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Here an efficient Laguerre-based finite-difference time-domain iterative algorithm is proposed. Different from the previously developed iterative procedure used in the efficient FDTD algorithm, a new perturbation term combined with the Gauss–Seidel iterative procedure is introduced to form the new Laguerre-based FDTD algorithm in the 3D cylindrical coordinate system. Such a treatment scheme can reduce the splitting error to a low level and obtain a good convergence; in other words, it can improve the efficiency and accuracy than other algorithms. To verify the performance of the proposed algorithm, two scattering numerical examples are given. The computation results show that the proposed algorithm can be better than the ADI-FDTD algorithm in terms of efficiency and accuracy. Meanwhile, the proposed algorithm is extremely useful for the problems with fine structures in the 3D cylindrical coordinate system.

In recent years, the unconditionally stable Laguerre-based finite-difference time-domain (FDTD) algorithm has been applied to simulate transient electromagnetic problems in the Cartesian coordinate. By transforming the time-domain problem to the Laguerre domain using the temporal Galerkin’s testing procedure, the transient solution is independent of time discretization. Thus, Laguerre-based FDTD formulation has the advantage of less numerical error when a larger time step is used.

The main drawback of the conventional Laguerre-based FDTD algorithm is that it requires solving a very large sparse matrix. To overcome this problem, a factorization-splitting scheme [

However, in many applications, we have to deal with 3D cylindrical structures such as in optical fiber communication, integrated optics, and defense industry. Moreover, the geometry of interest may consist of fine structures. If we adopt the conventional FDTD method to discretize the cylindrical structure with the Cartesian grid, a significant staircasing error appears. In fact, due to the existence of the

Therefore, in order to expand the research field of the FDTD method, we propose an efficient Laguerre-based finite-difference time-domain iterative algorithm in the 3D cylindrical coordinate system. Firstly, according to the ideology of the weighted Laguerre polynomial (WLP) FDTD scheme, the WLP-FDTD equations of the proposed algorithm in the 3D cylindrical coordinate system are deduced by introducing the new perturbation term and the nonphysical intermediate variables. Secondly, the field components on the

Introducing the WLP technology [

Equations (

Obviously, solving equation (

Referring to the brief form of the 3-D FDTD equation in [

In addition, the

In fact, equation (

In accordance with the design theory of the proposed algorithm in this paper, we introduce the nonphysical intermediate variables

Expanding (

Substituting (

Expanding (

Obviously, the necessary nonphysical variables

Next is how to solve the fields of the nonphysical variables

Clearly, the nonphysical variables

Substituting (

Expanding (

At this point, the implementation scheme of the proposed algorithm has been elaborated, and the overall execution process is described in Figure

Overall execution process of the proposed algorithm.

Because of the strangeness of the cylindrical coordinate system, drastic change of the field nearby the axis will lead the splitting error to be great [

The positions of the electric/magnetic fields of the 3D cylindrical coordinate are shown in Figure

Position of field in cylindrical coordinates.

Top view of the field nearby the axis.

According Stokes’s theorem, we start from the following integral form of Maxwell’s equations in the time domain:

The FDTD difference equation of

Obviously, the form of the right side of equation (

In this way, the form of the right side of equation (

Introducing the WLP technology [

In order to solve equation (

Substituting (

In a similar way, the FDTD difference equation of

Obviously, when

In order to verify the performance of the proposed algorithm, two scattering numerical examples are given.

First example: an irregular scatter with two grooves is calculated, and the angle of the groove is

Illustration of the irregular cylinder with two grooves under the oblique incident wave.

The observation points of the simulated electric fields are set at

Computational results of

Computational results of

Computational results of

In addition, to demonstrate the higher accuracy of the proposed algorithm, here two error formulations are provided to describe the accuracy, which are defined as

Figures

Errors obtained by

Errors obtained by

Errors obtained by

Simulation results of the two algorithms in the first example.

ADI-FDTD algorithm [ |
Proposed algorithm | |||
---|---|---|---|---|

cfln = 4 | cfln = 5 | cfln = 6 | Nt = 3 | |

CPU time (s) | 337.27 | 270.14 | 224.26 | 218.95 |

Observing Figures

Second example: another irregular scatter with one wedge-shaped bulge is shown in Figure

Illustration of the irregular cylinder with one wedge-shaped bulge under the oblique incident wave.

The observation points of simulated electric fields are set at

Computational results of

Computational results of

Computational results of

To demonstrate the accuracy of the proposed algorithm again, formulations (

Errors obtained by

Errors obtained by

Errors obtained by

Simulation results of the two algorithms in the second example.

ADI-FDTD algorithm [ |
Proposed algorithm | |||
---|---|---|---|---|

cfln = 4 | cfln = 5 | cfln = 6 | Nt = 3 | |

CPU time (s) | 335.48 | 271.93 | 226.06 | 222.99 |

Observing Figures

In this paper, an efficient Laguerre-based finite-difference time-domain iterative algorithm in the 3D cylindrical coordinate system is proposed. By adopting a new perturbation term

The simulation data for the 3D cylindrical coordinate structure used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported in part by the National Science Research Foundation of Institutions of Higher Learning in Jiangsu Province of China under Grant 18KJB510017, University-Level Science Foundation of Nanjing Insititute of Technology under Grant ZKJ201801, and National Science Foundation of China under Grant 51477182.