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.

Download Full-text

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

Electronics ◽

10.3390/electronics9101575 ◽

2020 ◽

Vol 9 (10) ◽

pp. 1575 ◽

Cited By ~ 2

Author(s):

Zhen Kang ◽

Ming Huang ◽

Weilin Li ◽

Yufeng Wang ◽

Fang Yang

Keyword(s):

Wave Propagation ◽

Time Domain ◽

Fdtd Method ◽

Magnetized Plasma ◽

Numerical Dispersion ◽

Step Size ◽

Time Step ◽

Precise Integration ◽

Dispersion Error ◽

Time Step Size

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.

Download Full-text

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

COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering ◽

10.1108/compel-10-2012-0272 ◽

2013 ◽

Vol 33 (1/2) ◽

pp. 116-125

Author(s):

Zhongming Bai ◽

Xikui Ma ◽

Xu Zhuansun ◽

Qi Liu

Keyword(s):

Finite Difference ◽

Numerical Solution ◽

Fourth Order ◽

Time Domain ◽

Fdtd Method ◽

Time Domain Method ◽

Integration Time ◽

Precise Integration ◽

Content Type ◽

Domain Method

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.

Download Full-text

A Precise Time Step Integration Method

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/pime_proc_1994_208_148_02 ◽

1994 ◽

Vol 208 (6) ◽

pp. 427-430 ◽

Cited By ~ 161

Author(s):

W X Zhong ◽

F W Williams

Keyword(s):

High Precision ◽

Integration Method ◽

Linear Time ◽

Step Size ◽

Time Step ◽

Structural Dynamic ◽

Time Step Size ◽

Linear Time Invariant ◽

Wide Range ◽

Time Invariant

A high-precision numerical time step integration method is proposed for a linear time-invariant structural dynamic system. Its numerical results are almost identical to the precise solution and are almost independent of the time step size for a wide range of step sizes. Numerical examples illustrate this high precision.

Download Full-text

A Scale-Invariant Formulation of the Anticipated Potential Vorticity Method

Monthly Weather Review ◽

10.1175/mwr-d-10-05004.1 ◽

2011 ◽

Vol 139 (8) ◽

pp. 2614-2629 ◽

Cited By ~ 23

Author(s):

Qingshan Chen ◽

Max Gunzburger ◽

Todd Ringler

Keyword(s):

Potential Vorticity ◽

Climate Models ◽

Optimal Parameter ◽

Grid Resolution ◽

Step Size ◽

Time Step ◽

Time Step Size ◽

Uniform Grids ◽

Wide Range ◽

New Formulation

AbstractThe long-term success of climate models that operate on multiresolution grids depends on access to subgrid parameterizations that act appropriately across a wide range of spatial and temporal scales. As the first step in a series of efforts to obtain such scale-aware subgrid parameterizations, the authors focus on the anticipated potential vorticity method (APVM) on a sequence of quasi-uniform grids with varying resolutions. Through a scale analysis technique and phenomenological theories for two-dimensional turbulent flows, they derive a new formulation of the APVM that depends on a single parameter that is formally independent of the time-step size, the grid resolution, and the flow itself. Results of numerical experiments with this new formulation demonstrate that the optimal parameter of the new APVM formulation is invariant with respect to the time-step size, is insensitive to the flows, and is only weakly dependent on the grid resolution.

Download Full-text

Optimum time-step size for 2D (2, 4) FDTD method

Electronics Letters ◽

10.1049/el.2010.3729 ◽

2011 ◽

Vol 47 (5) ◽

pp. 317 ◽

Cited By ~ 1

Author(s):

T.T. Zygiridis

Keyword(s):

Fdtd Method ◽

Step Size ◽

Time Step ◽

Optimum Time ◽

Time Step Size

Download Full-text

Fully Coupled Time Domain Modelling of 3D Floating Bodies and Mooring Systems in Regular and Irregular Sea States

Volume 1: Offshore Technology ◽

10.1115/omae2012-83464 ◽

2012 ◽

Author(s):

Kameswara S. Vepa ◽

Diederik Van Nuffel ◽

Wim Van Paepegem ◽

Joris Degrieck

Keyword(s):

Time Domain ◽

Domain Size ◽

Floating Body ◽

Computational Time ◽

Floating Bodies ◽

Step Size ◽

Coupled Models ◽

Time Step ◽

Time Step Size ◽

Fully Coupled

Research on floating bodies like Wave Energy Converters (WECs) and Laser Imaging Detection And Ranging (LIDAR) systems has recently known a large growth. To study the minute details of the working model, it is important to study the effect of interactions between the waves, floating bodies and the mooring systems that are controlling the motion of the floating body. To achieve a more realistic numerical model in the time domain, a number of programs are linked together. The idea is to use the strength of each individual program for better results and also reduce the computational time. This paper provides a solution in the direction of using a fully coupled time domain coupling code that controls the data flow between a fluid solver, a structural solver, and a kinematic system simulator. Two- and three-dimensional fully coupled models are studied for calculation times and accuracy of results, and scaling is tested through parallelization on a large HPC cluster. The time step size of the whole model can be controlled by the user. Calculation times and memory requirements vary largely based on the factors like: domain size, SPH particle size, material model used for the floating body and the mooring system, complexity of the mechanical system inside the floating body. As a test case, a rigid body model is presented in this paper.

Download Full-text

A method for solving heat transfer with phase change in ice or soil that allows for large time steps while guaranteeing energy conservation

The Cryosphere ◽

10.5194/tc-15-2541-2021 ◽

2021 ◽

Vol 15 (6) ◽

pp. 2541-2568

Author(s):

Niccolò Tubini ◽

Stephan Gruber ◽

Riccardo Rigon

Keyword(s):

Heat Transfer ◽

Phase Change ◽

Model Performance ◽

Grid Method ◽

Frozen Soil ◽

Integration Time ◽

Step Size ◽

Time Step ◽

Conservation Of Energy ◽

Time Step Size

Abstract. The accurate simulation of heat transfer with phase change is a central problem in cryosphere studies. This is because the non-linear behaviour of enthalpy as function of temperature can prevent thermal models of snow, ice, and frozen soil from converging to the correct solution. Existing numerical techniques rely on increased temporal resolution in trying to keep corresponding errors within acceptable bounds. Here, we propose an algorithm, originally applied to solve water flow in soils, as a method to solve these integration issues with guaranteed convergence and conservation of energy for any time step size. We review common modelling approaches, focusing on the fixed-grid method and on frozen soil. Based on this, we develop a conservative formulation of the governing equation and outline problems of alternative formulations in discretized form. Then, we apply the nested Newton–Casulli–Zanolli (NCZ) algorithm to a one-dimensional finite-volume discretization of the energy–enthalpy formulation. Model performance is demonstrated against the Neumann and Lunardini analytical solutions and by comparing results from numerical experiments with integration time steps of 1 h, 1 d, and 10 d. Using our formulation and the NCZ algorithm, the convergence of the solver is guaranteed for any time step size. With this approach, the integration time step can be chosen to match the timescale of the processes investigated.

Download Full-text

The impact of the inverse chirp z-transform on breast microwave radar image reconstruction

International Journal of Microwave and Wireless Technologies ◽

10.1017/s1759078720000379 ◽

2020 ◽

Vol 12 (9) ◽

pp. 848-854

Author(s):

Tyson Reimer ◽

Mario Solis-Nepote ◽

Stephen Pistorius

Keyword(s):

Image Reconstruction ◽

Time Domain ◽

Tumor Response ◽

Imaging System ◽

Step Size ◽

Time Step ◽

Time Step Size ◽

Inverse Discrete Fourier Transform ◽

Microwave Radar ◽

The Impact

This work examines the impact of the inverse chirp z-transform (ICZT) for frequency-to-time-domain conversion during image reconstruction of a pre-clinical radar-based breast microwave imaging system operating over 1–8 GHz. Two anthropomorphic breast phantoms were scanned with this system, and the delay-multiply-and-sum beamformer was used to reconstruct images of the phantoms, after using either the ICZT or the inverse discrete Fourier transform (IDFT) for frequency-to-time domain conversion. The contrast, localization error, and presence of artifacts in the reconstructions were compared. The use of the IDFT resulted in prominent ring artifacts that were not present when using the ICZT, and the use of the ICZT resulted in higher contrast between the tumor and clutter responses. In one of the phantoms, the tumor response was only visible in reconstructions that used the ICZT. The use of the ICZT evaluated with a time-step size of 11 ps resulted in the reduction of prominent artifacts present when using the IDFT and the successful identification of the tumor response in the reconstructed images.

Download Full-text

A method for solving heat transfer with phase change in ice or soil that allows for large time steps while guaranteeing energy conservation

10.5194/tc-2020-293 ◽

2020 ◽

Author(s):

Niccolò Tubini ◽

Stephan Gruber ◽

Riccardo Rigon

Keyword(s):

Heat Transfer ◽

Phase Change ◽

Model Performance ◽

Frozen Soil ◽

Integration Time ◽

Nonlinear Behaviour ◽

Step Size ◽

Time Step ◽

Conservation Of Energy ◽

Time Step Size

Abstract. The accurate simulation of heat transfer with phase change is a central problem in cryosphere studies. This is because the nonlinear behaviour of enthalpy as function of temperature can prevent thermal models of snow, ice and frozen soil from converging to the correct solution. Existing numerical techniques rely on increased temporal resolution in trying to keep corresponding errors withing acceptable bounds. Here, we propose an algorithm, originally applied to solve water flow in soils, as a method to solve these integration issues with guaranteed convergence and conservation of energy for any time step size. We review common modeling approaches, focusing on the fixed-grid method and on frozen soil. Based on this, we develop a conservative formulation of the governing equation and outline problems of alternative formulations in discretized form. Then, we apply the nested Newton-Casulli-Zanolli (NCZ) algorithm to a one-dimensional finite-volume discretization of the energy-enthalpy formulation. Model performance is demonstrated against the Neumann and Lunardini analytical solutions and by comparing results from numerical experiments with integration time steps of one hour, one day, and ten days. Using our formulation and the NCZ algorithm, the convergence of the solver is guaranteed for any time step size. With this approach, the integration time step can be chosen to match the time scale of the processes investigated.

Download Full-text