Precise Integration Time Domain Research Articles

A Switched-Huygens-Subgridding-Based Combined FDTD–PITD Method for Fine Structures

A new combined method is proposed for the problems containing fine structures. In this method, the precise-integration time-domain (PITD) method is adopted for fine structures with fine cells, and the finite-difference time-domain (FDTD) method is used for the rest with coarse cells. The fields in the coarse and fine cells are updated synchronously, which makes the proposed method fast. Besides, since the PITD method is only applied locally, the limitation of its large memory requirement is also alleviated. The numerical results confirm that this combined method has a fast computational speed.

Precise-Integration Time-Domain Formulation for Optical Periodic Media.

A numerical formulation based on the precise-integration time-domain (PITD) method for simulating periodic media is extended for overcoming the Courant-Friedrich-Levy (CFL) limit on the time-step size in a finite-difference time-domain (FDTD) simulation. In this new method, the periodic boundary conditions are implemented, permitting the simulation of a wide range of periodic optical media, i.e., gratings, or thin-film filters. Furthermore, the complete tensorial derivation for the permittivity also allows simulating anisotropic periodic media. Numerical results demonstrate that PITD is reliable and even considering anisotropic media can be competitive compared to traditional FDTD solutions. Furthermore, the maximum allowable time-step size has been demonstrated to be much larger than that of the CFL limit of the FDTD method, being a valuable tool in cases in which the steady-state requires a large number of time-steps.

Low-Memory Implementation of PITD Method Using a Thresholding Scheme

A precise-integration time-domain (PITD) method with a thresholding scheme is proposed to reduce the computation cost of the matrix exponential for solving electromagnetic (EM) problems. The finiteness of EM wave speed is used to analyze the structure of the matrix exponential, revealing that the dense matrix exponential can become sparse. A thresholding scheme for eliminating tiny and thus unimportant elements is established to obtain the sparse matrix exponential, where the threshold value is selected by Von Neumann stability criterion and induced spurious solutions are avoided. Theoretical analyses and numerical results confirm that the proposed method can save a large amount of memory and the accuracy of the original method is inherited.

An Efficient Numerical Formulation for Wave Propagation in Magnetized Plasma Using PITD Method

A modified precise-integration time-domain (PITD) formulation is presented to model the wave propagation in magnetized plasma based on the auxiliary differential equation (ADE). The most prominent advantage of this algorithm is using a time-step size which is larger than the maximum value of the Courant–Friedrich–Levy (CFL) condition to achieve the simulation with a satisfying accuracy. In this formulation, Maxwell’s equations in magnetized plasma are obtained by using the auxiliary variables and equations. Then, the spatial derivative is approximated by the second-order finite-difference method only, and the precise integration (PI) scheme is used to solve the resulting ordinary differential equations (ODEs). The numerical stability and dispersion error of this modified method are discussed in detail in magnetized plasma. The stability analysis validates that the simulated time-step size of this method can be chosen much larger than that of the CFL condition in the finite-difference time-domain (FDTD) simulations. According to the numerical dispersion analysis, the range of the relative error in this method is 10−6 to 5×10−4 when the electromagnetic wave frequency is from 1 GHz to 100 GHz. More particularly, it should be emphasized that the numerical dispersion error is almost invariant under different time-step sizes which is similar to the conventional PITD method in the free space. This means that with the increase of the time-step size, the presented method still has a lower computational error in the simulations. Numerical experiments verify that the presented method is reliable and efficient for the magnetized plasma problems. Compared with the formulations based on the FDTD method, e.g., the ADE-FDTD method and the JE convolution FDTD (JEC-FDTD) method, the modified algorithm in this paper can employ a larger time step and has simpler iterative formulas so as to reduce the execution time. Moreover, it is found that the presented method is more accurate than the methods based on the FDTD scheme, especially in the high frequency range, according to the results of the magnetized plasma slab. In conclusion, the presented method is efficient and accurate for simulating the wave propagation in magnetized plasma.

A Memory-Efficient Formulation of Precise-Integration Time-Domain Method With Riccati Matrix Differential Equations

A novel numerical method, referred to as the Riccati precise-integration time-domain (Riccati-PITD) method, is proposed to reduce the memory requirement of the conventional PITD method. In the Riccati-PITD method, the electromagnetic field components of the transverse electric/transverse magnetic (TE/TM) wave are arranged in a matrix for constructing the Riccati matrix differential equations (RDEs) about Maxwell's curl equations. Then the precise integration technique is adopted for the numerical integration of the RDEs. Theoretical analyses about the memory requirements of the conventional and the Riccati-PITD methods are presented. It is found that the memory requirement can be more greatly reduced by solving the electromagnetic-field components in the form of the matrices in the Riccati-PITD method rather than in the form of the vectors in the conventional PITD method. Numerical examples are also given to confirm that the proposed method has significant advantages over the conventional PITD method with respect to the execution time and the memory requirement.

A Precise-Integration Time-Domain Formulation Based on Auxiliary Differential Equation for Transient Propagation in Plasma

The electromagnetic wave propagation through plasma medium is one of the most important research fields in computational electromagnetics. A numerical formulation based on both the auxiliary differential equation (ADE) and the precise-integration time-domain (PITD) method for solving the plasma problems is proposed to break through the Courant-Friedrich-Levy (CFL) limit on the time-step size in a finite-difference time-domain (FDTD) simulation. In this new method, the current density J is introduced as the auxiliary variable to deal with the complex permittivity of the plasma which is dependent on the frequency, and the precise integration (PI) technique makes the selectable maximum time-step size become much larger and removes the impact of the time-step size to the numerical dispersion error. Numerical experimentations of the typical plasma problems verify and validate the reliability of the proposed formulation. Through the numerical results, it can be found that the maximum allowable time-step size of the new method is much larger than that of the CFL limit of the FDTD method, and the calculation error of the new method is nearly independent of the time-step size. As a consequence, the execution time is significantly reduced by using a larger time-step size.

Numerical Treatment for Electromagnetic Wave in Time-Variant Medium Using Generalized PITD Method

In this letter, the propagation of an electromagnetic (EM) wave in a time-variant medium is treated with a novel methodology stemming from the precise integration time domain (PITD) method. This methodology is a generalized version of the conventional PITD method (thus GPITD, for short). In the GPITD method, the electric displacement and the magnetic induction serve as the iteratively calculated components. In order to overcome the difficulty in implementing the precise integration (PI) routine caused by the time variation of the original coefficient matrix, an effective piecewise constant coefficient matrix is constructed. Both the numerical dispersion relation and the numerical stability condition of the GPITD method are described. Successful application of the GPITD method to solve the EM propagation problem associated with a layer whose permittivity suffers: 1) an abrupt change to a new value and 2) a sinusoidal modulation, confirms its effectiveness.

Analysis of the propagation properties of photonic crystals with defect by the precise integration time domain method

ABSTRACTPropagation properties of photonic crystals with defect are analyzed by the precise integration time domain (PITD) method in the paper. The precise integration is applied to the finite difference time domain in this method. The Yee cell differential discretization technique is used in space, and the precise integration method is adopted in time. The correctness of the method is demonstrated by calculating the one-dimensional photonic crystals with defect. The transmission coefficients of the transverse electric (TE) and transverse magnetic (TM) modes in the two-dimensional photonic crystals with defect are also obtained by the method. Furthermore, the stability, precision and efficiency of the method are discussed. Numerical results reveal that the PITD method is not limited by the Courant stability condition and has higher calculation precision and efficiency, which provides a new and effective analytical method for studying the propagation properties of photonic crystals with defect.

Stability Condition and Numerical Dispersion of Fourth-Order Compact Precise-Integration Time-Domain Method

In order to mitigate the numerical dispersion errors of the compact precise integration time-domain (CPITD) algorithm, the fourth-order CPITD [CPITD(4)] method is proposed recently which is based on the fourth-order finite-difference scheme and the precise integration technique. Both the stability condition and the numerical dispersion equation have been obtained. In this paper, the numerical dispersion characteristics of the CPITD(4) method, especially the improvement of the numerical dispersion by using the fourth-order finite-difference scheme are discussed in detail. It is found that the numerical dispersion errors of the CPITD(4) method are much less than those of the CPITD, compact finite-difference time-domain (CFDTD), and CFDTD(2,4) methods and can be made nearly independent of the time-step size. Furthermore, the numerical experiments validate the accuracy and the efficiency of the CPITD(4) method, and also verify our analysis of the numerical dispersion characteristics of the CPITD(4) method.

A Numerical Method Without Coordinate Transformations to the Electromagnetic Problem Involving Objects in Arbitrary Translational Motion

A novel numerical time-domain method, referred to as Maxwell–Minkowski precise integration time-domain (MM-PITD) method which combines MM equations and the precise integration (PI) technique, is developed in this paper to solve the problems involving moving objects. In the proposed method, the computation cells in different regions bear different governing equations for electromagnetic fields. With the suggested criterion to distinguish the governing equations in these cells, the discretized formulations for the whole computation domain are obtainable. Since the coefficient matrix used in the PI technique is required to be constant or piecewise constant, an approximation approach to locally linearizing the time-variant equation system is provided. Two appealing superiorities of the proposed method to other methods are: 1) the speed of the object can be either constant or time variant and 2) multiple objects can be taken into account conveniently. In addition, the proposed method is more efficient compared with the other methods wherein two or more coordinate frames are used. Several numerical experiments are carried out to test the MM-PITD method. Its effectiveness, efficiency, and generality are thereby confirmed.

Conformal Precise-Integration Time-Domain Method for Analyzing Electromagnetic Fields in Fine-Structured Device With Moving 3-D Part

Herein, a novel method, referred to as conformal precise-integration time-domain (CO-PITD) method, which combines the precise-integration time-domain (PITD) method and the conformal mesh technique, is proposed to numerically calculate the evolving electromagnetic (EM) fields in the fine-structured device with moving part in time domain, particularly for 3-D case. The EM parameters in the computation cells partially occupied by moving medium are evaluated with the aid of the conformal mesh technique. For the implementation of the PITD method, the discretized Maxwell equations in those cells are modified due to the temporal variation, caused by the movement, of the EM parameters. Furthermore, the challenge from time-variant coefficient matrix is overcome by constructing an effective piecewise time-constant substitute. Meanwhile, the stability condition and the dispersive relation of the proposed method are both theoretically derived. The validity of the CO-PITD method is verified in the context of several numerical experiments. In addition, the comparisons with the established methods show that the proposed method is of excellent efficiency since a much large time step size is permitted in it.

A Memory-Saving Realization of the Perfectly Matched Layer in the Precise-Integration Time-Domain Method

The memory demand of the precise-integration time-domain (PITD) method is enlarged quite a lot by the implementation of the perfectly matched layers following the conventional way, which is enclosing the field components of the perfectly matched layer (PML) regions and the main region into one field component vector. In this letter, a region-splitting (RS) technique, for the memory-saving realization of the PML in the PITD method, is proposed. In the RS technique, the field components in the main region and each PML region are calculated separately. The accurate mutual information transmissions between any two regions are guaranteed by advancing the sampling time points of the connecting field components by half-time-step. Several numerical experiments are performed to verify the effectiveness and the efficiency of the proposed technique.

Numerical analysis of electromagnetic field in devices with the high-speed unit

PurposeThe purposes of this paper are to numerically analyse the distribution of the electromagnetic field in the electromagnetic device wherein a high-speed unit exists and to develop a strong tool to analyse the evolution of an electromagnetic field tangled with moving parts.Design/methodology/approachThe precise integration time domain (PITD) method and parameter weighted averaging approximation scheme.FindingsIt is shown that that the electromagnetic field in the device is significantly affected by the velocity of the moving unit and the parameters of the base material. The computation resources of the proposed method are saved and the efficiency is enhanced.Originality/valueThe parameter approximation (PA)-PITD method can be an effective and efficient time domain method to analyse the evolution of the electromagnetic field in electromagnetic devices with moving parts and similar problems.

Numerical Analysis of Electromagnetic Scattering From a Moving Target by the Lorentz Precise Integration Time-Domain Method

A novel numerical method, referred to as Lorentz precise integration time-domain (Lorentz-PITD) method, is proposed to deal with the scattering problem from a moving conducting slab. Both the overset grid generation technique and the Lorentz transformation are employed in this method. By using the Lorentz transformation and the linear interpolation technique, the incident plane wave in the rest frame is introduced to the moving frame; the scattered fields are transformed from the moving frame to the rest frame. Numerical experiments validate the Lorentz-PITD method and show that the Lorentz-PITD method is more computationally efficient compared with the Lorentz finite-difference time-domain method. An accompanied find is that there must be a tradeoff between the applicable frequency range and the computation cost in the proposed method.

A Low-Dispersion Realization of a Rectangular Grid With PITD Method Through Artificial Anisotropy

Through using a nonuniform rectangular grid in the time-domain algorithm, the computational efficiency can be improved obviously, but the numerical anisotropic dispersion error is seriously deteriorative. In this letter, a rectangular grid with the precise-integration time-domain method through artificial anisotropy is proposed to reduce the numerical dispersion error of a rectangular grid. Both the stability condition and the numerical dispersion equation are obtained analytically, and the numerical anisotropic dispersion of the proposed method is examined in detail. It is found that the numerical dispersion error can be reduced obviously and can also be made nearly independent of the time-step size. The numerical experiments validate and verify that the proposed method is of higher accuracy and efficiency.

Analysis of photonic crystal transmission properties by the precise integration time domain

Photonic crystals are materials patterned with a periodicity in the dielectric constant, which can create a range of forbidden frequencies called as a photonic band gap. The photonic band gap of the photonic crystal indicates its primary property, which is the basis of its application. In recent years, photonic crystals have been widely used to design optical waveguides, filters, microwave circuits and other functional devices. Therefore, the study on the transmission properties in photonic crystal is significantly important for constructing the optical devices. The finite difference time domain (FDTD) is a very useful numerical simulation technique for solving the transmission properties of the photonic crystals. However, as the FDTD method is based on the second order central difference algorithm, its accuracy is relatively low and the Courant stability condition must be satisfied when this method is used, which may restrict its application. To increase the accuracy and the stability, considerable scientific interest has been attracted to explore the schemes to improve the performance of the FDTD. The fourth order Ronge-Kutta (RK4) method has been applied to the FDTD method, which improves the accuracy and eliminates the influence of accumulation errors of the results, but the stability remains very poor if the time step is large. An effective time domain algorithm based on the high precision integration is proposed to solve the transmission properties of photonic crystals. The Yee cell differential technique is used to discretize the first order Maxwell equations in the spatial domain. Then the discretized Maxwell equations with the absorption boundary conditions and the expression of excitation source are rewritten in the standard form of the first order ordinary differential equation. According to the precise division of the time step and the additional theorem of exponential matrix, the high precision integration is used to obtain the homogeneous solution. To obtain the discretized electric and magnetic fields, the particular solution must be solved based on the excitation and then be added to the homogeneous solution. The transmission properties of photonic crystals are obtained by the Fourier transform. Practical calculation of photonic crystals is carried out by the precise integration time domain, and the accuracy and the stability are compared with those from the FDTD and the RK4 methods. The numerical results show that the precise integration time domain has a higher calculation precision and overcomes the restriction of stability conditions on the time step, which provides an effective analytical method of studying the transmission properties of photonic crystals.

A High-Order 2-D CPITD Method for Electrically Large Waveguide Analysis

Based on the fourth-order finite difference (FD) scheme, a modified compact two dimensional precise integration time domain (CPITD) method, called CPITD(4), is proposed to mitigate the numerical dispersion errors of the CPITD algorithm when modeling electrically large and longitudinally invariant wave-guiding structures. Both the stability condition and the dispersion equation of the CPITD(4) method are derived analytically. It is found that the dispersion error of the CPITD(4) method is much smaller than that of the CPITD method. In particular, it should be pointed out that the CPITD(4) method can improve the performance of the CPITD method when β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</sub> /β leaves away from 1. Numerical experiments of typical waveguide structures validate and verify that the CPITD(4) method can decrease the calculated error without increasing the memory requirement and the execution time for all the cases.

2D CPITD method for extracting attenuation constant of longitudinally invariant lossy transmission lines

Owing to the fact that the conventional compact two-dimensional (2D) precise integration time-domain (CPITD) method ignores the attenuation along the propagation direction for lossy invariant transmission lines, the calculation results are unsatisfactory. To extract the exact attenuation constant, a novel CPITD method that considers the attenuation constant along the propagation direction and describes an iterative process to seek out the attenuation constant is proposed. Compared with previous research related to the compact 2D finite-difference time-domain (CFDTD) method, the analytical or approximate analytical solution of the attenuation constant as the initial input value is used followed by the golden division method to find the interval of the exact attenuation constant in the iterative process. Numerical experimentations validate that the results of the novel method are much more accurate than those of the conventional CPITD method. Furthermore, the proposed method is of higher computational efficiency than the CFDTD method.

A split-step-scheme-based precise integration time domain method for solving wave equation

Purpose – This paper aims to present a modified precise integration time domain (PITD) method for the numerical solution of 2D scalar wave equation. Design/methodology/approach – The split step (SS) scheme is applied to factorize the conventional PITD calculation into two sub-steps procedures and then field components can be updated along one spatial direction only in each sub-step. The perfectly matched layer (PML) absorber is extended to this method for modeling open region problems by using the stretched coordinate approach. Findings – It is shown that this method requires less computation time and storage space in comparison with the conventional PITD method, yet maintains the numerical stability despite using large time steps. Research limitations/implications – The WE-PITD method requires the divergence free region, which may be a limit on its usage. Hence, there is a challenge of using this technique in the 3D problems. Originality/value – Based on the SS scheme, the PITD method is used to solve the scale wave equation rather than Maxwell's equations, leading to a significant reduction in the computation time and memory usage.

An efficient application of PML in fourth-order precise integration time domain method for the numerical solution of Maxwell's equations

Purpose – The purpose of the paper is to introduce a perfectly matched layer (PML) absorber, based on Berenger's split field PML, to the recently proposed low-dispersion precise integration time domain method using a fourth-order accurate finite difference scheme (PITD(4)). Design/methodology/approach – The validity and effectiveness of the PITD(4) method with the inclusion of the PML is investigated through a two-dimensional (2-D) point source radiating example. Findings – Numerical results indicate that the larger time steps remain unchanged in the procedure of the PITD(4) method with the PML, and meanwhile, the PITD(4) method employing the PML is of the same absorbability as that of the finite-difference time-domain (FDTD) method with the PML. In addition, it is also demonstrated that the later time reflection error of the PITD(4) method employing the PML is much lower than that of the FDTD method with the PML. Originality/value – An efficient application of PML in fourth-order precise integration time domain method for the numerical solution of Maxwell's equations.