Matched Layer Absorbing Boundary Condition Research Articles

Propagation Characteristics of Lightning Radiation Field on Three-Layer Vertical Layered Ground Based on CPML Absorption Boundary

The perfectly matched layer absorbing boundary condition based on coordinate scaling transformation (CPML) has a good absorption effect in finite-difference time-domain method (FDTD) calculation. In this study, we derive the CPML update equations in the 2-D cylindrical coordinate system by setting an appropriate coordinate-stretching variable, which can more easily achieve CPML and obtain good results in the calculation of lightning electromagnetic field. Then, the propagation characteristics of the lightning return stroke radiation field on the three-layer vertical layered ground structure are studied. The results show the following: 1) at 10 m above the ground, the vertical electric field and azimuthal magnetic field are almost not affected by the vertically layered ground structure, while the horizontal electric field is slightly affected by the vertical layered ground structure and 2) at 3 m below the ground, the lightning radiation fields are all affected by the vertically layered ground structure, which the vertical electric field is the most affected, and it can also use two-layer vertical layered ground structure instead of three-layer vertical layered ground structure for FDTD calculation under certain conditions, which can simplify the calculation model. All the results are obtained at a close distance from the lighting channel.

A procedure for 3D simulation of seismic wave propagation considering source‐path‐site effects: Theory, verification and application

AbstractThis paper aims at obtaining a semi‐analytical and semi‐numerical 3D model of source‐to‐site seismic wave propagation due to kinematic finite‐fault sources. To this end, a two‐step procedure integrating the frequency‐wavenumber (FK) approach with the spectral element method (SEM) is proposed based on the concept of domain reduction. First, the broadband responses of a stratified crust are accurately calculated by using a novel FK approach and are converted into effective seismic inputs around the region of interest. After that, the seismic wavefields at local and regional scales arising from complex geological and topographical conditions are finely simulated using the SEM, and a perfectly matched layer absorbing boundary condition is simultaneously applied to realize the absorption of outgoing waves. In this procedure, a hybrid source modelling scheme that combines the low‐wavenumber deterministic and high‐wavenumber stochastic components on the fault plane is introduced, effectively addressing the high‐frequency motion radiated from the source rupture process. Subsequently, the proposed FK‐SEM procedure is verified step‐by‐step using the point source and finite‐fault source models. To illustrate the feasibility of the procedure, 3D physics‐based numerical simulations (PBSs) of two seismic events, including a historical Sanhe‐Pinggu earthquake and a well‐recorded Yangbi earthquake, are performed. The case studies validate that the proposed FK‐SEM procedure allows a significant reduction in computational effort and a substantial improvement in modelling resolution and can be applied to the source‐to‐site broadband synthetics of earthquake scenarios with limited resources. In addition, this coupled geophysics‐engineering simulation meets the requirements of time‐history analysis for engineering structures, which facilitates the study of soil‐structure interactions and regional‐scale building damage assessment.

The effect of soil improvement and auxiliary rails at railway track transition zones

Railway track transition zones are areas where there is a sudden change in the track-ground structure. They include changes between ballasted and slab track, bridge approaches, and tunnel entry/exits. They are often the location of rapid track deterioration, and therefore this paper investigates the use of auxiliary rails and soil improvement to minimise train-track-ground dynamic effects. To do so, a 3D finite element model is developed using eight-node solid elements and a perfectly matched layer absorbing boundary condition. A moving train load is simulated using a sprung mass model to represent train-track interaction. After presenting the model, it is validated against field data collected on both a plain line and at a transition zone. Once validated, a sensitivity study is performed into auxiliary rails and soil improvement. It is found that auxiliary rails can improve the dynamic characteristics of the track across the transition, and that more widely spaced auxiliary rails provide greater benefit compared to closely spaced ones. Regarding soil improvement, a large benefit is found, and for the material properties under investigation, the effect of soil stiffening is greater than using auxiliary rails.

Full-Vector Analysis of Bending Waveguides by Using Meshless Finite Cloud Method in a Local Cylindrical Coordinate System

By using a meshless finite cloud method, a full-vector mode solver in terms of transverse-magnetic-field components for optical dielectric waveguide bends is developed in a local cylindrical coordinate system. To construct the interpolation function, both the point collocation and the fixed reproducing kernel technique are utilized, and then with the help of the continuity conditions of the longitudinal field components and the perfectly matched layer absorbing boundary conditions, a standard matrix eigenvalue is obtained. Compared with the conventional meshed numerical techniques, the distributions of solution points of the present meshless method can be applied to the area of complexity in a completely free manner. Consequently, excellent convergence with high accuracy can be realized. Moreover, the present method can be applied to both the uniform and non-uniform nodal distributions, thus the computational procedure is relatively flexible. To validate the proposed method, both fundamental and high-order leaky modes for a typical rib bending waveguide are analyzed and their modal field distributions and corresponding effective refractive indexes are presented. Results are in good agreement with those published earlier, showing the effectiveness of the present method.

Theoretical analysis of large negative dispersion photonic crystal fiber with small confinement loss.

Solid core circular and octagonal photonic crystal fibers (CPCF and OPCF) are proposed for analyzing different guiding properties such as dispersion, effective mode area, nonlinearity, and confinement loss from 0.8 to 2.6 µm wavelength. The proposed structures use three different types of background materials: SF10, BK7, and silica. Moreover, the fill fraction is varied by changing the diameter of the air hole where the lattice pitch is unchanged. The proposed PCFs show a high negative dispersion with low confinement loss and small effective mode area. In the proposed design, the finite element method with a perfectly matched layer absorbing boundary condition is used. At 1.8 µm wavelength with 0.6 fill fraction, the maximum negative dispersion of -922.5ps/(nm.km) is observed for CPCF when the background material is SF10. In addition, at this particular wavelength, the confinement loss is observed to be very small. Moreover, -560.12ps/(nm.km) dispersion is found for the similar condition at 1.55 µm wavelength. On the other hand, using BK7 as the background material, -706.77ps/(nm.km) dispersion is found at 1.55 µm wavelength for CPCF. Results also imply that CPCF shows better performance than OPCF for a wide wavelength range. Furthermore, at 1.55 µm wavelength, silica-based glass exhibits maximum dispersion, whereas increasing wavelength flint type glass shows the similar result. Analyzing different guiding properties of PCFs has significant impact on broadband dispersion compensation applications, especially using SF10.

PML absorbing boundary condition for structure-preserving seismic wave modeling

Some efficient structure preserving algorithms have been proposed for high precision seismic exploration and detection in recent years. However, feasible artificial boundary conditions to truncate unbounded medium are never discussed in structure preserving seismic modeling. We pay our emphasis on application of perfectly matched layer absorbing boundary condition (PML ABC) to fit the structure preserving algorithms. In the domain of interest, a typical structure preserving algorithm is presented to discretize the conservative dynamic system, while in the PML region, traditional central difference is applied to solve the dissipative system. In the adjacent region of the interior and exterior domains, the temporary values of wave field between two time steps are approximated by interpolation using the present and the updated value of the wave fields in the PML region. A numerical experiment is adopted to demonstrate compatibility of the algorithms and no visible reflections of outgoing waves occur on the edge of interior domain.

Perfectly matched layer absorbing boundary conditions for Euler equations with oblique mean flows modeled with smoothed particle hydrodynamics.

Absorbing boundary conditions (ABCs) play a critical role in the simulation of sound or wave propagation problems. This paper proposes a technique of space-time transformed perfectly matched layer (PML) boundary condition implemented in a widely used mesh-free method called smoothed particle hydrodynamic (SPH) method, to absorb the outgoing sound waves with oblique shear mean flow. Special consideration is given to the particle features of the SPH, and the PMLs are formulated to correct the truncation error of SPH and absorb the outgoing wave at the same time, aiming to reduce the storage and computational cost in the infinite computational domain. Because the group velocity and phase velocity of the outgoing sound waves in the PMLs may be in different directions, exponentially growing pseudo reflections can result. The authors thus employ space-time transformation to eliminate the reflections effectively in PML boundaries for stable solutions. Moreover, a uniform framework of PML absorbing boundary conditions for Euler equations in the cases of arbitrary oblique mean flow and static media is derived. Finally, the present PML-SPH method with this stable absorbing boundary is applied to simulate sound waves propagating with mean flow. The obtained numerical results agree very well with the reference results.

Symplectic MRTD for solving three‐dimensional Maxwell equations based on optimised operators

A new set of symplectic operators for the symplectic multi-resolution time-domain (S-MRTD) method is obtained by using the time reversible constraint and the growth factor. Then, the perfectly matched layer absorbing boundary condition based on splitting field technique is introduced into the S-MRTD method and its iterative formula is derived. The stability and numerical dispersion of S-MRTD method with optimised symplectic operators are discussed in detail. Finally, the optimal S-MRTD method is applied to the calculation of electromagnetic radiation and scattering. Numerical calculation and simulation results show that the S-MRTD is more accurate and efficient than the traditional FDTD method.

Perfectly matched layer boundary conditions for the second‐order acoustic wave equation solved by the rapid expansion method

ABSTRACTWe derive a governing second‐order acoustic wave equation in the time domain with a perfectly matched layer absorbing boundary condition for general inhomogeneous media. Besides, a new scheme to solve the perfectly matched layer equation for absorbing reflections from the model boundaries based on the rapid expansion method is proposed. The suggested scheme can be easily applied to a wide class of wave equations and numerical methods for seismic modelling. The absorbing boundary condition method is formulated based on the split perfectly matched layer method and we employ the rapid expansion method to solve the derived new perfectly matched layer equation. The use of the rapid expansion method allows us to extrapolate wavefields with a time step larger than the ones commonly used by traditional finite‐difference schemes in a stable way and free of dispersion noise. Furthermore, in order to demonstrate the efficiency and applicability of the proposed perfectly matched layer scheme, numerical modelling examples are also presented. The numerical results obtained with the put forward perfectly matched layer scheme are compared with results from traditional attenuation absorbing boundary conditions and enlarged models as well. The analysis of the numerical results indicates that the proposed perfectly matched layer scheme is significantly effective and more efficient in absorbing spurious reflections from the model boundaries.

Complex frequency‐shifted multi‐axial perfectly matched layer for frequency‐domain seismic wavefield simulation in anisotropic media

ABSTRACTSeismic anisotropy has an important influence on seismic data processing and interpretation. Although the frequency‐domain seismic wavefield simulation has a problem of solving the large scale linear sparse matrix due to the computational limitations, it has some advantages over the time‐domain seismic wavefield simulation including efficient inversion using only a limited number of frequency components and easy implementation of multiple sources. To accurately simulate seismic wave propagation in the frequency domain, we also need to choose the absorbing boundary conditions to absorb artificial reflections from edges of the model as we do in the time domain. Compared with the classical boundary conditions including the perfectly matched layer and complex frequency‐shifted perfectly matched layer, the complex frequency‐shifted multi‐axial perfectly matched layer has been proven to effectively suppress the unwanted reflections at grazing incidence and solve the instability problem in the time‐domain seismic numerical modelling in anisotropic elastic media. In this paper, we propose to extend the complex frequency‐shifted multi‐axial perfectly matched layer absorbing boundary condition to the frequency‐domain seismic wavefield simulation in anisotropic elastic media. To test the validity of our proposed algorithm, we compare the results (snapshots and seismograms) of the frequency‐domain seismic wavefield simulation with those of the time‐domain modelling. The model studies indicate that the complex frequency‐shifted multi‐axial perfectly matched layer absorbing boundary condition is stable in the frequency‐domain seismic wavefield simulation in anisotropic media, and provides better absorbing performance than the complex frequency‐shifted perfectly matched layer boundary condition.

Smoothed particle hydrodynamics (SPH) model for coupled analysis of a damaged ship with internal sloshing in beam seas

The flooding of a damaged ship in waves is a complex process, often coupled with the internal and external liquid motion together with the ship hull motion. Paramount to the operation safety, in order to improve the prediction accuracy of ship motion during the flooding process, the strip theory is applied to study the dynamic response of the damaged ship in beam seas; a smoothed particle hydrodynamics (SPH) model is developed to consider the coupling effects of various factors including internal sloshing of intact cabins and damaged cabins and external waves. The numerical wave tank with a perfectly matched layer absorbing boundary condition is established and validated by the experimental results. The detailed sensitivity study is carried out focusing on the effects of damaged opening sizes, the relative position of opening, and the incident wave and the liquid loading conditions on the dynamic response of the damaged ship in regular beam waves. It is observed that the flooding process was slowed down and interrupted by the water exchanges at the damaged opening due to the dynamic motion. Compared with the opening facing the incident wave, the back one endangered the ship pronouncedly with large amplitude and frequency roll motion. It is also revealed that the liquid tank in the damaged ship imposes a significant influence on its rolling response. It is further demonstrated that the present SPH model is capable of handling the nonlinear phenomenon in a flooding process of a damaged ship.

Unsplit perfectly matched layer absorbing boundary conditions for second-order poroelastic wave equations

An unsplit perfectly matched layer (PML), an absorbing boundary condition, is proposed for the second-order Biot’s equations to simulate wave propagation in unbounded poroelastic media. The PML formulas are derived using a two-parameter complex stretching approach, resulting in a two-dimensional (2D) time-domain system which consists of four second-order displacement equations and six first-order auxiliary differential equations. The proposed PML was validated using three 2D benchmarks: (i) spurious reflection of PML at grazing incidence; (ii) application of PML in stratified porous media and (iii) the relationship between the material property and the stability of PML. The numerical results show that the outgoing body waves and surface waves can be effectively attenuated by the proposed PML with appropriate scaling factors and damping profiles. The proposed formulas are much more compact, making it more efficient in studying poroelastic wave phenomena in open domain.

Element-by-element parallel spectral-element methods for 3-D teleseismic wave modeling

Abstract. The development of an efficient algorithm for teleseismic wave field modeling is valuable for calculating the gradients of the misfit function (termed misfit gradients) or Fréchet derivatives when the teleseismic waveform is used for adjoint tomography. Here, we introduce an element-by-element parallel spectral-element method (EBE-SEM) for the efficient modeling of teleseismic wave field propagation in a reduced geology model. Under the plane-wave assumption, the frequency–wavenumber (FK) technique is implemented to compute the boundary wave field used to construct the boundary condition of the teleseismic wave incidence. To reduce the memory required for the storage of the boundary wave field for the incidence boundary condition, a strategy is introduced to efficiently store the boundary wave field on the model boundary. The perfectly matched layers absorbing boundary condition (PML ABC) is formulated using the EBE-SEM to absorb the scattered wave field from the model interior. The misfit gradient can easily be constructed in each time step during the calculation of the adjoint wave field. Three synthetic examples demonstrate the validity of the EBE-SEM for use in teleseismic wave field modeling and the misfit gradient calculation.

Near-zero dispersion flattened, low-loss porous-core waveguide design for terahertz signal transmission

We demonstrate a photonic crystal fiber with near-zero flattened dispersion, ultralower effective material loss (EML), and negligible confinement loss for a broad spectrum range. The use of cyclic olefin copolymer Topas with improved core confinement significantly reduces the loss characteristics and the use of higher air filling fraction results in flat dispersion characteristics. The properties such as dispersion, EML, confinement loss, modal effective area, and single-mode operation of the fiber have been investigated using the full-vector finite element method with the perfectly matched layer absorbing boundary conditions. The practical implementation of the proposed fiber is achievable with existing fabrication techniques as only circular-shaped air holes have been used to design the waveguide. Thus, it is expected that the proposed terahertz waveguide can potentially be used for flexible and efficient transmission of terahertz waves.

Implementation of Delayed Detached Eddy Simulation method to a high order spectral difference solver

Abstract The Delayed Detached Eddy Simulation (DDES) method based on Spalart–Allmaras (SA) turbulence model is implemented to a high order spectral difference solver for aeroacoustics problems in this study. A modified SA model by Crivellini et al. is used to eliminate the instability of the nonlinear source terms in the turbulence model. To speed up the unsteady simulation, the multi-time-step method based on the optimized Adams–Bashforth scheme is utilized for time marching. The perfectly matched layer absorbing boundary condition for Navier–Stokes equations is applied at the far-field and outflow boundary regions. The SA model is firstly validated with the high Reynolds number flow over a flat plate and the flow around a NACA0012 airfoil at 10° and 15° angles of attack. The numerical results are compared with the experimental or reference data by other researchers. The good agreement indicates that the high order spectral difference solver with SA model is accurate to compute the turbulence boundary layer. Then the developed DDES solver is validated with the backward facing step flow and the flow over a NACA0012 airfoil at 60° angle of attack. The computed results are compared with the experimental data, as well as the URANS results. The validation results show that the DDES solver is capable of computing the turbulence flow with boundary layer separations. Finally the noise from the flow past two cylinders in tandem is computed with the DDES solver. The computed results, including the time mean flow field, dynamic pressures on both cylinders and far field noise spectra are compared thoroughly with the experimental data provided by NASA, as well as the reference numerical results by other researchers. An assessment is made of the accuracy of the present high order SD solver versus the low order method. The results show that the high order SD solver with the DDES method is more accurate than low order solver for aeroacoustics numerical simulation with similar resolution, and capable of predicting accurately the noise for aeroacoustics problems with turbulence separations.

FDTD Modelling of 1D Photonic Crystal comprised of Double Positive and Double Negative Metamaterial for Thermal Masking Application with CPML Absorbing Boundary Condition.

<p>Periodic structures of double positive and double negative metamaterial of thickness of λ/4 is designed to stop long-wave infrared and mid-wave infrared frequencies for masking from infrared detection devices. Band-gaps are obtained by calculating reflection and transmission coefficients at probe point close to the front and back faces of the periodic structure. 1D- finite difference time domain method is implemented in Matlab to study the electromagnetic wave propagation which is incident normal to a periodic stack of double positive and double negative metamaterial of having refractive indices of 9 and -6 respectively, at centre wavelength. Drude model is adapted to model double negative medium. Band-gap obtained are compared with the conventional photonic crystal by replacing the double negative medium with a double positive medium with the magnitude of refractive index same as that of double negative medium. Band-gap obtained confirms the presence of Zero-n͂ band-gap in DPS-DNG photonic crystal which is wider than the reflection band in conventional photonic crystal; nearly twice in mid-wave infrared region and five times in long-wave infrared region. A novel and highly efficient convolutional perfectly matched layer absorbing boundary condition is used to terminate the infinite computational finite difference time domain lattices.</p>

Implemention of perfectly matched layer absorbing boundary condition for CN‐RPIM method

In this Letter, the perfectly matched layer (PML) absorbing boundary condition is incorporated into the unconditionally stable radial point interpolation meshless method based on Crank–Nicolson radial point interpolation method scheme. By splitting field variables in the whole computational domain, all variables are defined in the support domain of nodes near PML boundary, and an implicit time-stepping formulation is derived. Numerical experiment is implemented to demonstrate the performance of the implemented PML.

Fundamental Scheme-Based Nonorthogonal LOD-FDTD for Analyzing Curved Structures

In this letter, nonorthogonal locally 1-D finite-difference time domain based on fundamental scheme (F-LOD-NFDTD) is presented for analyzing electromagnetic curved structures. The formulation based on fundamental scheme with curvilinear coordinate is performed in convenient matrix-operator-free forms in the right-hand side of resultant equations. The convolutional perfectly matched layer absorbing boundary condition is also incorporated. The nonorthogonal grids are used to fully mesh the computational domain that leads to efficient computation. A comparison to the conventional LOD-NFDTD in terms of CPU time and memory requirements reveals the merits of the proposed F-LOD-NFDTD method in terms of lighter calculation burden and higher efficiency. Numerical results are presented to illustrate the significance of the proposed approach.

Reverse‐time migration from rugged topography using irregular, unstructured mesh

ABSTRACTWe developed a reverse‐time migration scheme that can image regions with rugged topography without requiring any approximations by adopting an irregular, unstructured‐grid modelling scheme. This grid, which can accurately describe surface topography and interfaces between high‐velocity‐contrast regions, is generated by Delaunay triangulation combined with the centroidal Voronoi tessellation method. The grid sizes vary according to the migration velocities, resulting in significant reduction of the number of discretized nodes compared with the number of nodes in the conventional regular‐grid scheme, particularly in the case wherein high near‐surface velocities exist. Moreover, the time sampling rate can be reduced substantially. The grid method, together with the irregular perfectly matched layer absorbing boundary condition, enables the proposed scheme to image regions of interest using curved artificial boundaries with fewer discretized nodes. We tested the proposed scheme using the 2D SEG Foothill synthetic dataset.

Reflection seismic waveform tomography of physical modelling data

Waveform tomography is commonly tested using numerically generated synthetic seismic data, before the method is applied to field seismic data. However, there are often noticeable differences between idealized synthetic data and real field data, and many factors in the field data, such as noise, irregular source/receiver geometry, affect the inversion solutions. For exploring the potential of reflection seismic waveform tomography, we presented a more realistic test than the synthetic data test, by applying it to physical modelling data, to reconstruct a laboratorial model with complex velocity variation. First, we provided a formulation of the perfectly matched layer absorbing boundary condition, associated with the second-order acoustic wave equation, in order to suppress artificial reflections from subsurface model boundaries in seismic waveform simulation and tomography. Then, we demonstrated the successful implementation of a layer-striping inversion scheme applicable to reflection seismic waveform tomography. Finally, we confirmed the effectiveness of frequency grouping, rather than a single frequency at each iteration, a strategy specifically for the frequency-domain waveform tomography.