High-order Finite-difference Time-domain Method Research Articles

Crosstalk Analysis of Printed Circuit Board Traces With Right-Angled Bent Corners via Time Domain Hybrid Method

Discontinuous impedance exists when bent corners are introduced into the traces of printed circuit board (PCB). By combining transmission line (TL) equations, higher order finite-difference time-domain (FDTD) method, state variable method, and Norton's theorem, an efficient time domain hybrid method is proposed to realize the fast crosstalk simulation of the traces with right-angled bent corners (RABCs) in this work. First, the whole structure of the traces is decomposed into a cascade network, which consists of multiple equal length parallel microstrip lines (EL-PMSLs) and the RABC structures. Then, the crosstalk responses on these EL-PMSLs can be quickly solved by applying TL equations combined with the higher order FDTD method. With the application of Norton's theorem, the equivalent circuit model of whole PCB traces is constructed. Finally, the equivalent circuits of the RABC structures are extracted, and the port voltages of these RABCs are calculated via state variable method and fed back to these EL-PMSLs as boundaries, thus the round-trip transmission of interference signals on the PCB traces is realized. Numerous crosstalk simulations of PCB traces with typical RABC structures, as well as extracted from a practical PCB, are presented and compared with the results of commercial software CST to verify the correctness and efficiency of the proposed method.

A high-order finite-difference scheme for time-domain modeling of time-varying seismoelectric waves

Simulation of the seismoelectric effect serves as a useful tool to capture the observed seismoelectric conversion phenomenon in porous media, thus offering promising potential in underground exploration activities to detect pore fluids such as water, oil, and gas. The static electromagnetic (EM) approximation is among the most widely used methods for numerical simulation of the seismoelectric responses. However, the static approximation ignores the accompanying electric field generated by the shear wave, resulting in considerable errors when compared to analytical results, particularly under high-salinity conditions. To mitigate this problem, we have adopted a spatial high-order finite-difference time-domain method based on Maxwell’s full equations of time-varying EM fields to simulate the seismoelectric response in 2D mode. To improve the computational efficiency influenced by the velocity differences between seismic and EM waves, different time steps are set according to the stability conditions, and the seismic feedback values of EM time nodes are obtained by linear approximation within the seismic unit time step. To improve the simulation accuracy of the seismoelectric response with the time-varying EM calculation method, finite-difference coefficients are obtained by solving the spatial high-order difference approximation based on the Taylor expansion. Our method yields consistent simulation results compared to those obtained from the analytical method under different salinity conditions, thus indicating its validity for simulating seismoelectric responses in porous media. We further apply our method to layered and anomalous body models and extend our algorithm to three dimensions. Results indicate that the time-varying EM calculation method can effectively capture the reflection and transmission phenomena of the seismic and EM wavefields at the interfaces of contrasting media. This may allow for the identification of abnormal locations, thus highlighting the capability of seismoelectric response simulation to detect subsurface properties.

Numerical modeling for karst cavity sonar detection beneath bored cast in situ pile using 3D staggered grid finite difference method

Karst cavities beneath bored cast in situ piles constitute substantial dangers to the stability and quality of underground construction projects. Therefore, it is important to detect and image these karst cavities before the construction of piles for such projects. New sonar detection methods, which have advantages such as low cost and high efficiency, can ensure the quality of the bedrock for underground construction projects. In this study, a three-dimensional (3D) hole-bedrock-cave model for the sonar detection of karst cavities was developed using the finite difference time domain (FDTD) method. The modeling parameters were calibrated with measured data, and the numerical results were then compared with measured field detection data to validate the reliability of the proposed method. The numerical results revealed the following findings. (1) The waves transmitted to the center of the pile hole bottom can be divided into three types: the waves traveling upward into the slurry that are absorbed by a perfectly matched layer (PML) and can be ignored, the waves traveling horizontally along the hole bottom that constitute the primary interference during the detection process, and the waves traveling downward into the bedrock that represent the effective detection signal. To minimize the negative effects of multiple reflections from the pile hole wall, we deployed a receiver at the midpoint between the transmission point and the hole wall, at which location the signals from the multiple reflections are symmetrical and easy to identify. (2) The reflected signals generated from different depths, sizes and directions of cavities have different travel time features that make it possible to estimate the depth, size and direction of a cavity beneath a pile. (3) The velocities of multiple surface wave reflections can be predicted in the frequency spectrum of a test signal and then used to predict the arrival times of multiple surface wave reflections and identify reflected P-waves. (4) Cavities have a focusing effect on waves when they are completely filled with a low-velocity medium. The energy of a wave reflected from the floor of a karst cavity after focusing is theoretically stronger than that reflected from the roof of a karst cavity. (5) PMLs can effectively absorb waves reflected from artificial boundaries to avoid false interference. High-order FDTD methods can eliminate numerical dispersion and reduce the computational costs. In general, the sonar detection of karst cavities beneath bored piles can be simulated using a 3D high-order staggered grid finite difference method and PMLs. This numerical modeling scheme is reliable and can improve the accuracy and feasibility of practical detection experiments.

A Kirchhoff approximation-based numerical method to compute multiple acoustic scattering of a moving source

Within the scope of a study of external noise propagation from moving ground vehicles, a numerical method is developed to compute the acoustic field emitted by a moving source in the presence of scattering objects such as roads, buildings or noise-shields. This method is developed with the purpose of being used in a vehicle design process and therefore it must have a low computational cost, which requires a certain number of approximations. The case of a fixed point source is studied first then the effect of a movement of the source is taken into account through the introduction of a retarded time. The acoustic source is assumed to be represented by one or many harmonic monopoles of possibly different frequency moving with a constant speed in a quiescent flow field. Scattering from nearby perfectly reflecting objects is computed through a Kirchhoff–Helmholtz integral equation applying the Kirchhoff approximation. A ray-surface intersection algorithm to compute shadow areas is proposed. The method is validated against analytical solutions and experimental results for a fixed source, and against a higher-order finite difference time-domain method for the multiple scattering of a moving source. Results are good and show that this method can potentially be used to predict urban noise.

Study and analysis of a novel Runge–Kutta high‐order finite‐difference time‐domain method

In this study, a novel high-order finite-difference time-domain (HO-FDTD) method, Runge–Kutta (RK)-HO-FDTD, is proposed. The RK-HO-FDTD method employs the strong stability preserving RK algorithm to substitute time derivates and the Taylor series to replace spatial derivates. The characteristics of the stability, dispersion and convergence are studied. The proposed new method presents a better numerical dispersion and a faster convergence rate both in time and space domain. Compared with the HO-FDTD method, if the mesh size is fixed, it is found that the computational memory of the RK-HO-FDTD method is more than two times of the HO-FDTD method for the same mesh size; but if keeping the same accuracy, the computational cost of the RK-HO-FDTD is the controllable times that of the HO-FDTD method. And compared with the HO-FDTD and RK-MRTD methods, the new scheme presents more accuracy and great potential in electromagnetic problems.

High order FDTD methods for transverse magnetic modes with dispersive interfaces

A new finite-difference time-domain (FDTD) algorithm is introduced to solve two dimensional (2D) transverse magnetic (TM) modes with a straight dispersive interface. Driven by the consideration of simplifying interface jump conditions, the auxiliary differential equation of the Debye constitution model is rewritten to form a new Debye–Maxwell TM system. Interface auxiliary differential equations are utilized to describe the transient changes in the regularities of electromagnetic fields across a dispersive interface. The resulting time dependent jump conditions are rigorously enforced in the FDTD discretization by means of a matched interface and boundary scheme. Higher order convergences are numerically achieved for the first time in the literature in 2D FDTD simulations of dispersive inhomogeneous media.

GPR simulation based on complex frequency shifted recursive integration PML boundary of 3D high order FDTD

When applying the finite difference time domain (FDTD) method in Ground Penetrating Radar (GPR) simulation, the absorbing boundary conditions (ABC) are used to mitigate undesired reflection that can arise at the model's truncation boundaries. The classical PML boundary can make spurious reflection for the waves, such as reaching to the PML interface with near-grazing angles, low frequency waves or evanescent waves. The non-split complex frequency shifted PML which base on recursive integration (CFS-RIPML) has a good absorption effect for these interference waves. Meanwhile, the recursive integration, which does not need split field component, can overcome the shortcoming of CFS technique that needs more intermediate variable and large memory. In addition, the high-order FDTD can improve calculation accuracy and reduce the error caused by numerical dispersion effectively. In this paper, we derive the 3D high-order FDTD method with CFS-RIPML boundary and apply it in GPR simulation. The results show that the CFS-RIPML has significantly better absorption effect and lower reflections error than UPML and PML boundary. Compared with the two-order, the high-order FDTD can improve calculation accuracy effectively with the same grid size. Combination with CFS-RIPML boundary and high-order FDTD can improve the reliability and calculation accuracy of GPR and other geophysics numerical simulation.

An absorbing boundary condition for general dispersive medium and general FDTD spatial scheme

Based on the stretched coordinate perfectly matched layer (SC-PML) formulation and the auxiliary differential equation method, an absorbing boundary condition for general dispersive medium is presented, and applied to both the standard finite difference time domain (FDTD) method and the high-order FDTD method. The proposed D-H formulations are completely independent of the material properties of the FDTD computational domain. Thus they can be directly applied to the simulations involving arbitrary dielectrics. Numerical results show that compared with the CPML, the proposed method is versatile, has an improved absorbing performance and low computational complexity, and can substantially reduce the computational time.

An efficient GPU-based time domain solver for the acoustic wave equation

An efficient algorithm for time-domain solution of the acoustic wave equation for the purpose of room acoustics is presented. It is based on adaptive rectangular decomposition of the scene and uses analytical solutions within the partitions that rely on spatially invariant speed of sound. This technique is suitable for auralizations and sound field visualizations, even on coarse meshes approaching the Nyquist limit. It is demonstrated that by carefully mapping all components of the algorithm to match the parallel processing capabilities of graphics processors (GPUs), significant improvement in performance is gained compared to the corresponding CPU-based solver, while maintaining the numerical accuracy. Substantial performance gain over a high-order finite-difference time-domain method is observed. Using this technique, a 1 s long simulation can be performed on scenes of air volume 7500 m 3 till 1650 Hz within 18 min compared to the corresponding CPU-based solver that takes around 5 h and a high-order finite-difference time-domain solver that could take up to three weeks on a desktop computer. To the best of the authors’ knowledge, this is the fastest time-domain solver for modeling the room acoustics of large, complex-shaped 3D scenes that generates accurate results for both auralization and visualization.

High-order FDTD methods for transverse electromagnetic systems in dispersive inhomogeneous media

This Letter introduces a novel finite-difference time-domain (FDTD) formulation for solving transverse electromagnetic systems in dispersive media. Based on the auxiliary differential equation approach, the Debye dispersion model is coupled with Maxwell's equations to derive a supplementary ordinary differential equation for describing the regularity changes in electromagnetic fields at the dispersive interface. The resulting time-dependent jump conditions are rigorously enforced in the FDTD discretization by means of the matched interface and boundary scheme. High-order convergences are numerically achieved for the first time in the literature in the FDTD simulations of dispersive inhomogeneous media.

High-Order Interface Treatment Techniques for Modeling Curved Dielectric Objects

Three modified conformal techniques combined with high-order finite-difference time-domain method (FDTD (2, 4)) are proposed to investigate the interaction of electromagnetic waves with three-dimensional (3-D) curved dielectric surfaces. These proposed conformal techniques utilize an effective average dielectric constant to modify the update equations in the FDTD (2, 4) scheme, which is derived by area, linear, and volume averages of different dielectric regions in twenty seven spatial discrete cells at most, respectively. Some numerical results are presented to show the accuracies of linear and area average techniques. Good agreements are obtained with those of the method of moments and other analytical ones, even with coarse meshes adopted for handling electrically large or complex geometries effectively. Further, using our proposed techniques, the RCS of some typical 3-D objects are predicted and studied in detail.

A General Procedure for Introducing Structured Nonorthogonal Discretization Grids Into High-Order Finite-Difference Time-Domain Methods

This paper describes a general procedure for introducing structured nonorthogonal discretization grids into any high-order finite-difference time-domain method originally formulated on a uniform rectangular grid. The proposed procedure employs a coordinate transformation to map a conformal grid-a grid where all cells conform to the material boundaries of the given problem-onto a uniform rectangular grid where any high-order finite-difference scheme can be easily applied.

A Modified High Order FDTD Method Based on Wave Equation

A modified (2M, 4) scheme of the high-order two-dimensional finite-difference time-domain (FDTD) method based on wave equation is proposed. It has the fourth-order accuracy in the time domain using the symplectic integrator propagator, and the 2M-order accuracy in the space domain using the discrete singular convolution method. The distinctive features between the modified scheme and the traditional (2M, 4) FDTD based on Yee algorithm are listed as follows. First, the modified scheme is based on the wave equation. Second, the computational region is discretized by uniform mesh rather than the Yee mesh. Third, the modified scheme costs less memory than the Yee algorithm because fewer field elements are involved in computation. Numerical examples are provided to validate its accuracy and effectiveness

A higher order FDTD method in Integral formulation

We present a fourth-order (4, 4) finite-difference time-domain (FDTD)-like algorithm based on the integral form of Maxwell's equations. The algorithm, which is called the integro-difference time-domain (IDTD) method, achieves its fourth-order accuracy in space and time by taking into account the spatial and temporal variations of electromagnetic fields within each computational cell. In the algorithm, the electromagnetic fields within each cell are represented by space and time integrals (or integral averages) of the fields, i.e., the electric and magnetic fluxes (D,B) are represented by the surface-integral average, and the electric and magnetic fields (E,H) by the line and time integral average. In order to relate the integral average fields in the staggered update equations, we have obtained constitutive relations for these fields. It is shown that the IDTD update equations combined with the constitutive relations are fourth-order accurate both in space and time. The fourth-order correction terms are represented by the modified coefficients in the update equations; the numerical structure remains the same as the conventional second-order update equations and more importantly does not require the storage of field variables at the previous time steps to obtain the fourth-order accuracy in time. Furthermore, the Courant-Friedrichs-Lewy (CFL) stability criteria of this fourth-order algorithm turns out to be identical to the stability limits of conventional second-order FDTD scheme based on differential formulation.

High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

This paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high-order finite-difference time-domain (FDTD) schemes of computational electromagnetics (CEM) with material interfaces in one (1D) and two spatial dimensions (2D). By making use of fictitious points, systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage, to arbitrarily high orders of accuracy in principle. While often limited by numerical instability, orders up to 16 and 12 are achieved, respectively, in 1D and 2D. Detailed stability analyses are presented for the present approach to examine the upper limit in constructing embedded FDTD methods. As natural generalizations of the high-order FDTD schemes, the proposed derivative matching methods automatically reduce to the standard FDTD schemes when the material interfaces are absent. An interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code. Another feature of the present approach is that it can be robustly implemented with other high accuracy time-domain approaches, such as the multiresolution time-domain method and the local spectral time-domain method, to cope with material interfaces. Numerical experiments on both 1D and 2D problems are carried out to test the convergence, examine the stability, access the efficiency, and explore the limitation of the proposed methods. It is found that operating at their best capacity, the proposed high-order schemes could be over 2000 times more efficient than their fourth-order versions in 2D. In conclusion, the present work indicates that the proposed hierarchical derivative matching methods might lead to practical high-order schemes for numerical solution of time-domain Maxwell's equations with material interfaces.

A generalized higher order finite-difference time-domain method and its application in guided-wave problems

In this paper, a (2M,4) scheme of the finite-difference time-domain (FDTD) method is proposed, in which the time differential is of the fourth order and the spatial differential using the discrete singular convolution is of order 2M. Compared with the standard FDTD and the scheme of (4, 4), the scheme of (2M, 4) has much higher accuracy. By choosing a suitable M/spl ges/2, the (2M, 4) scheme can arrive at the highest accuracy. In addition, an improved approximation of the symplectic integrator propagator is presented for the time differential. On the one hand, it can directly simulate unlimited conducting structures without the air layer between the perfectly matched layer and inner structure; on the other hand, it needs only a quarter of the memory space required by the Runge-Kutta time scheme and requires one third of the meshes in every direction of the standard FDTD method. By choosing suitable meshes and bandwidth M, our scheme not only retains higher accuracy but also saves memory space and CPU time. Numerical examples are provided to show the high accuracy and effectiveness of the proposed scheme.