Matched Layer Method Research Articles

Seismic metamaterial barriers for ground vibration mitigation in railways considering the train-track-soil dynamic interactions

With the rapid development of high speed rail system, ground vibration mitigation solutions are desperately needed. Based on the concepts of phononic crystals, seismic metamaterial, which is a novel vibration mitigation method, can theoretically yield excellent performance in shielding dynamic propagation waves in broad frequency bands. However, the application of seismic metamaterials in railway-induced vibration mitigation is a very recent and ongoing topic. Therefore, this study is the world's first to establish new and practical contribution towards a better understanding into the mitigation effects by seismic metamaterials for railway-induced ground vibrations. The seismic metamaterials are made of an array of concrete inclusions in this study. The dispersion theory for seismic metamaterials is proposed for analyzing the theoretical band gaps. A 3D coupled train-track-soil interaction model is also developed based on the multi-body simulation principle, finite element theory, and perfectly matched layers method using LS-DYNA. The dimensions of seismic metamaterials are determined based on the dominant frequencies of vibration accelerations in natural ground. When the seismic metamaterials are adopted in railway ground, the vibration responses are investigated in both time and frequency domains to illustrate the mitigation effects. Finally, the numbers of inclusions, initial distances, and train speeds are changed to investigate their influences on shielding effects. The novel insight stemmed from this study incites an original and better understanding into the attenuation of ground vibrations using seismic metamaterials in high speed railways.

Numerical investigation of sound transmission loss of circular plates embedded with acoustic black hole

The acoustic black hole (ABH) can be utilized to achieve aggregation of flexural wave in structures with the feature that the thickness gradually reduced to zero with a power exponent no less than 2. These characteristics could be applied in vibration reduction, noise attenuation or improving sound insulation. To investigate the sound transmission loss of circular plates embedded with acoustic black hole, series of vibro-acoustic coupling finite element models (FEM) for TL analysis of ABH circular plates were established by automatic matched layer (AML) method in this paper and the simulation results show that the circular plate with a single ABH embedded in the center position can significantly improve the sound transmission loss in stiffness-controlled region, and at the first-order resonance frequency.

Anti-Plane Wave in a Piezoelectric Viscoelastic Composite Medium: A Semi-Analytical Finite Element Approach Using PML

In the present study, anti-plane wave propagation has been considered in a piezoelectric viscoelastic composite medium where the surface has been coated with an infinitesimally thin and perfectly conducting electrode which is grounded. The frequency equation has been obtained analytically and the result has been found in well agreement with the classical B-G wave frequency equation which validates the result. Moreover, Semi-Analytical Finite Element using Perfectly Matched Layer (SAFE-PML) method has been adopted in order to deduce frequency equation. Numerical computation and graphical illustration have been done in order to study some influencing parameters. Both frequency equations obtained through analytical method and SAFE-PML method have been compared through graphs which also validate the obtained results.

Influences of piles on the ground vibration considering the train-track-soil dynamic interactions

Abstract High-speed trains can induce significant amplification of dynamic responses of components in railway tracks especially when the train travels at the so-called ‘critical speed’. Based on a critical literature review, most previous studies with respect to train-track-soil interactions have merely been focused on the simplified natural ground vibrations. Accordingly, there exists no investigation into the influences of piles on the ground responses despite the fact that the pile-reinforced ground improvement has been widely adopted in soft soil regions for high-speed railway with slab track systems. In order to highlight the influences of piles on ground vibrations, a 3D fully coupled train-track-soil model has been developed based on the multi-body simulation principle, finite element theory, and perfectly matched layers method using LS-DYNA, in which the dynamic material properties of slab tracks have been adopted. This model has been validated by comparing its results of ground vibrations and train-track interactions with field-test results. This is thus the world’s first to investigate the critical speeds of slab-track railway with natural and pile-reinforced ground improvement. The dynamic displacements, vibration velocities, and dynamic stresses of soils with natural and pile-reinforced grounds have then been evaluated under normal and critical train speeds. The accelerations of car body and dynamic impact factors with the increasingly train speed have also been presented. The piles influences on the wave propagations in the soils have been highlighted. The novel insight from this study provides a new and better understanding of ground vibrations in high-speed railway systems using slab tracks in practice.

Developing Perfectly Matched Layer method to solve Heat Equation numerically

Perfectly Matched Layer (PML) techniques, although having been studied extensively to study systems with nonrestrictive boundaries in many physical fields, are not readily adaptable to the study of thermodynamic systems governed by the heat equation. Using the explicit finite difference method, we can easily describe systems with perfectly absorbing or reflecting boundaries. These, however, are highly idealized physical states, so we have begun extending our abilities to simulating more realistic physical situations by defining arbitrary spatial differential operators, which govern how heat at the boundaries of the system of interest propagates. As a first attempt, we transformed the linear spatial domain to a trigonometric domain to combat reflection from boundaries, but this is quite crude. We then approached the problem using Fourier Transforms in order to define the problem in a heat distribution's frequency space, damp excessive heat past the boundaries, and transform back to position space. Initial results using the finite difference method verify the current computer simulation's ability to solve problems with ideal circumstances, and the development of a PML method which accurately simulates nonrestrictive boundaries and can be easily translated into a computer algorithm is in progress.

Perfectly Matched Layer Method for Acoustic Scattering Problem by a Locally Perturbed Line with Impedance Boundary Condition

Perfectly Matched Layer Method for Acoustic Scattering Problem by a Locally Perturbed Line with Impedance Boundary Condition

Effects of Acoustic Black Hole Parameters and Damping Layer on Sound Insulation Performance of ABH Circular Plate

The acoustic black hole (ABH) can be utilized to achieve aggregation of flexural wave in structures with the feature that the thickness gradually reduced to zero with a power exponent no less than 2. The above characteristics could be applied in vibration reduction, noise attenuation or improving sound insulation. Previous literatures on vibration and acoustic characteristics of ABH structures mainly focus on the structural response under mechanical force excitation, while the transmission loss (TL) of circular plates embedded with two-dimensional ABHs investigated in this paper is a vibro-acoustic coupling procedure under excitation of diffuse sound field. Series of vibro-acoustic coupling finite element models (FEM) for TL analysis of ABH circular plates were established by automatic matched layer (AML) method in this paper and an experimental platform for measuring TLs of ABH circular plates and uniform plates was constructed. The accuracy of the FEM analysis was verified by experimental measurements. To systematically analyze the influence mechanism of parameters of the ABH on TLs of ABH circular plates, the effects of diameter, orientation, number, and truncation thickness of ABHs on TLs of ABH circular plates were further studied. The effect of the damping layer on TLs of circular plates embedded with 1 and 19 ABHs was also analyzed and it reveals that the influence of damping layer mainly concentrates on the first-order resonance frequency and damping-controlled region of the plate, and at some frequencies, the greater the damping layer thickness, the worse the sound insulation performance, despite that the modal damping loss factor has been increased in the whole frequency domain.

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.

A local collocation method to construct Dirichlet-type absorbing boundary conditions for transient scalar wave propagation problems

This paper introduces an effective way to equip the standard finite element method (FEM) for the solution of transient scalar wave propagation problems in unbounded domains. Similar to many other methods, we truncate the unbounded domain at an artificial boundary and convert the problem into a bounded one by prescribing appropriate absorbing boundary conditions (ABCs) at the truncating boundary. In the present method, the ABCs are time-dependent, and they are constructed by a simple collocation approach which is local in space and time. Therefore, the method does not make use of any routine schemes such as Fourier and Laplace transform. We shall show that the method is simple, and it can be easily applied to an explicit time domain FEM approach so that the sparsity of the FEM scheme (as well as its efficiency) can be preserved. The proposed method does not require any auxiliary variables as well as any approximating differential operators. This feature roots from the fact that here the ABCs are Dirichlet-type (or first-type) and thus they can be easily imposed to the corresponding boundaries. Therefore, the method shares some similarities with the conventional 1st and 2nd order ABC methods in terms of the simplicity of implementation. The method employs basis functions that exactly satisfy the governing and dispersion wave equations. The basis function can be easily adjusted to act as outgoing waves transmitting energy from the interior domain (near field) towards exterior domain (far field); i.e, they can cope with satisfaction of radiation boundary conditions. Several numerical examples are presented to evaluate the performance and to demonstrate the effectiveness of the approach. We shall show that the present method is capable of yielding results with a proper level of accuracy, similar to that of the perfectly matched layers method (PMLs), and it performs stably even in the case of long-term computations.

An efficient approach for prediction of subway train-induced ground vibrations considering random track unevenness

In this paper, an efficient and robust approach is presented for the prediction of subway train-induced ground vibrations considering random track unevenness. The proposed approach is developed by a two-step approach. Firstly, the power spectral density functions of the wheel–rail interaction forces are obtained through the combination of the vehicle–track–tunnel–soil theoretical model and the pseudo-excitation method (PEM). Secondly, by assuming the geometry of the track, tunnel, and soil to be homogeneous along the track, the nonstationary random ground vibrations are solved by combining the PEM and the 2.5D finite element–perfectly matched layers method. To improve computational efficiency, an efficient wavenumber sampling scheme is proposed. In the numerical examples, the proposed approach is validated by comparing the results obtained with the published results. The influence of the train speed on the ground vibrations are also investigated, in which two types of track structure, namely the direct fixation track and floating slab track, are considered.

Time-domain PML formulation for modeling viscoelastic waves with Rayleigh-type damping in an unbounded domain: Theory and application in ABAQUS

The Perfectly Matched Layer (PML) method is a powerful approach to absorb outgoing waves from a finite computational domain in various media, such as elastic, poroelastic, anisotropic, and viscoelastic, by means of layers of artificial material placed at the finite domain boundaries. However, its use in most finite-element method (FEM) codes, which take into account viscous damping by employing only the Rayleigh and Lindsday damping formulation, is inadequate. This paper introduces a viscoelastic Perfectly Matched Layer (PML) formulation with Rayleigh-type damping for finite-element time-domain analyses. The PML formulations in the frequency- and time-domain are derived using a two-parameter complex coordinate stretching function. The displacement field is the only unknown variable in the formulation, so that the approach can be readily implemented in general finite-element codes. The weak form formulation in the time-domain is implemented in ABAQUS/Standard by merging it with a user-defined element (UEL) subroutine in Fortran90. The UEL is developed for 2D plane-strain (PE4ML) and plane-stress (PS4ML) four-node, linear, isoparametric, quadrilateral elements. The Hilber-Hughes-Taylor implicit time integration scheme is used to evaluate the unknown displacements, velocities, and accelerations. The validity and efficiency of the viscoelastic PML formulation and the UEL subroutine are examined with two numerical examples, namely a single-layer and a multi-layered soil profile with different damping ratios and properties. The results also highlight the poor performance of elastic PML layers, when they are placed at the boundaries of a viscoelastic interior domain, because, as the damping ratio increases, the reflection at the interface of the interior domain and the elastic PML domain increases. Long-time stability of the model is also examined, and no instabilities are observed.

Eigenmode and frequency domain analysis of the third-order microring filters

In this paper we compare results of the eigenmode analysis of the third-order add/drop filter with the transfer functions (S-parameters) calculated in wide spectral range covering telecommunication C band. Both type of calculations are performed using 2D finite elements (FEM) method. The computational domain is closed using standard perfectly matched layer method. Also, the model of add/drop microring filter based on the temporal coupled mode is presented, since such models are usually used as a starting point in filter design procedure. The obtained results show excellent agreement between the two approaches. It is well known that when single microring is coupled to access waveguides or another rings, the resonance frequency will deviate from its original isolated resonator value. This effect, known as coupling-induced resonance frequency shift, causes resonance frequency mismatches between individual resonators and thus significantly impacts eigen spectra, as well as transfer characteristics, of the coupled-resonator systems. FEM calculations show that this effect has no significant importance when coupling between the access waveguide and the microring is strong enough, i.e. when the conditions for flat filter response are satisfied.

Wavefield simulation of 3D borehole dipole radiation

Reflections in acoustic borehole-logging data can be used to image near-borehole geologic structures, using either monopole or dipole measurements, the latter of which resolve azimuthal ambiguities. Dipole methods in particular can characterize small and subtle fractures in reservoir environments. Accurate tools for wavefield simulation are required to account for wave modes in and out of the borehole, and for the receiver responses to these modes. We have evaluated a 3D elastic staggered-grid finite-difference method for isotropic and anisotropic media in the context of the borehole acoustic imaging problem. Based on the convolutional perfectly matched layer and the multiaxial perfectly matched layer schemes, a hybrid perfectly matched layer method was developed and implemented to reduce artificial boundary reflections. The superiority of this hybrid perfectly matched layer beyond the convolutional perfectly matched layer and the multiaxial perfectly matched layer was proven based on numerical benchmarks in isotropic and anisotropic media. Numerical simulation of radiation, reflection, and multipole reception of elastic waves, excited by a dipole source, was carried out. To optimize azimuthal detection, the relationships between S-wave polarization as well as source-receiver offset and source-reflector angle was analyzed. Results indicated that the S-S reflection is most sensitive to the angle between the incident ray and the normal to the reflector. Its maximum amplitude occurs as the incident angle reaches its critical value, a fact that can be used to calculate the total propagation distance of the S-S wave. The critical angle as well as the SH-wave velocities of geologic structures outside the borehole can thus be determined.

Modeling of guided circumferential SH and Lamb-type waves in open waveguides with semi-analytical finite element and Perfectly Matched Layer method

The circumferential guided waves (CGW) are of increasing interest for non-destructive inspecting pipes or other cylindrical structures. If such structures are buried underground, these modes can also deliver some valuable information about the surrounding medium or the quality of the contact between the pipe and the embedding medium. Toward this goal, the detailed knowledge of the dispersive characteristics of CGW is required; henceforth, the robust numerical method has to be established, which allows for the extensive study of the propagation of these modes under different loading conditions. Mathematically, this is the problem of the propagation of guided waves in an open waveguide. This problem differs significantly from the corresponding problem of a closed waveguide both in physical and numerical aspect. The paper presents a combination of semi-analytical finite element method with Perfectly Matched Layer technique for a class of coupled acoustics/elasticity problems with application to modeling of CGW. We discuss different aspects of our algorithm and validate the proposed approach against other established methods available in the literature. The presented numerical examples positively verify the robustness of the proposed method.

An Adaptive Perfectly Matched Layer Method for Multiple Cavity Scattering Problems

AbstractA uniaxial perfectly matched layer (PML) method is proposed for solving the scattering problem with multiple cavities. By virtue of the integral representation of the scattering field, we decompose the problem into a system of single-cavity scattering problems which are coupled with Dirichlet-to-Neumann maps. A PML is introduced to truncate the exterior domain of each cavity such that the computational domain does not intersect those for other cavities. Based on the a posteriori error estimates, an adaptive finite element algorithm is proposed to solve the coupled system. The novelty of the proposed method is that its computational complexity is comparable to that for solving uncoupled single-cavity problems. Numerical experiments are presented to demonstrate the efficiency of the adaptive PML method.

Convergence of Hardy Space Infinite Elements for Helmholtz Scattering and Resonance Problems

We perform a convergence analysis for discretizations of Helmholtz scattering and resonance problems obtained by Hardy space infinite elements. Superalgebraic convergence rates with respect to the number of Hardy space degrees of freedom are achieved. We consider spheres and piecewise polytopes as transparent boundaries. The analysis is based on a G\aa rding-type inequality and standard operator theoretical results. While the obtained results are related to those for the radial perfectly matched layer method, different techniques are used in the analysis.

A Cell-Based Smoothed Radial Point Interpolation—Perfectly Matched Layer Method For Two-Dimensional Acoustic Radiation Problems

We present a cell-based smoothed radial point interpolation method (CS-RPIM) model for two-dimensional acoustic radiating problem by incorporating the perfectly matched layer method (PML). In this work, the computational region, truncated by PML, is discretized into triangular background cells. Each cell is further divided into several smoothing cells, and then the cell-based gradient smoothing operation is implemented throughout the smoothing cells. The pressure field function is approximated using the RPIM shape functions. The supporting node selection for shape function construction uses the T2L-scheme associated with edges of the background cells. The cell-based gradient smoothing operation provides proper softening effect, and makes the acoustic stiffness of the CS-RPIM model much softer than that of the FEM (finite element method)/PML model, which in turn significantly reduces the numerical dispersion error. Numerical results show that, compared with FEM–PML, the CS-RPIM achieves better absorbing effect in the PML, and higher accuracy in the computational region. This enables us to conclude that the CS-RPIM model with the PML can be well applied in solving acoustic radiation problems.

Hardy space infinite elements for time harmonic wave equations with phase and group velocities of different signs

We consider time harmonic wave equations in cylindrical wave-guides with physical solutions for which the signs of group and phase velocities differ. The perfectly matched layer methods select modes with positive phase velocity, and hence they yield stable, but unphysical solutions for such problems. We derive an infinite element method for a physically correct discretization of such wave-guide problems which is based on a Laplace transform in propagation direction. In the Laplace domain the space of transformed solutions can be separated into a sum of a space of incoming and a space of outgoing functions where both function spaces are Hardy spaces of a curved domain. The Hardy space is constructed such that it contains a simple and convenient Riesz basis with small condition numbers. In this paper the new method is only discussed for a one-dimensional fourth order model problem. Exponential convergence is shown. The method does not use a modal separation and works on an interval of frequencies. Numerical experiments confirm exponential convergence.

Modification of exhaust muffler of a diesel engine based on finite element method acoustic analysis

In order to improve the acoustic attenuation performance of an exhaust muffler of a 175 series of agricultural diesel engine, automatic matched layer method of finite element is adopted on the basis of LMS Virtual.Lab software to simulate the non-reflecting boundary conditions, which can avoid the complex calculation and then figure out the value of propagated sound power directly and finally obtain the transmission loss of the exhaust muffler. Compared with the experimental data, it can be found that the error between the simulation and measured values is small, and it can be accurately simulated for the acoustic performance of the exhaust muffler at the frequencies smaller than 3000 Hz, which verifies the validity of the acoustic solution. An improved design that properly distributes the insertion length of intubation, increases the length–diameter ratio, and adds the length of the first expansion cavity is proposed for the poor acoustic attenuation performance in low and medium frequencies. Compared with the original design, the transmission loss value at low and medium frequencies obviously increases, so the acoustic attenuation performance at the frequencies becomes better.

Numerical modeling of three-dimensional open elastic waveguides combining semi-analytical finite element and perfectly matched layer methods

Among the numerous techniques of non-destructive evaluation, elastic guided waves are of particular interest to evaluate defects inside industrial and civil elongated structures owing to their ability to propagate over long distances. However for guiding structures buried in large solid media, waves can be strongly attenuated along the guide axis due to the energy radiation into the surrounding medium, usually considered as unbounded. Hence, searching the less attenuated modes becomes necessary in order to maximize the inspection distance. In the numerical modeling of embedded waveguides, the main difficulty is to account for the unbounded section. This paper presents a numerical approach combining a semi-analytical finite element method and a perfectly matched layer (PML) technique to compute the so-called trapped and leaky modes in three-dimensional embedded elastic waveguides of arbitrary cross-section. Two kinds of PML, namely the Cartesian PML and the radial PML, are considered. In order to understand the various spectral objects obtained by the method, the PML parameters effects upon the eigenvalue spectrum are highlighted through analytical studies and numerical experiments. Then, dispersion curves are computed for test cases taken from the literature in order to validate the approach.