Semi-Analytical Solutions of the 3-D Homogeneous Helmholtz Equation by the Method of Connected Local Fields

Abstract

Abstract-We advance the theory of the two-dimensional method of connected local fields (CLF) to the three-dimensional cases. CLF is suitable for obtaining semi-analytical solutions of Helmholtz equation. The fundamental building block (cell) of the 3-D CLF is a cube consisting of a central point and twenty six points on the cube's surface. These surface points form three symmetry groups: six on the planar faces, twelve on the edges and eight on the vertices (corners). The local field within the unit cell is expanded in a truncated spherical FourierBessel series. From this representation we develop a closed-form, 3D local field expansion (LFE) coefficients that relate the central point to its immediate neighbors. We also compute the CLF-based FD-FD numerical solutions of the 3D Green's function in free space. Compared with the analytic solution, we found that even at a low three points per wavelength spatial sampling, the accumulated phase errors of the CLF 3D Green's function after propagating a distance of ten wavelengths are well under ten percent.

Broadband CPW-Fed Circularly Polarized Planar Monopole Antenna With Inverted-L Strip and Asymmetric Ground Plane for Wlan Application

“…In this framework, related semi-analytical techniques from [17][18][19] provide correct initial guesses for the optimization process which are considered to be used and cited. So as listed in Table 1, all the dimensions of the proposed antenna have been optimized according to the results of numerical analysis.…”

Dispersion and Local-Error Analysis of Compact Lfe-27 Formula for Obtaining Sixth-Order Accurate Numerical Solutions of 3d Helmholtz Equation

“…They produce block tri-diagonal matrices which require much less computing resources compared to other non-compact stencils. LFE-27 is a 3D compact FD-like stencil used in the method of connected local fields (CLF) [1]. In this paper, we show that LFE-27 possesses such good numerical quality that it is accurate to the sixth order.…”

“…The work on 3D CLF formulae has been detailed in our prior accompanying paper [1] except for the study of their global plane wave dispersion characteristics and local truncation error, which are the main focuses of this paper. We list below the four closely related 3D CLF stencils derived from using a local SFBS.…”

“…Their 3D stencil was given (see Equation (6) below) with truncated Taylor series with numerical coefficients. Recently, we derived and published a sixth-order accurate, 3D compact Helmholtz stencil [1]. Like our 2D formula, it is given in closed-form expression as weighted products of spherical Bessel functions (see Equation (10) below).…”

“…Dispersion characteristics of the classical 7/27-point, secondorder accurate formulae FD2-7, FD2-27 [24] and six-order accurate FD6-27 formula [8] will be briefly reviewed. Our focus will be on the in-depth investigation of the numerical phase and group velocity dispersion characteristics of the proposed SFBS-based formulae including three second-order accurate LFE-FC-7, LFE-EC-13, LFE-CR-9 equations, and the main sixth-order accurate LFE-27 formula [1,25]. Here FC, EC and CR respectively denote face-centered, edge-centered and corner points on the twenty-seven point CLF basic cube/patch.…”