On the use of spatio-temporal wavelet expansions for transient analysis of wire antennas and scatterers

To analyze a wire antenna excited by a time varying voltage source or a wire scatterer excitated by transient electromagnetic incident wave, the problem is formulated in terms of a time-domain integral equation for the induced current. To solve the integral equation, we reduce it to matrix equation via the method of moments using the known-to-be-stable implicit scheme. However, rather than directly constructing and solving the relatively large matrix equation, we propose an iterative procedure which allows us to gradually obtain a solution of refined accuracy both everywhere and simultaneously at any time instance. To render this procedure rapidly converging, we use a basis of spatio-temporal wavelet functions. This basis facilitates a good approximation of the induced current using far less basis functions than would be needed if other expansions, such as standard-pulse or Fourier basis functions were chosen. The use of this basis further enables the iterative procedure to increase the temporal and spatial resolutions where required without unnecessarily affecting their levels elsewhere.     

Analysis of transient scattering from composite arbitrarily shapedcomplex structures

A time-domain surface integral equation approach based on the electric field formulation is utilized to calculate the transient scattering from both conducting and dielectric bodies consisting of arbitrarily shaped complex structures. The solution method is based on the method of moments (MoM) and involves the modeling of an arbitrarily shaped structure in conjunction with the triangular patch basis functions. An implicit method is described to solve the coupled integral equations derived utilizing the equivalence principle directly in the time domain. The usual late-time instabilities associated with the time-domain integral equations are avoided by using an implicit scheme. Detailed mathematical steps are included along with representative numerical results     

A study of the numerical dispersion relation for the 2-D ADI-FDTD method

This letter presents a numerical dispersion relation for the two-dimensional (2-D) finite-difference time-domain method based on the alternating-direction implicit time-marching scheme (2-D ADI-FDTD). The proposed analytical relation for 2-D ADI-FDTD is compared with those relations in the previous works. Through numerical tests, the dispersion equation of this work was shown as correct one for 2-D ADI-FDTD.     

Transient analysis of EM pulse penetration into a conducting layer using wavelet-based implicit TDIE and iterative IMC technique

The penetration of a wide-band electromagnetic pulse into a conducting layer is formulated in terms of an implicit time-domain integral equation and cast into matrix form via the method of moments. Though such a wide-band problem indeed calls for a time-domain solution, one may argue that the frequency-dependent attenuation of the field penetrating the conductor could suggest a frequency-domain approach that would greatly reduce the number of unknowns. The proposed via media is to use spatio-temporal wavelet functions instead of the standard pulse basis. Owing to their multiresolution property, both in the spatial and temporal domains, these functions can span the field propagating inwardly with substantially fewer terms. The reduction in the number of basis functions used is effected by the impedance matrix compression technique, which automatically omits the basis functions whose coefficients would be insignificant due to the attenuation. Reducing the number of basis functions renders the matrix equation much smaller and the overall solution far more efficient.     

TIME-DOMAIN VOLUME INTEGRAL EQUATION FOR TRANSIENT SCATTERING FROM INHOMOGENEOUS OBJECTS-2D TM CASE

This letter proposes a time-domain volume integral equation based method for analyzing the transient scattering from a 2D inhomogeneous cylinder by involking the volume equivalence principle for the transverse magnetic case. This integral equation is solved by using an MOT scheme. Numerical results obtained using this method agree very well with those obtained using the FDTD method.     

Time-domain volume integral equation for transient scattering from inhomogeneous objects—2D TE case

This letter proposes a time-domain volume integral equation based method for analyzing the transient scattering from a 2D inhomogeneous cylinder by involking the volume equivalence principle for the transverse electric case. This integral equation is solved by using an MOT scheme. Numerical results obtained using this method agree very well with those obtained using the FDTD method.     

Time-domain volume integral equation for transient scattering from inhomogeneous objects—2D TM case

This letter proposes a time-domain volume integral equation based method for analyzing the transient scattering from a 2D inhomogeneous cylinder by involking the volume equivalence principle for the transverse magnetic case. This integral equation is solved by using an MOT scheme. Numerical results obtained using this method agree very well with those obtained using the FDTD method.     

TIME-DOMAIN VOLUME INTEGRAL EQUATION FOR TRANSIENT SCATTERING FROM INHOMOGENEOUS OBJECTS-2D TE CASE

This letter proposes a time-domain volume integral equation based method for analyzing the transient scattering from a 2D inhomogeneous cylinder by involking the volume equivalence principle for the transverse electric case. This integral equation is solved by using an MOT scheme. Numerical results obtained using this method agree very well with those obtained using the FDTD method.     

A hybrid implicit-explicit FDTD scheme for nonlinear opticalwaveguide modeling

A hybrid implicit-explicit finite-difference time-domain (FDTD) method for solving the wave equation in nonlinear optical waveguiding structures is proposed. The new scheme combines the computational simplicity of the explicit scheme in linear medium regions with the superior stability property of the partially implicit scheme in regions of nonlinear materials, thus eliminating potential problems of instability associated with nonlinearity. Simulation results for Kerr-type nonlinear slab waveguides and corrugated waveguides are presented and compared with those obtained using the conventional noniterative FDTD scheme     

A finite-difference time-domain beam-propagation method for TE- and TM-wave analyses

The application of the existing time-domain beam-propagation method (TD-BPM) based on the finite-difference (FD) formula has been limited to the TE-mode analysis. To treat the TM mode as well as the TE mode, an improved TD-BPM is developed using a low-truncation-error FD formula with the aid of the alternating-direction implicit scheme. To improve the accuracy in time, a Pade (2,2) approximant is applied to the time axis. Although the truncation error in time is found to be O(/spl Delta/t/sup 2/), as in the case of the Pade (1,1) approximant, this method allows us to use a large time step. A substantial reduction in CPU time is found when compared to the conventional method in which a broadly banded matrix is solved by the Bi-CGSTAB. The effectiveness in evaluating the TE- and TM-mode waves is shown through the analysis of the power reflectivity from a waveguide facet. This method is also applied to the analysis of a waveguide grating. The accuracy and efficiency of the TD-BPM are assessed in comparison with the finite-difference time-domain method.     

Space-domain finite-difference and time-domain moment method for electromagnetic simulation

A hybrid time-domain numerical method based on finite-difference technique and moment method is proposed. Starting from Maxwell's differential equations, our method uses Yee's finite difference scheme in the space domain, but does not utilize the customary explicit leap-frog time scheme. Instead, in the time domain, the fields are expanded in a series of basis functions and treated by a moment method procedure. By choosing appropriate basis functions and testing functions, the conventional finite-difference time-domain (FDTD) formulation and the order-marching unconditionally stable FDTD scheme can be derived from our method as two special cases. Finally, we use triangle basis functions and Galerkin's testing procedure to get an implicit formulation. To verify the accuracy and efficiency of the new formulation, we compare the results with the FDTD method. Our method improves computational efficiency notably, especially for multiscale problems with fine geometric structures, which is restricted by stability constrain in the FDTD method.     

Time-domain reflective beam propagation method

A time-domain reflective beam propagation method (TD-RBPM) is presented for the analysis of optical pulse propagation and reflection in waveguide structures with multiple discontinuities. By utilizing an implicit scheme, the method allows relatively large time steps and, therefore, is highly efficient. The accuracy of the method is verified by simulating a time-domain Gaussian pulse reflected and transmitted from a multiple period grating and a four-layer antireflection coating structure.     

ADI-FDTD algorithm in curvilinear co-ordinates

The alternating direction implicit (ADI) scheme has been successfully applied to the finite-difference time-domain (FDTD) method to achieve an unconditionally stable algorithm. The ADI-FDTD method is extended to the curvilinear co-ordinate system to form an alternating direction implicit nonorthogonal FDTD (ADI-NFDTD) method. The numerical results show that the proposed ADI-NFDTD algorithm demonstrates better late time stability compared to the conventional NFDTD scheme.     

On the current source implementation for the ADI-FDTD method

The current source implementation for the alternating-direction implicit finite-difference time-domain (ADI-FDTD) method is examined in this letter. By regarding the ADI-FDTD method as an approximate factorization of the Crank-Nicolson scheme, current source is naturally coupled into simulations. As a result, field distribution is much more improved and free of asymmetry.     

Transient electromagnetic scattering from dielectric objects using the electric field Integral equation with Laguerre polynomials as temporal basis functions

In this paper, we propose a time-domain electric field integral equation (TD-EFIE) formulation for analyzing the transient electromagnetic response from three-dimensional (3-D) dielectric bodies. The solution method in this paper is based on the Galerkin's method that involves separate spatial and temporal testing procedures. Triangular patch basis functions are used for spatial expansion and testing functions for arbitrarily shaped 3-D dielectric structures. The time-domain unknown coefficients of the equivalent electric and magnetic currents are approximated using a set of orthonormal basis function that is derived from the Laguerre functions. These basis functions are also used as the temporal testing functions. Use of the Laguerre polynomials as expansion functions for the transient portion of response enables one not only to handle the time derivative terms in the integral equation in an analytic fashion but also completely separates the space and the time variables. Thus, the time variable along with the Courant condition can be eliminated in a Galerkin formulation using this procedure. We also propose an alternative formulation using a different expansion of the magnetic current. The total computational cost for this new method is similar to that of an implicit marching-on in time (MOT)-EFIE scheme, even though at each step this procedure requires more computations. Numerical results involving equivalent currents and far fields computed by the two proposed methods are presented and compared.     

An unconditionally stable scheme for the finite-difference time-domain method

In this work, we propose a numerical method to obtain an unconditionally stable solution for the finite-difference time-domain (FDTD) method for the TE/sub z/ case. This new method does not utilize the customary explicit leapfrog time scheme of the conventional FDTD method. Instead we solve the time-domain Maxwell's equations by expressing the transient behaviors in terms of weighted Laguerre polynomials. By using these orthonormal basis functions for the temporal variation, the time derivatives can be handled analytically, which results in an implicit relation. In this way, the time variable is eliminated from the computations. By introducing the Galerkin temporal testing procedure, the marching-on in time method is replaced by a recursive relation between the different orders of the weighted Laguerre polynomials if the input waveform is of arbitrary shape. Since the weighted Laguerre polynomials converge to zero as time progresses, the electric and magnetic fields when expanded in a series of weighted Laguerre polynomials also converge to zero. The other novelty of this approach is that, through the use of the entire domain-weighted Laguerre polynomials for the expansion of the temporal variation of the fields, the spatial and the temporal variables can be separated.     

Dispersion analysis of the three-dimensional complex envelope ADI-FDTD method

In this paper, numerical dispersion properties of the three-dimensional complex envelope (CE) alternate-direction implicit finite-difference time-domain (ADI-FDTD) method are studied. The variations of dispersion errors with propagation direction, ratio of carrier to envelope frequencies, and spatial and temporal steps are presented. It is found that the CE ADI-FDTD scheme have much better accuracy and efficiency over the ADI-FDTD, especially with a higher ratio of carrier to envelope frequencies. Therefore, the CE ADI-FDTD is recommended for use in efficient narrow bandwidth electromagnetic modeling.     

Efficient compact 2-D time-domain method with weighted Laguerre polynomials

An efficient time-domain method based on a compact two-dimensional (2-D) finite-difference time-domain (FDTD) method combined with weighted Laguerre polynomials has been proposed to analyze the propagation properties of uniform transmission lines. Starting from Maxwell's differential equations corresponding to the compact 2-D FDTD method, we use the orthonormality of weighted Laguerre polynomials and Galerkin's testing procedure to eliminate the time variable. Thus, an implicit relation, which results in a marching-on-in-degree scheme, can be obtained. To verify the accuracy and efficiency of the hybrid method, we compare the results with those from the conventional compact 2-D FDTD and compact 2-D alternating-direction-implicit (ADI) FDTD methods. The hybrid method improves the computational efficiency notably, especially for complex problems with fine structure details that are restricted by stability constrains in the FDTD method.     

An unconditionally stable 3-D ADI-MRTD method free of the CFLstability condition

In this paper, an alternating direction implicit (ADI) technique is applied to the recently developed multiresolution time-domain (MRTD) method, resulting in an unconditionally stable ADI-MRTD scheme free of the Courant-Friedich-Lecy (CFL) stability condition. The unconditional stability is theoretically proved, and preliminary numerical results are presented to validate the scheme. Because the scheme is now free of the stability condition, its time step is determined only by modeling accuracy. The price for having the unconditional stability is, however, that the required computation memory becomes almost twice of that for the original MRTD     

A domain decomposition finite-difference method for parallelnumerical implementation of time-dependent Maxwell's equations

A domain decomposition technique together with an implicit finite-difference scheme is used to design a parallel algorithm to solve for electromagnetic scattering in the time-domain by an infinite square metallic cylinder. The implicit difference scheme yields second-order accuracy, unconditional stability, and at each time step, a large system of linear equations. The domain decomposition technique reduces the solution of this large system to that of many independent smaller subsystems. The present algorithm can be easily implemented on coarse-grain parallel vector supercomputers to obtain a speedup close to the number of available central processing units (CPUs)