Locally One-dimensional Finite-difference Time-domain Research Articles

An Unconditionally Stable Conformal LOD-FDTD Method for Curved PEC Objects and its Application to EMC Problems

The traditional finite-difference time-domain (FDTD) method is constrained by the Courant–Friedrich–Levy condition and suffers from the notorious staircase error in electromagnetic simulations. This article proposes a 3-D conformal locally one-dimensional FDTD (CLOD-FDTD) method to address the two issues for modeling perfectly electrical conducting (PEC) objects. By considering the partially filled cells, the proposed CLOD-FDTD method can significantly improve the accuracy compared with the traditional locally one-dimensional FDTD (LOD-FDTD) method and the FDTD method. At the same time, the proposed method preserves unconditional stability, which is analyzed and numerically validated using the von Neumann method. Significant gains in central processing unit time are achieved by using large time steps without sacrificing accuracy. Two numerical examples, including a PEC cylinder and a missile, are used to verify its accuracy and efficiency with different meshes and time steps. It can be found from these examples that the CLOD-FDTD method shows better accuracy and can improve the efficiency compared with those of the traditional FDTD method and the traditional LOD-FDTD method.

From Time-Collocated to Leapfrog Fundamental Schemes for ADI and CDI FDTD Methods

The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence implicit (CDI) FDTD method. In this paper, the formulations from time-collocated to leapfrog fundamental schemes are presented for ADI and CDI FDTD methods. For the ADI FDTD method, the time-collocated fundamental schemes are implemented using implicit E-E and E-H update procedures, which comprise simple and concise right-hand sides (RHS) in their update equations. From the fundamental implicit E-H scheme, the leapfrog ADI FDTD method is formulated in conventional form, whose RHS are simplified into the leapfrog fundamental scheme with reduced operations and improved efficiency. For the CDI FDTD method, the time-collocated fundamental scheme is presented based on locally one-dimensional (LOD) FDTD method with complying divergence. The formulations from time-collocated to leapfrog schemes are provided, which result in the leapfrog fundamental scheme for CDI FDTD method. Based on their fundamental forms, further insights are given into the relations of leapfrog fundamental schemes for ADI and CDI FDTD methods. The time-collocated fundamental schemes require considerably fewer operations than all conventional ADI, LOD and leapfrog ADI FDTD methods, while the leapfrog fundamental schemes for ADI and CDI FDTD methods constitute the most efficient implicit FDTD schemes to date.

Error-Controllable Scheme for the LOD-FDTD Method

The implicit locally one-dimensional finite-difference time-domain (LOD-FDTD) method is useful for designing plasmonic devices and waveguide structures. By using a large timestep size, the implicit LOD-FDTD method can reduce the computational time; however, this involves a trade-off with accuracy. To overcome this trade-off, we propose an error-controllable scheme for the LOD-FDTD method, wherein the fast inverse Laplace transform is employed to generate the electromagnetic field in arbitrary time domain from that in complex frequency domain. Compared to the conventional LOD-FDTD method, our scheme provides higher accuracy with more efficient calculations.

Unconditionally stable five‐step LOD‐FDTD method in plasma medium

AbstractA three‐dimensional unconditionally stable five‐step locally one‐dimensional finite‐difference time‐domain (LOD‐FDTD) method for anisotropic magnetized plasma medium is presented. Firstly, we search for a diagonal matrix to transform Maxwell coefficient matrix of plasma into skew symmetric matrix which is beneficial to analyze the properties of the proposed algorithm. Then, the detailed formula derivation and numerical stability analysis is presented. At last, the resonance frequency of a microwave cavity partially filled with a plasma slab and the rotation angle of an infinite plasma slab are simulated to verify the accuracy and stability of the proposed method. The effectiveness of the proposed method is further demonstrated by comparing with the three‐step LOD‐FDTD method.

Hybrid sub-gridding ADE–FDTD method of modeling periodic metallic nanoparticle arrays**Project supported by the National Natural Science Foundation of China (Grant Nos. 61471105 and 61331007).

In this paper, a modified sub-gridding scheme that hybridizes the conventional finite-difference time-domain (FDTD) method and the unconditionally stable locally one-dimensional (LOD) FDTD is developed for analyzing the periodic metallic nanoparticle arrays. The dispersion of the metal, caused by the evanescent wave propagating along the metal-dielectric interface, is expressed by the Drude model and solved with a generalized auxiliary differential equation (ADE) technique. In the sub-gridding scheme, the ADE–FDTD is applied to the global coarse grids while the ADE–LOD–FDTD is applied to the local fine grids. The time step sizes in the fine-grid region and coarse-grid region can be synchronized, and thus obviating the temporal interpolation of the fields in the time-marching process. Numerical examples about extraordinary optical transmission through the periodic metallic nanoparticle array are provided to show the accuracy and efficiency of the proposed method.

Development of 2D LOD‐FDTD method with low numerical dispersion in lossy media

An efficient 2D locally one-dimensional finite-difference time-domain (LOD-FDTD) method in lossy material is presented in this Letter. The weighted factor and scaling factors of the isotropic dispersion FDTD method are reformed for the LOD-FDTD method in lossy media. Compared with the conventional LOD-FDTD method, the relative attenuation and phase errors of the proposed method are significantly reduced. The numerical anisotropy is also greatly improved by using the proposed method.

Numerical Dispersion Relation for the 2-D LOD-FDTD Method in Lossy Media

A closed-form expression is derived for the numerical dispersion relation of the 2-D locally one-dimensional finite-difference time-domain (LOD-FDTD) method in lossy media. In contrast to the lossless formulation, we found that transverse-electric (TE $_{z}$ ) and transverse-magnetic (TM $_{z}$ ) waves in lossy media exhibit different numerical dispersion relations. Moreover, when the material relaxation-time constant is not well resolved by the integration time-step, the TM $_{z}$ case shows much worse accuracy than the TE $_{z}$ case. To remove this limitation, a split-field LOD-FDTD formulation for TM $_{z}$ waves is then considered, which exhibits the same dispersion relation as the LOD-FDTD method for TE $_{z}$ waves. The validity of the theoretical results is illustrated through numerical simulations.

Stability Analyses of Nonuniform Time-Step LOD-FDTD Methods for Electromagnetic and Thermal Simulations

This paper presents the stability analyses of nonuniform time-step (NUTS) locally one-dimensional finite-difference time-domain (LOD-FDTD) methods for electromagnetic (EM) and thermal simulations. The overall (spatial domain) transition matrix for the whole 3-D computational domain is considered for NUTS, which takes into consideration general inhomogeneous and lossy media. Rigorous stability analyses of NUTS LOD-FDTD methods are provided for both EM and thermal simulations. The analytical proofs of unconditional stability are performed through careful assertion of respective matrix definiteness, along with spectral radius and induced matrix norm analyses. Proper transformations and manipulations are carried out differently for EM and thermal analyses to suit different matrix properties. In each analysis, the fundamental form of the transition matrix is utilized with only one main inverse term, which results in much simpler and concise analysis. It is shown that the NUTS LOD-FDTD methods are unconditionally stable for both EM and thermal simulations.

Analysis of Extraordinary Optical Transmission With Periodic Metallic Gratings Using ADE-LOD-FDTD Method

In this paper, a frequency-dependent locally one-dimensional finite-difference time-domain (LOD-FDTD) method is developed for the extraordinary optical transmission (EOT) analysis of periodic metallic gratings. The dispersion of the metal, caused by the evanescent waves propagating along the interface between the metal and the dielectric materials in the visible and near infrared regions, is expressed by the Drude model and solved with a generalized auxiliary differential equation (ADE) technique. With efficient preprocessing for the lower–upper (LU) decomposition in ADE-LOD-FDTD, the periodic boundary condition (PBC) is applied to the 2-D metallic grating structure. Two numerical examples with different subwavelength slits are calculated, and the mechanism of the EOT phenomenon is investigated. Compared with the standard ADE-FDTD method, the results from the proposed method show its good efficiency for the nanostructures.

Second-Order Temporal-Accurate Scheme for 3-D LOD-FDTD Method With Three Split Matrices

This letter presents a second-order temporal-accurate scheme for three-dimensional (3-D) locally one-dimensional finite-difference time-domain (LOD-FDTD) method with three split matrices. The main iterations of the proposed scheme comprise only one more procedure in addition to the existing three procedures. Using proper (initial only) input and (often infrequent, field-point only) output processings, the second-order temporal accuracy is achieved and verified analytically. To further enhance the efficiency, the proposed scheme is implemented based on the principle of fundamental schemes for unconditionally-stable FDTD methods, which feature a matrix-operator-free right-hand side. Stability analysis is provided to ascertain the unconditional stability of the proposed scheme and numerical results are demonstrated to justify its higher accuracy.

A Three-Dimensional Unconditionally Stable Five-Step LOD-FDTD Method

A three-dimensional unconditionally stable five-step locally one-dimensional finite-difference time-domain (LOD-FDTD) method is presented. Unlike the two-step LOD-FDTD and three-step LOD-FDTD methods, the proposed method has second order temporal accuracy. Hence, it gives less numerical dispersion than the two-step LOD-FDTD and three-step LOD-FDTD methods. It also gives less numerical dispersion than the alternating direction implicit finite-difference time-domain (ADI-FDTD) method. Moreover, for every propagation angle θ, it provides very small anisotropy error than the above-mentioned FDTD methods. Effects of the time step and the mesh size on the performance of the proposed method are discussed in detail. In this paper, validation of the stability and the accuracy of the proposed method is done with the help of simulation results. To further show the advantage of the proposed method, performance of the proposed method with artificial coefficients (control parameters) is also discussed in this paper.

Efficient Parallel LOD-FDTD Method for Debye-Dispersive Media

The locally one-dimensional finite-difference time-domain (LOD-FDTD) method is a promising implicit technique for solving Maxwell's equations in numerical electromagnetics. This paper describes an efficient message passing interface (MPI)-parallel implementation of the LOD-FDTD method for Debye-dispersive media. Its computational efficiency is demonstrated to be superior to that of the parallel ADI-FDTD method. We demonstrate the effectiveness of the proposed parallel algorithm in the simulation of a bio-electromagnetic problem: the deep brain stimulation (DBS) in the human body.

Singularity Processing Method of Microstrip Line Edge Based on LOD-FDTD

In order to improve the performance of the accuracy and efficiency for analyzing the microstrip structure, a singularity processing method is proposed theoretically and experimentally based on the fundamental locally one-dimensional finite difference time domain (LOD-FDTD) with second-order temporal accuracy (denoted as FLOD2-FDTD). The proposed method can highly improve the performance of the FLOD2-FDTD even when the conductor is embedded into more than half of the cell by the coordinate transformation. The experimental results showed that the proposed method can achieve higher accuracy when the time step size is less than or equal to 5 times of that the Courant-Friedrich-Levy (CFL) condition allowed. In comparison with the previously reported methods, the proposed method for calculating electromagnetic field near microstrip line edge not only improves the efficiency, but also can provide a higher accuracy.

Efficiency-Improved LOD-FDTD Method for Solving Periodic Structures at Oblique Incidence

This letter presents an efficiency-improved locally one-dimensional finite-difference time-domain (LOD-FDTD) method for solving the problems of oblique incident wave on two-dimensional periodic structures. By expressing the extra time derivatives caused by the field transformation technique as the form of the coefficient matrix, and adopting the LOD scheme, the transformed Maxwell's equations were split into two sub-steps. Then, by introducing an auxiliary field variable denoted by ez, two sub-step procedures were reformulated in a much simpler and more concise scheme. The proposed method is unconditionally stable, and more efficient validated by numerical example.

Efficient Modeling and Simulation of Graphene Devices With the LOD-FDTD Method

The intraband term of the graphene electronic model is incorporated into Maxwell equations, and then the locally one-dimensional finite difference time domain (LOD-FDTD) method is applied to simulate graphene devices efficiently. Numerical results of the approach are compared with the explicit FDTD method. At Courant Friedrich Levy number (CFLN) equal to 100, the proposed approach is approximately 60% faster in terms of simulation time and with reasonable accuracy as compared to the FDTD method.

Locally one-dimensional finite-difference time-domain scheme for the full-wave semiconductor device analysis

The application of an unconditionally stable locally one-dimensional finite-difference time-domain (LOD-FDTD) method for the full-wave simulation of semiconductor devices is described. The model consists of the electron equations for semiconductor devices in conjunction with Maxwell's equations for electromagnetic effects. Therefore the behaviour of an active device at high frequencies is described by considering the distributed effects, propagation delays, electron transmit time, parasitic elements and discontinuity effects. The LOD-FDTD method allows a larger Courant'Friedrich'Lewy number (CFLN) as long as the dispersion error remains in the acceptable range. Hence, it can lead to a significant time reduction in the very time consuming full-wave simulation. Numerical results show the efficiency of the presented approach.

An Improvement for the Locally One-Dimensional Finite-Difference Time-Domain Method

To reduce the memory usage of computing, the locally one-dimensional reduced finite-difference time-domain method is proposed. It is proven that the divergence relationship of electric-field and magnetic-field is non-zero even in charge-free regions, when the electric-field and magnetic-field are calculated with locally one-dimensional finite-difference time-domain (LOD-FDTD) method, and the concrete expression of the divergence relationship is derived. Based on the non-zero divergence relationship, the LOD-FDTD method is combined with the reduced finite-difference time-domain (R-FDTD) method. In the proposed method, the memory requirement of LOD-R-FDTD is reduced by1/6 (3D case) of the memory requirement of LOD-FDTD averagely. The formulation is presented and the accuracy and efficiency of the proposed method is verified by comparing the results with the conventional results.

A 3-D LOD-FDTD Method for the Wideband Analysis of Optical Devices

An improved three-dimensional (3-D) locally one-dimensional finite-difference time-domain (LOD-FDTD) method is developed and applied to the wideband analysis of waveguide gratings. First, the formulation is presented, in which dispersion control parameters are introduced to reduce the numerical dispersion error and perfectly matched layers are simply implemented without the field components being split. Next, as a preliminary calculation, the wavelength response of the waveguide grating is analyzed in a two-dimensional problem. The dispersion control contributes to the accuracy improvement even with a large time step beyond the Courant-Friedrich-Levy limit. Finally, a 3-D waveguide grating is analyzed. The use of the dispersion control parameters only in the propagation direction enables us to employ a large time step for efficient calculations, i.e., the computation time can be reduced to about half that of the explicit counterpart. In the Appendix, maximum time step for providing a highly accurate result is also predicted using the numerical dispersion analysis.

Numerical Dispersion Analysis of the Unconditionally Stable Three-Dimensional LOD-FDTD Method

The numerical dispersion characteristics of the recently developed three-dimensional unconditionally stable locally- one-dimensional finite-difference time-domain (LOD-FDTD) method are derived analytically. The effect of grid size and the Courant Friedrich Lewy (CFL) limits on dispersion are studied in detail. The LOD-FDTD method allows larger time steps as compared to the conventional FDTD method (CFL limit). The analysis shows that the unconditionally stable three-dimensional LOD-FDTD method has an advantage over the conventional FDTD method when modeling structures that require fine grids. The LOD-FDTD method allows larger CFL numbers as long as the dispersion error remains in acceptable range.

Acceleration of LOD-FDTD Method Using Fundamental Scheme on Graphics Processor Units

This letter presents the acceleration of locally one-dimensional finite-difference time-domain (LOD-FDTD) method using fundamental scheme on graphics processor units (GPUs). Compared to the conventional scheme, the fundamental LOD-FDTD (denoted as FLOD-FDTD) scheme has its right-hand sides cast in the simplest form without involving matrix operators. This leads to a substantial reduction in floating-point operations as well as field and coefficient memory access. To reap further advantages of FLOD-FDTD, certain field updates are embedded in the implicit solutions while exploiting the reuse of field data. Using FLOD-FDTD, it is found that significant speed-up is achievable for the method on GPUs.