Pseudospectral Time-domain Method Research Articles

Staggered Grid Pseudo-spectral Time-domain Method for Light Scattering Analysis

A pseudo-spectral time-domain (PSTD) method has been widely used for light scattering in a random medium because of its e-ciency to compute electromagnetic flelds in a large scale structure. However, artifacts appear when simulating a small particle scattering problem with the PSTD method. The particle size must be large enough compared to a cell size to reduce the appearance of the artifacts. In this paper, we propose a staggered grid PSTD method to overcome this problem. Being difierent from the PSTD method reported so far, electric and magnetic flelds are sampled on a Yee's lattice. We found that the artifacts are caused by neglecting a Nyquist spatial frequency component in spatial difierentiation and can be eliminated by using the staggered grid.

The Optimal Spatially-Smoothed Source Patterns for the Pseudospectral Time-Domain Method

Spatially-smoothed sources are often utilized in the pseudospectral time-domain (PSTD) method to suppress the associated aliasing errors to levels as low as possible. In this work, the explicit conditions of the optimal source patterns for these spanning sources are presented based on the fact that the aliasing errors are mainly attributed to the high spatial-frequency parts of the time-stepped source items and the source patterns are thereafter demonstrated to be exactly corresponding to the normalized rows of the Pascal's triangle. The excellent performance of these optimal sources is verified by the practical 1-D, 2-D and 3-D PSTD simulations and also compared with that of non-optimal sources.

Application of Alternating Direction Implicit (ADI) Algorithm to Staggered-grid PSTD Modeling of Electromagnetic Waves

By applying the fast Fourier transform (FFT) and inverse transform to represent the spatial derivatives, the pseudo-spectral time-domain (PSTD) method has achieved a spatial grid of only two points per wavelength while maintaining a high accuracy. Its computational speed can be further accelerated by applying alternating-direction implicit (ADI), which makes PSTD works even when the time step exceeds the maximum time step allowed under the Courant- Friedrich-Levy (CFL) condition. However, due to the unsuitability of FFT on non-continuou condition, conventional ADI-PSTD turns to be unstable in media with high property contrast. We have expanded current ADI-PSTD into staggered-grid ADI-PSTD. After applying the staggered- grid approach into our algorithm by shifting the spatial derivatives halfway between 2 adjacent nodes, the stability of the difierentiation operators is enhanced. We will discuss the staggered- grid ADI-PSTD method in detail and apply it to the simulation of a half-space model. The analysis and discussion of the results show its advantages and further potential.

Pseudospectral time-domain method for pressure-release rough surface scattering using a surface transformation and image method

When numerically analyzing acoustic scattering at a pressure-release rough surface, the conventional pseudospectral time domain (PSTD) method using Fourier transform requires rigorous stability conditions in order to solve the spatial derivative in the wave equation on the irregular boundaries between the two media due to the Gibbs phenomenon and short wavelength in air. To eliminate such disadvantages, a new algorithm is proposed based on the Fourier PSTD method utilizing a surface boundary transformation and an image method. Irregular surface boundaries are flattened by transformation and then an image method is applied to the half-space domain. The efficiency and accuracy of the proposed PSTD method are better than the conventional Fourier PSTD method. Numerical results are presented for a sloped and a sinusoidal pressure-release surface.

Simulation of shear wave propagation in a soft medium using a pseudospectral time domain method

Elastography applications require the use of efficient models to simulate the propagation of shear waves in soft media such as human tissues. These models are needed to improve understanding of the measured displacement field, to reconstruct the viscoelasticity of heterogeneous tissues, and to test inversion algorithms. This paper reports a numerical model based on a pseudospectral time domain method developed to simulate shear and compression wave propagation in an axisymmetric heterogeneous viscoelastic medium. This model was adapted to the study of soft tissues where the ratio between the compression and the shear wave velocity was about a thousand and validated in the homogeneous situation by comparison with an analytical model based on elastodynamic Green's functions. Displacements obtained experimentally using transient elastography are presented, compared with simulation results, and discussed.

An analytical derivation of the optimum source patterns for the pseudospectral time-domain method

In the computational electromagnetics and acoustics, spatially smoothed sources are often utilized to alleviate the aliasing errors in the pseudospectral time-domain (PSTD) algorithms. In our work, an analytical derivation of the optimum source patterns is presented according to the accurately derived expressions of the dominant source-introduced aliasing errors according to the circular discrete convolution and Tailor series expansion method. We quantitatively demonstrate, for the first time in literature, that the aliasing errors can be optimally suppressed and rapidly reduced to the negligible levels by these optimum patterns and with the increment of source cells. We also provide the different implementation schemes of the optimal patterns both for the soft and hard source cases. The numerical calculation and 1D PSTD transient simulations are conducted to verify the excellent performance of these optimum sources.

Application of time reversal mirror technique in microwave-induced thermo-acoustic tomography system

Microwave-induced thermo-acoustic tomography (MITAT) is a promising technique with great potential in biomedical imaging. It has both the high contrast of the microwave imaging and the high resolution of the ultrasound imaging. In this paper, the proportional relationship between the absorbed microwave energy distribution and the induced ultrasound source distribution is derived. Further, the time reversal mirror (TRM) technique based on the pseudo-spectral time domain (PSTD) method is applied to MITAT system. The simulation results show that high contrast and resolution can be achieved by the TRM technique based on PSTD method even for the received signals with very low signal-to-noise ratio (SNR) and the model parameter with random fluctuation.

Simulations of thermally induced photoacoustic wave propagation using a pseudospectral time-domain method

Most physical models used to evaluate thermally induced photoacoustic waves in biomedical applications are approximations based on assumptions necessary to obtain analytical results, such as thermal and stress confinements. In contrast, using numerical methods to solve the general photoacoustic wave equations gives detailed information on the wave phenomenon without requiring as many assumptions to be made. The photoacoustic wave generated by thermal expansion is characterized by the heat conduction theorem and the state, continuity, and Navier-Stokes equations. This study developed a numerical solution in axis-symmetric cylindrical coordinates using a pseudospectral time-domain scheme. The method is efficient for large-scale simulations since it requires only 2 grid points per minimum wavelength, in contrast to conventional methods such as the finite-difference time-domain method requiring at least 10-20 grid points. The numerical techniques included Berenger's perfectly matched layers for free wave simulations, and a linear-perturbation analytical solution was used to validate the simulation results. The numerical results obtained using 4 grid points per minimum wavelength in the simulation domain agreed with the theoretical estimates to within an absolute difference error of 3.87 x 10(-2) for a detection distance of 3.1 mm.

A Novel Unconditionally Stable PSTD Method Based on Weighted Laguerre Polynomial Expansion

A novel unconditionally stable pseudo-spectral time-domain (US-PSTD) method is implemented by virtue of a weighted Laguerre polynomial expansion. This novel algorithm preserves both unconditional stability and less mandatory spatial resolution, and hence is proved to be an efficient and accurate method. Numerical examples are included to validate the proposed algorithm.

Analysis of Dielectric Resonator Antenna Array by Using Unconditionally Stable Pseudospectral Time-Domain Method

To simulate the triangular dielectric resonator antennas (DRAs), an unconditionally stable technique called the alternating direction implicit (ADI) is developed for the pseudospectral time-domain (PSTD) algorithm in this letter. This time-domain method is used for solving Maxwell's equations, which is comparative higher efficiency over the traditional finite-difference time-domain (FDTD) method. Also, the multidomain strategy is employed to allow flexible treatment for internal inhomogeneities. Numerical results are demonstrated with the unconditional stability and the second-order accuracy for the ADI-PSTD algorithm. To verify the DRA array design, reasonable agreement is obtained between theory and experiment.

Monte Carlo simulation of rough surface scattering using a modified pseudospectral time‐domain method.

A pseudospectral time‐domain (PSTD) method using a surface flattening transformation and image method is applied to a randomly rough pressure‐release surface scattering problem. Above, the PSTD method efficiently avoids the Gibbs phenomenon appearing in the conventional PSTD method when treating irregular boundaries with large impedance contrast. The pressure‐release surface is generated by Pierson–Moskowitz spectrum and Monte Carlo method is used to calculate the statistical properties of the rough surface. The Monte Carlo results for elemental scattering area will be investigated by comparing with Chapman–Harris formula.

A Hybrid Fast Multipole Pseudo-Spectral Time Domain Method

The major computation cost of pseudo-spectral method comes from the evaluation of differentiation matrix multiplication. In the past, uniform or Chebyshev collocation points are used for sampling. The differentiation matrix multiplication was evaluated by fast Fourier transform (FFT) or fast cosine transform (FCT), in order to reduce the computation complexity from O(N2) to O(N log(N)). However, the intrinsic properties of FFT or FCT may cause the wraparound effect and Gibbs phenomenon. Moreover, FFT or FCT is not applicable to other collocation points such as Legendre and Hermite. In order to improve the accuracy and applicability of the pseudo-spectral method, the fast multipole method (FMM) is exploited to substitute the FFT or FCT. By making use of the similarity of the N-body problem and the collocation problem, a new FMM-based pseudo-spectral time domain method is developed in this paper.

Application of the pseudospectral time-domain method to the scattering of light by nonspherical particles

The pseudospectral time-domain (PSTD) method is a powerful approach for computing the single-scattering properties of arbitrarily shaped particles with small-to-moderate-sized parameters. In the PSTD method, the spatial derivative approximation based on the spectral method is more accurate than its counterpart based on the finite-difference technique. Additionally, the PSTD method can substantially diminish accumulated errors that increase with the spatial scale and temporal duration of simulation. We report on the application of the PSTD method to the scattering of light by nonspherical ice particles. The applicability of the PSTD method is validated against the Lorenz-Mie theory and the T-matrix method. The phase functions computed from the PSTD method and the Lorenz-Mie theory agree well for size parameters as large as 80. Furthermore, the PSTD code is also applied to the scattering of light by nonspherical ice crystals, namely, hollow hexagonal columns and aggregates, which are frequently observed in cirrus clouds. The phase functions computed from the PSTD method are compared with the counterparts computed from the finite-difference time-domain (FDTD) method for a size parameter of 20 and an incident wavelength of 3.7 microm. The comparisons show good agreement between the two methods.

An improved acoustical wave propagator method and its application to a duct structure

The pseudospectral time-domain method has long been used to describe the acoustical wave propagation. However, due to the limitation and difficulties of the fast Fourier transform (FFT) in dealing with nonperiodic problems, the dispersion error is inevitable and the numerical accuracy greatly decreases after the waves arrive at the boundary. To resolve this problem, the Lagrange-Chebyshev interpolation polynomials were used to replace the previous FFT, which, however, brings in an additional restriction on the time step. In this paper, a mapped Chebyshev method is introduced, providing the dual benefit of preserving the spectral accuracy and overcoming the time step restriction at the same time. Three main issues are addressed to assess the proposed technique: (a) Spatial derivatives in the system operator and the boundary treatment; (b) parameter selections; and (c) the maximum time step in the temporal operator. Furthermore, a numerical example involving the time-domain evolution of wave propagation in a duct structure is carried out, with comparisons to those obtained by Euler method, the fourth-order Runge-Kutta method, and the exact analytical solution, to demonstrate the numerical performance of the proposed technique.

PSTD Method for Transient Electromagnetics and its Application to Optical Waveguide Analysis

A pseudo-spectral time-domain method, abbreviated as PSTD method, plays an important role in analyzing many kinds of scientific and engineering fields, especially large scale area problems such as weather forecast and scattering. The employment of the PSTD method is owed to the usage of the fast Fourier transform (FFT) for the spatial derivatives of Maxwell's equations. In this paper we discuss the PSTD method for the electromagnetic transient problem and its application to the analysis of optical waveguides with a long periodic grating, and show the experimental result for the waveguides fabricated based on the PSTD method discussed here.

Pseudo-spectral time-domain simulation of the transmission and the group delay of photonic devices

The pseudo-spectral time-domain (PSTD) method is applied to the simulation of photonic structures. We study transmission and group delay properties of dielectric and dispersive devices. An auxiliary differential equation technique is used to model the dispersive behavior of metallic features. Comparisons with the finite-difference time-domain method show the superiority of the PSTD method in particular when extracting phase information from a photonic device. The validity of the algorithm is demonstrated by investigating a ring resonator, Bragg gratings and metallic nano-particle arrangements.

High Precision Time Domain Forward Modeling for Crosshole Electromagnetic Tomography

To improve the resolution of crosshole electromagnetic tomography, high precision of forward modeling is necessary. A pseudo-spectral time domain (PSTD) forward modeling was used to simulate electromagnetic wave propagation between two boreholes. The PSTD algorithm is based on the finite difference time domain (FDTD) method and uses the fast Fourier transform (FFT) algorithm for spatial derivatives in Maxwell's equations. Besides having the strongpoint of the FDTD method, the calculation precision of the PSTD algorithm is higher than that of the FDTD method under the same calculation condition. The forward modeling using the PSTD method will play an important role in enhancing the resolution of crosshole electromagnetic tomography.

Strong ground motion simulation for seismic hazard assessment in an urban area

Seismic wave propagation and strong ground motion in a number of urban settings are simulated by using the pseudospectral time domain method with staggered grid real fast Fourier transform differentiation. For urban seismic study and damage prevention, simulations of the seismic wave propagation and ground motion need a comprehensive consideration of earthquake type, building structure and three-dimensional velocity structures of the crust and especially the near-surface sediments. The distribution of ground motion in an urban area varies greatly in response to geological structures. Wave interferences between the body wave and the secondary surface wave may result in peak ground motions and high collapse ratio of buildings in urban regions far away from the seismogenic fault. The simulated waveforms using an impulse point source in a 3D velocity structure can be employed to synthesize seismograms of ground motion as Green's function for any prescribed rupture scenario and source functions. The simulated results show that the synthesized seismograms coincide well with the observed seismic waveforms for the 1995 Hyogo-ken Nanbu M7.2 earthquake, which occurred in a heavily urbanized area. The simulation Green's function method is useful for predicting the strong ground motion in an urban region without observed seismic waveforms to provide a useful strong ground motion and hazard scenario for urban planning and earthquake hazard prevention. The present technique is an important supplement to the empirical Green's function method that heavily depends on the availability of local and regional seismicity.

A Hybrid PSTD/ADI-CFDTD Method for Mixed-Scale Electromagnetic Problems

We propose a hybrid technique combining the pseudospectral time-domain (PSTD) method with the alternating-direction implicit conformal finite-difference time-domain (ADI-CFDTD) method to solve 3-D mixed-scale problems in computational electromagnetics. A mixed-scale problem contains both electrically large and relatively homogeneous regions and electrically small fine details, thus poses a significant computational challenge to any single computational method if it is utilized alone. In particular, the ADI-CFDTD method is an unconditionally stable time-domain method with second-order spatial accuracy, and allows the time step to be increased beyond the Courant-Friedrichs-Levy limit; it is suitable for electrically small problem (structure details much smaller than a wavelength) but is inefficient and suffers from large numerical errors for electrically large-scale regions. The PSTD method, on the other hand, is accurate and efficient for regions with large, relatively homogeneous materials, but loses its efficiency for electrically small structures. The hybrid PSTD/ADI-CFDTD method overcomes these disadvantages and is potentially more useful than the individual solvers. The implementation details and numerical accuracy of this hybrid method are examined. Numerical examples demonstrate the advantages of the hybrid PSTD/ADI-CFDTD method

A 3-D Spectral-Element Time-Domain Method for Electromagnetic Simulation

A spectral-element time-domain (SETD) method is proposed to solve 3-D transient electromagnetic problems based on Gauss-Lobatto-Legendre polynomials. It has the advantages of spectral accuracy and block-diagonal mass matrix. With the inexpensive inversion of the block-diagonal mass matrix, the proposed method requires only a trivial sparse matrix-vector product at each time step, thus significantly reducing CPU time and memory requirement. Galerkin's method is used for spatial discretization, and a fourth-order Runge-Kutta scheme is employed for the time integration. The perfectly matched layer (PML) is employed to truncate the boundary in unbounded problems. The pseudospectral time-domain method is used to simplify the treatment of the PML inside the proposed SETD method. Numerical examples are shown to verify the efficiency and the spectral accuracy with the order of basis functions