Stable non-orthogonal FDTD method

A novel approach for implementing a stable nonorthogonal finite-difference time-domain algorithm is proposed. Instead of representing a vector field with its contravariant and covariant components, this approach only makes use of the covariant components to annihilate any unstable behaviour. The new algorithm is validated by numerical experiments.     

Z-transform theory and the FDTD method

In implementing the finite-difference time-domain (FDTD) method on materials which are dispersive or nonlinear, the relationship between the flux density and the electric field can be the most complicated part of the problem. Because the FDTD method is a sampled time-domain method, this relationship can be can be looked upon as a digital filtering problem. The Z transform is typically used in digital filtering and signal processing problems. The paper illustrates the use of the Z transform in implementing the FDTD method where complicated dispersive or nonlinear materials are involved     

Optimization of nonlinear dispersive APML ABC for the FDTD analysis of optical solitons

We have investigated the parameter optimization for the nonlinear dispersive anisotropic perfectly matched layer (A-PML) absorbing boundary conditions (ABCs) for the two- and the three-dimensional (2D and 3D) finite-difference time-domain (FDTD) analyses of optical soliton propagation. The proposed PML is applied to the FDTD method of the standard and the high-spatial-order schemes. We first searched for the optimum values of the loss factor, permittivity, and the order of polynomial grading for particular numbers of APML layers in a two-dimensional (2-D) setting with Kerr and the Raman nonlinearity and Lorentz dispersion, and then we applied the optimized APML to a full three-dimensional (3-D) analysis of nonlinear optical pulse propagation in a glass substrate. An optical pulse of spatial and temporal soliton profile has been launched with sufficient intensity of electric field to yield a soliton pulse, and a reflection of -60dB has been typically obtained both for the 2-D and the 3-D cases.     

Error between unconditionally stable FDTD methods and conventional FDTD method

Both the ADI-FDTD method and the CN-FDTD method can be viewed as approximate factorisation of the conventional FDTD scheme, so the error between these methods and the conventional FDTD method is obtained. It is shown that the error is associated with the time step size, the permittivity and the permeability of the medium, and the spatial variation rate of field. The space discretisation has no relation with the error, which is demonstrated by numerical examples     

A multiwire formalism for the FDTD method

The thin-wire formalism is a widely used subcell model that allows the finite-difference time-domain (FDTD) method to take account of wires thinner than the cell size. In this paper, the original formalism is generalized to a multiwire formalism that allows the FDTD method to take account of bundles composed of arbitrarily close wires     

二维无条件稳定时域有限差分方法

基于半隐式的Crank-Nicolson差分格式给出了一种无条件稳定时城有限差分方法。和传统FDTD法中采用的显式差分格式不同,对Maxwell方程组采用半隐式差分格式,在时间和空间上仍然是二阶精确的。但时间步长不再受稳定性条件的限制,只需考虑数值色散误差对其取值的制约。利用分裂场完全匹配层吸收边界截断计算空间,为保证PML空间的无条件稳定性,其方程也采用半隐式差分格式。数值结果表明相同条件下US-FDTD方法与传统FDTD方法的计算精度是相同的,而且在增大时间步长时US-FDTD方法是稳定的和收敛的。可以预见US-FDTD方法在模拟具有电小结构问题时具有实际意义。     

Compact higher-order split-step FDTD method

A compact higher-order split-step FDTD (SS-FDTD) method is developed, that achieves higher computation efficiency than the previous non-compact higher-order SS-FDTD by reducing the bandwidth of the system involved. From the stability and dispersion analysis, it can be concluded that this compact higher-order SS-FDTD is also unconditionally stable and has the same level of accuracy as the non-compact method.     

The complementary operators method in FDTD simulations

The complementary operators method (COM) was developed for the purpose of canceling the reflections that arise from the artificial boundaries that terminate the computational domain in FDTD simulations. This article presents a discussion and an evaluation of COM. The theoretical background is developed in the analytical and discretized domains. Five numerical experiments are presented to show the effectiveness and high efficiency of COM. Applications to antenna radiation and absorbing boundary conditions are discussed     

The implementation of time-domain diakoptics in the FDTD method

The time-domain diakoptics is implemented in the finite-difference time-domain (FDTD) method with two types of connecting interfaces: i) directional interface (TLM-type), and ii) total-field interface (FDTD-type). The FDTD-type interface provides a more efficient way to realize time-domain diakoptics than TLM, especially for device optimization problems. To emulate the TLM-type interface in FDTD, two novel algorithms are developed in this paper. One is to implement an ultra-wideband absorbing boundary on the excitation plane during excitation. The other is to separate directional waves without the knowledge of incident waves. For a large circuit with cascaded modules, sequential and parallel algorithms are provided to connect them. With these connecting algorithms, time-domain diakoptics is one candidate method to realize modular and parallel computation in FDTD simulations. The validity of these algorithms is confirmed by comparison with simulated results from Microwave SPICE     

Off-grid perfect boundary conditions for the FDTD method

We implement off-grid boundary conditions within Yee's finite-difference time-domain (FDTD) method without disturbing the existing uniform mesh or changing the standard FDTD code. Both perfect electric conductor and perfect magnetic conductor walls are considered. Examples of straight, slanted, and curved walls are shown, the latter two being represented by an enhanced staircase approximation. We show that: 1) offsets comparable to the spatial step size lead to instabilities and 2) this issue is easily resolved by stepping into the neighboring cell. The method enhances the flexibility of the FDTD method with respect to complex geometrical domains without reducing the spatial step.     

Higher-order alternative direction implicit FDTD method

A second-order in time and fourth-order in space alternative direction implicit (ADI) finite difference time domain (FD-TD) method has been developed. The dispersion relation is derived and compared with the ADI-FDTD method. Numerical results demonstrated that the higher-order ADI-FDTD has a better accuracy compared to the ADI-FDTD method.     

Linear lumped loads in the FDTD method using piecewise linear recursive convolution method

This letter gives a new algorithm to include linear lumped elements into finite difference time domain algorithm. The proposed method can efficiently account for two-terminal networks made of several lumped elements. The piecewise linear recursive convolution (PLRC) technique is used to implement lumped loads into Yee cells. Using this method, it is possible to simulate equivalent circuits of terminations for microstrip structures, integrated circuits, or digital devices. The advantage of the PLRC technique is addressed and simulation results validating this method are presented.     

FDTD ABC for waveguides with a new discrete Green's function andits effective implementation

The authors propose a new discrete Green's function and an effective implementation for the absorbing boundary condition for the FDTD simulation of waveguide problems. The Green's function is given analytically in a simple, closed-form formula. A simple exponential approximation method is used for an efficient ABC implementation     

A higher order FDTD method in Integral formulation

We present a fourth-order (4, 4) finite-difference time-domain (FDTD)-like algorithm based on the integral form of Maxwell's equations. The algorithm, which is called the integro-difference time-domain (IDTD) method, achieves its fourth-order accuracy in space and time by taking into account the spatial and temporal variations of electromagnetic fields within each computational cell. In the algorithm, the electromagnetic fields within each cell are represented by space and time integrals (or integral averages) of the fields, i.e., the electric and magnetic fluxes (D,B) are represented by the surface-integral average, and the electric and magnetic fields (E,H) by the line and time integral average. In order to relate the integral average fields in the staggered update equations, we have obtained constitutive relations for these fields. It is shown that the IDTD update equations combined with the constitutive relations are fourth-order accurate both in space and time. The fourth-order correction terms are represented by the modified coefficients in the update equations; the numerical structure remains the same as the conventional second-order update equations and more importantly does not require the storage of field variables at the previous time steps to obtain the fourth-order accuracy in time. Furthermore, the Courant-Friedrichs-Lewy (CFL) stability criteria of this fourth-order algorithm turns out to be identical to the stability limits of conventional second-order FDTD scheme based on differential formulation.     

Signal-processing techniques to reduce the sinusoidal steady-stateerror in the FDTD method

Techniques to improve the accuracy of the finite-difference time-domain (FDTD) solutions employing sinusoidal excitations are developed. The FDTD computational domain is considered as a sampled system and analyzed with respect to the aliasing error using the Nyquist sampling theorem. After a careful examination of how the high-frequency components in the excitation cause sinusoidal steady-state errors in the FDTD solutions, the use of smoothing windows and digital low-pass filters is suggested to reduce the error. The reduction in the error is demonstrated for various cases     

Sensitivity analysis with the FDTD method on structured grids

We propose an adjoint-variable approach to design-sensitivity analysis with time-domain methods based on structured grids. Unlike conventional adjoint-based methods, it does not require analytical derivatives of the system matrices. It is simple to implement with existing computational algorithms such as the finite-difference time-domain (FDTD) technique. The resulting FDTD algorithm produces the response and its gradient in the design parameter space with two simulations regardless of the number of design parameters. The proposed method is validated by the adjoint-based FDTD analysis of waveguide structures with metallic boundaries.     

A frequency-dependent FDTD method for biological applications

A frequency-dependent finite-difference time-domain (FD)² TD method for calculating the response of pulses in plasma or water has recently been described. This method is an advance over the traditional finite-difference time-domain (FDTD) method in that it allows for the frequency dependence of these two media. The modification of the (FD)²TD method to obtain broadband frequency information in 3D biological applications is discussed. The implementation of this method is described, and its accuracy is verified by comparison with analytic solutions using the Bessel function expansion. The use of this method is illustrated by an example of the 3D simulation of a hyperthermia treatment using two applicators over a frequency range of 40 to 200 MHz     

Cavity losses modeling using lossless FDTD method

The impulse response of a lossless resonant system, usually obtained using the finite-difference time-domain method, permits us to determine the resonant frequencies through the Fourier transform. However, the obtained spectrum has no physical meaning since the losses have not been implemented. Rather than modeling physically the losses, we propose to apply a specific time-domain window to the already simulated signal of the lossless system. This Losses window depends on a user-defined quality factor. The advantage of this postsimulation losses implementation is a capability of parametric study of composite losses. Losses of various physical origins are found for example in the case of reverberation chambers.     

High-order accurate unconditionally-stable implicit multi-stage FDTD method

High-order accurate unconditionally-stable implicit multi-stage finite-difference time-domain (IMS-FDTD) methods are presented, among which the details of an IMS(6,4)-FDTD method are given for demonstration. The analysis of the numerical dispersion relation shows that their performance is much better than those of conventional unconditionally-stable implicit FDTD methods.     

A fictitious domain method for conformal modeling of the perfectelectric conductors in the FDTD method

We present a fictitious domain method to avoid the staircase approximation in the study of perfect electric conductors (PEC) in the finite-difference time-domain (FDTD) method. The idea is to extend the electromagnetic field inside the PEC and to introduce a new unknown, the surface electric current density to ensure the vanishing of the tangential components of the electric field on the boundary of the PEC. This requires the use of two independent meshes: a regular three-dimensional (3-D) cubic lattice for the electromagnetic field and a triangular surface-patching for the surface electric current density. The intersection of these two meshes gives a simple coupling law between the electric field and the surface electric current density. An interesting property of this method is that it provides the surface electric current density at each time step. Furthermore, this method looks like FDTD with a special model for the PEC. Numerical results for several objects are presented