FDTD METHOD UPDATE
To guarantee numerical stability for the common FDTD approach, the upper limit of the timestep Δt must be bounded by a criterion which restricts an update cycle’s EM field propagation from cell to cell from being faster than allowed by the phase velocity vp = 1/√εμ within any material (ε, μ). In the time domain, the spatial and temporal updating scheme of the associated Finite-Integration Technique (FIT) is similar to the FDTD method, including stability and dispersion properties. The algorithm can be expressed in a fully explicit form, i.e., equation 6 is written asĪ detailed description of the entire FDTD updating equations for electric and magnetic field components is given in Allen Taflove’s standard FDTD book. The x component of the electric field within equation 3 is formulated for FDTD by the application of central differences as follows: O and O indicate the spatial and temporal second order error terms. The indices i, j, k indicate integer values specifying a certain node in the spatial FDTD grid and u n any temporal dependent function at the time nΔt, i.e., the electric or magnetic field. Achieving 2nd order accuracy using a 2nd order finite differences approximation for the differential equations 2 and 3 leads toįor time and space as shown for the x direction. The original FDTD formulation proposed by Yee was intended for homogeneous, isotropic and lossless media on a uniform grid by application of a cartesian coordinate system. The computation of the magnetic components in FDTD is performed at t = (n + l + 1/2)Δt, (l = 0, 1, 2. ) with respect to a globally defined timestep Δt. Thereby the electric components are calculated at incrementing time t = (n+l)Δt, (l = 0, 1, 2. The discretization in time for FDTD is performed in a leap frog manner by application of a temporally shifted updating for E- and H-field components as shown in Fig. Each three E- and H-field components are assigned to a node i, j, k within the three dimensional (3-D) FDTD grid. The fields are located in a way in which each E component is surrounded by four H components and vice versa, which leads to a spatially coupled system of field circulations corresponding to the law of Faraday and Ampere. 2 depicts the position of the electric (yellow) and magnetic (green) field components for each primary and secondary Yee cell, respectively, allocated within the staggered cartesian grid. The FDTD algorithm proposed by Yee is based on a description of the temporarily coupled system as described in equations 2 and 3 on the basis of a finite central difference approximation. The exponentially increasing availability of computational power has made FDTD the most popular numerical method for a broad range of applications and has resulted in the release of several public and commercial FDTD-based simulation platforms.īy assuming a spatial environment without any electric or magnetic sources, the relationsĮnable a definition of the time dependent Maxwell’s curl equations in differential form as follows:
1, approximately 4500 scientific papers related to FDTD have been published, mainly from the late 90s to the present (1999-2001: counts not complete). Within the last two decades, FDTD gained rapidly increasing interest, mainly in electromagnetics (EM), for the simulation of complex and largely inhomogeneous structures due to its straightforward and explicit approach. The Finite-Difference Time-Domain (FDTD) method is based on a spatial and temporal discretization of Maxwell’s equations, commonly within a rectilinear cartesian grid originally proposed by Yee in 1966. The Finite-Difference Time-Domain (FDTD) Technique Introduction to FDTD
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