Time Solution of the Time Domain Magnetic Field Integral Equation Using a Predictor-Corrector Scheme

“…In (7), is the magnetic field due to impulsively excited RWG basis function and can be evaluated analytically [33]- [37].…”

“…iii) The fact that is a polynomial function allows for the analytical evaluation of the convolution in (7). As a result only the surface integral over testing functions is computed numerically [33]- [37]; this integration is the sole source of numerical error in the MOT matrix entries.…”