Perfectly Matched Layer Research Articles

Tuberculosis detection using a low-loss and highly sensitive photonic crystal fiber technique in the terahertz regime

This research paper introduces a design and numerical investigation of a photonic crystal fiber (PCF) sensor with a hollow core. The sensor is specifically designed to detect Mycobacterium tuberculosis in the frequency range of 1–2 THz. The cladding region of the PCF sensor incorporates six rectangular air holes and an absorbing layer known as the perfectly matched layer (PML), which is utilized to examine various optical properties. Through a thorough investigation, our designed PCF sensor demonstrates enhanced sensitivity of 98.692%, 98.721%, 98.75%, and 98.764% for four different infected samples of tuberculosis at a frequency of 1.6 THz. The sensor exhibits minimal confinement loss and effective material loss. Furthermore, the numerical aperture and effective mode index of the proposed sensor are analyzed. An effective area of 346,550, 345,110, 343,710, and 343,020µm2 for the four different infected samples of tuberculosis is obtained. The fabrication methods employed for this sensor are mentioned. Overall, this PCF sensor can be promising for a wide range of chemical, gas, and biosensing applications.

A novel algorithm of the perfectly matched layer based on the Runge–Kutta method of order 2 accuracy

AbstractThe perfectly matched layer (PML) has the ability to terminate the finite‐difference time‐domain (FDTD) lattice successfully. It is crucial to investigate the interaction between electromagnetic waves and the PML. Utilizing the Runge–Kutta method of order 2 accuracy, we derive a novel and efficient complex frequency‐shifted (CFS) PML implementation algorithm, which is named the RK‐PML. To validate the RK‐PML, we provide two numerical examples, which include rectangular waveguide and plasma cube truncated by the PML. By analyzing the time‐domain relative reflection errors, the frequency‐domain reflection coefficients, the computational time, and the memory footprint, the proposed RK‐PML outperforms slightly the convolutional PML (CPML).

Using Perfectly Matched Layer in a GPU simulation of ultrasound NDT

Full Waveform Inversion (FWI) is a state-of-the-art method of image reconstruction with promising results for seismology and NDT. FWI estimates the velocity of wave propagation along the inspected object by iteratively comparing acquired data with simulated data. This method is computationally demanding as at least three simulations are needed per iteration, which motivates the research effort for fast simulation methods. When simulating acoustic waves within a finite domain, artificial reflections created at the boundaries yield unrepresentative signal acquisition in the simulation. These reflections disturb the computation of residuals in FWI, creating artifacts that result in inaccurate positioning of flaws and discontinuities. Various absorbing boundary conditions (ABCs) have been proposed to avoid these reflections, e.g., Higdon, Reynolds, and Perfectly Matched Layers (PML). Although Reynolds and Higdon have been used in other applications, they are not well suited for FWI for reasons such as weak absorption rate and excessive introduction of amplitude errors. On the other hand, the PML method, an ABC that gradually attenuates boundary reflections with the addition of layers with matched acoustic impedance, provides adequate attenuation for FWI and can be readily extended to elastic waves. In this work, we describe an implementation of PML in a parallel GPU-accelerated CUDA simulator, which is intended to be integrated into an FWI regression in future works. Finally, we provide examples demonstrating that PML can significantly reduce the domain of simulation of an ultrasound NDT while maintaining simulation accuracy.

Design of Quasi-Shaped spectroscopy based optical sensor for the detection of alcohol

In this research, an entirely novel optical sensor for alcohol detection based on quasi-shaped spectroscopy is offered. A quasi shape with a tightly bonded structure in the cladding region and a hexahedron core region makes up the recommended design, which has a remarkable sensitivity of about 91.35%, 92.55%, and 90.40%and low confinement losses (CLs) of 5.44 × 10−08 dB/m, 6.75 × 10−08 dB/m, and 5.85 × 10−08 dB/m for detecting alcohol like ethanol (n = 1.354), butanol (n = 1.3993), and propanol(n = 1.384), at 1 THz. The numerical aperture, the effective area, and the V-parameter have all been identified and assessed. The 0.8THz to 3 THz operational wavelength range is stated. In the COMSOL Multiphysics (Version 5.6) environment, the properties of these suggested alcohol sensors are numerically examined utilizing the FEM (finite element method) with full vectors. In contrast to other efforts, the proposed sensor, which consists of a quasi-lattice PCF with a perfectly matched layer (PML) of a circular air hole shape and TOPAS as the background material, aims to improve sensitivity response. The proposed sensor might be crucial in the alcohol detection process due to its superior sensitivity response, a single mode of operation across the entire operational terahertz range, and shallow confinement loss. So, this PCF can proficiently be pragmatic in many biological sensing and other THz technology domains.

Sound Field Modelling and Noise Reduction for a Forklift Power Compartment Based on Perfectly Matched Layer and Acoustic Packaging Design

The core goal of this paper is to put forward a feasible scheme of noise reduction for a target forklift on the basis of solving the problem of vibration and acoustic radiation from complex structures in infinite domain. Based on the previous report and vibration acceleration tests, the acoustic virtual wind tunnel model of forklift power compartment was established using finite element method and boundary element method, in which the perfectly matched layer was first applied to simulate the attenuation propagation of sound waves in air. In addition, according to the distribution characteristics of sound pressure field with different frequencies, the acoustic energy mainly radiated through the bottom and right side, and concentrated in the low frequency. Consequently, the acoustic packaging design for the whole forklift power compartment was presented, and a satisfying noise reduction effect was achieved.

The use of FFT algorithm for circumferential coordinate in axisymmetric analysis of solids by generalized finite difference method

A formulation is presented for GFDM analysis of axisymmetric viscoelastic bodies under general loading conditions. The proposed formulation reduces the dimension of analysis by one by using complex Fourier series for circumferential coordinate ((θ) dependency) of field variables. For θ (Fourier) transform computations, the discrete Fourier transform formulas with FFT algorithm are employed. Radiational effects are accounted for through the use of perfectly matched layer (PML) with the equation being modified in accordance with the proposed formulation. For the assessment of the formulation, several sample problems are presented. The results indicate its capability in estimating the response even for that requiring sensitive analysis, such as, those in wave propagation analysis.

Auto-tuning numerical method for acoustic wave simulation using analytical solution.

Numerical wavefield simulation such as commercial simulation software enables an optimal design of an ultrasound computed tomography (USCT) system for clinical purpose of before prototyping. Such simulator, not developed for optimal design though, can provide rapid implementation for acoustic wave propagation but may lead to unexpected errors during the establishment of numerical model. Here, we propose an auto-tuning numerical method (ATNM), aiming to optimize physical parameters (e.g. grid size, Courant-Friedrichs-Lewy (CFL) number, perfectly matched layer (PML) absorption coefficient, etc) such that the enumerated wavefield computed on those converges to the corresponding analytical solution derived from acoustic scattering theory. We use genetic algorithm (GA) to automatically calibrate numerical wavefield. Our preliminary test is to investigate the best design of PML absorption coefficient for USCT to minimize mean relative error (MRE) between the k-Wave simulation and the analytic model and show its efficacy. The experimental results verify our hypothesis that this calibrated numerical simulator on a simple physical domain is generalizable to any other domains.

A high-order perfectly matched layer scheme for second-order spectral-element time-domain elastic wave modelling

In this study, we develop a new high-order perfectly matched layer (PML) formulation for the 3D spectral-element time-domain (SETD) method to solve the second-order elastic wave equations in terms of displacements. The high-order PML consists of multiple well-known frequency-dependent complex stretching functions, leading to the complicated temporal convolution which requires large computational resources. To avoid the evaluation of convolution, we deal with the high-order stretching functions by introducing the recursive auxiliary first-order differential equations which can be easily solved in time domain. In addition, an equivalent concept of PML is adopted for absorbing waves, preserving the original form of wave equations in physical domain. Finally, we obtain a high-order, non-convolutional and unchanged-form PML formulation, which can be easily incorporated into the SETD codes designed for the physical domain, to make them applicable for the second-order wave equations in the unbounded media. Numerical experiments demonstrate the accuracy and stability of the high-order PML through the comparison with the reference solution. It is found that the high-order PML provides a much better absorption performance than the first-order PML, particularly for the grazing-incidence waves and surface waves in the large-scale domain. We also give a fluid-filled borehole model to illustrate that the high-order PML works well for absorbing the complex guided waves in the presence of fluid-solid interaction.

Well-Posedness and Convergence Analysis of PML Method for Time-Dependent Acoustic Scattering Problems Over a Locally Rough Surface

Abstract We aim to analyze and calculate time-dependent acoustic wave scattering by a bounded obstacle and a locally perturbed non-selfintersecting curve. The scattering problem is equivalently reformulated as an initial-boundary value problem of the wave equation in a truncated bounded domain through a well-defined transparent boundary condition. Well-posedness and stability of the reduced problem are established. Numerically, we adopt the perfect matched layer (PML) scheme for simulating the propagation of perturbed waves. By designing a special absorbing medium in a semi-circular PML, we show the well-posedness and stability of the truncated initial-boundary value problem. Finally, we prove that the PML solution converges exponentially to the exact solution in the physical domain. Numerical results are reported to verify the exponential convergence with respect to absorbing medium parameters and thickness of the PML.

Improved accuracy of plane-wave electromagnetic modelling by application of the total and scattered field decomposition and perfectly matched layers

SUMMARY In 2-D magnetotelluric modelling, the standard application of Dirichlet boundary conditions (BC) may severely diminish the solution accuracy, because the unknown scattered part of the electromagnetic field is erroneously reflected at the domain boundary. Therefore, we adapt the total and scattered field decomposition (TSFD) to geophysical modelling, enabling the application of fully absorbing boundary methods, here perfectly matched layers (PML), to the scattered field. Our novel TSFD divides the modelling domain into two regions. In the total-field region containing the area of interest, the solution is computed for the total field. In the scattered-field region containing the boundaries, the solution is obtained for the scattered field, which is fully absorbed by PML at the boundaries. The plane-wave source is excited at the TSFD interface between both regions. Thus, boundary reflections are eradicated leading to superior solution accuracy, and boundaries can be placed closer to the receivers, shrinking the computational problem. Especially for challenging models with strong lateral changes, the solution accuracy of the TSFD is superior to that of the standard Dirichlet approach. Owing to the linearity of Maxwell’s equations, the inaccuracy introduced to the electric and magnetic fields by using Dirichlet BC can be expected to partly cancel out in the magnetotelluric transfer functions, for example the impedance tensor. In this work, we quantify this cancellation effect. The inaccuracy is less than typical measurement errors in the vast majority of apparent resistivity and phase data, even, when the primary fields are strongly inaccurate.

Convergence analysis of time-domain PMLS for 2D electromagnetic wave propagation in dispersive waveguides

This work is dedicated to the analysis of generalized perfectly matched layers (PMLs) for 2D electromagnetic wave propagation in dispersive waveguides. Under quite general assumptions on frequency-dependent dielectric permittivity and magnetic permeability we prove convergence estimates in homogeneous waveguides and show that the PML error decreases exponentially with respect to the absorption parameter and the length of the absorbing layer. The optimality of this error estimate is studied both numerically and analytically. Finally, we demonstrate that in the case when the waveguide contains a heterogeneity supported away from the absorbing layer, instabilities may occur, even in the case of the non-dispersive media. Our findings are illustrated by numerical experiments.

A hybrid artificial boundary combining perfectly matched layer and infinite elements for analysing semi-infinite problems

Artificial boundaries are crucial for simulating the dynamic response of semi-infinite problems. In this paper, a novel hybrid artificial boundary for the simulation of the far field is proposed by combining the perfectly matched layer (PML) and infinite element method (IEM), while the near field is simulated by the finite element method (FEM). The new hybrid artificial boundary (PML-IEM) is devised such that it can enjoy the relative advantage of each method. To start, a brief description is given on the fundamentals of the PML, IEM and the proposed PML-IEM. Then, the procedure for implementing the FEM-PML (using finite elements) and the IEM (using infinite elements) in the stretched coordinates is derived. In the parametric analysis, the factors affecting the accuracy of the PML, IEM, and the proposed PML-IEM are assessed in details. Based on this, optimal parameters are selected for implementing the proposed PML-IEM. Through the examples, the accuracy and reliability of the hybrid method proposed for analysing the semi-infinite problems under the surface line loads are demonstrated to be superior to those by the pure PML or IEM.

Unconditionally Stable System Incorporated Factorization-Splitting Algorithm for Blackout Re-Entry Vehicle

A high-temperature plasma sheath is generated on the surface of the re-entry vehicle through the conversion of kinetic energy to thermal and chemical energy across a strong shock wave at the hypersonic speed. Such a condition results in the forming of a blackout which significantly affects the communication components. To analyze the re-entry vehicle at the hypersonic speed, an unconditionally stable system incorporated factorization-splitting (SIFS) algorithm is proposed in finite-difference time-domain (FDTD) grids. The proposed algorithm shows advantages in the entire performance by simplifying the update implementation in multi-scale problems. The plasma sheath is analyzed based on the modified auxiliary difference equation (ADE) method according to the integer time step scheme in the unconditionally stable algorithm. Higher order perfectly matched layer (PML) formulation is modified to simulate open region problems. Numerical examples are carried out to demonstrate the performance of the algorithm from the aspects of target characteristics and antenna model. From resultants, it can be concluded that the proposed algorithm shows considerable accuracy, efficiency, and absorption during the simulation. Meanwhile, plasma sheath significantly affects the communication and detection of the re-entry vehicle.

Double-pole unsplit complex-frequency-shifted multiaxial perfectly matched layer combined with strong-stability-preserved Runge-Kutta time discretization for seismic wave equation based on the discontinuous Galerkin method

The classical complex-frequency-shifted perfectly matched layer (CFS-PML) technique has attracted widespread attention for seismic wave simulations. However, few studies have addressed the double-pole variant of the CFS-PML scheme. The double-pole CFS-PML has a stronger capacity to absorb near-grazing incident waves and evanescent waves than the classical CFS-PML. Using the discontinuous Galerkin (DG) method, we derive a double-pole unsplit auxiliary ordinary differential equation CFS-multiaxial PML (AODE CFS-MPML) formulation, which combines a fourth-order strong-stability-preserved Runge-Kutta time discretization for wavefield simulation on an unstructured grid. The double-pole unsplit CFS-MPML formulations are obtained by introducing auxiliary memory variables and AODEs. The original stress-velocity equations and the double-pole unsplit AODE CFS-MPML equations are all first-order hyperbolic systems and suitable for the DG method. The attenuative variables are added directly to the original seismic wave equations without changing their formats. In contrast to the split perfectly matched layer (PML), we avoid reformulating PML equations in the nonattenuative modeling region. The original seismic wave equation is solved in the nonattenuative modeling domain, whereas the double-pole unsplit AODE CFS-MPML equation is implemented in the PML absorbing region. Three numerical examples validate the performance of the double-pole unsplit AODE CFS-MPML technique. The isotropic and anisotropic experiments demonstrate that our developed double-pole unsplit AODE CFS-MPML is more stable and obtains more accurate solutions than the classical CFS-PML. The second example indicates the flexibility of the combination of the DG method with the double-pole CFS-MPML on undulating topography. The final example displays the applicability and effectiveness of our method in a 3D situation.

The Half-Space Matching method for elastodynamic scattering problems in unbounded domains

In this paper, the Half-Space Matching (HSM) method, first introduced for scalar problems, is extended to elastodynamics, to solve time-harmonic 2D scattering problems, in locally perturbed infinite anisotropic homogeneous media. The HSM formulation couples a variational formulation around the perturbations with Fourier integral representations of the outgoing solution in four overlapping half-spaces. These integral representations involve outgoing plane waves, selected according to their group velocity, and evanescent waves. Numerically, the HSM method consists in a finite element discretization of the HSM formulation, together with an approximation of the Fourier integrals. Numerical results, validating the method, are presented for different materials, isotropic and anisotropic. Comparisons with the Perfectly Matched Layers (PML) method are performed for several anisotropic materials. These results highlight the robustness of the HSM method compared to the sensitivity of the PML method with respect to its parameters.

An arbitrary Lagrangian–Eulerian method for analyzing finite-amplitude viscous acoustic waves radiated from vibrational solid boundaries: An implicit method

This paper is concerned with the development of an arbitrary Lagrangian–Eulerian (ALE) method for predicting nonlinear acoustic waves radiated from large-amplitude vibrational solid boundaries immersed in viscous fluids of infinite extent. The method is formulated based on a nonlinear acoustic model characterized by velocity potential and acoustic pressure. Unsplit implementations of three perfectly matched layer (PML) techniques are presented, namely conventional PML, convolution PML, and complex frequency-shifted PML (CFS-PML), to enable bounded-domain modeling of nonlinear acoustic waves propagating in an unbounded fluid medium. The long-time stability characteristics of the three distinct PML techniques are examined through theoretical analysis and numerical verification of their ability to absorb evanescent waves. Our results reveal that the CFS-PML technique exhibits superior long-time stability performance compared to the other two techniques. The CFS-PML is also compared to two different absorbing layer techniques: the Kosloff Absorbing Layer (KAL) technique and the Acoustic Black Hole (ABH) technique. The finite element method is adopted for spatial discretization of the finite-amplitude acoustic wave model, and an implicit predictor–corrector algorithm is proposed for time advancement. Several benchmark problems, including acoustic waves produced by a vibrational piston, and transversely oscillating and pulsating cylinders, are analyzed by the proposed method. It is found that the proposed method attains a second-order rate of convergence for spatial discretization, and the convergence rate in terms of the time step size is about first-order. The application of the method to acoustic levitation problems is also demonstrated, and the nonlinear phenomenon of acoustic waves in a single-axis acoustic levitator is discussed.

Helmholtz-equation solution in nonsmooth media by a physics-informed neural network incorporating quadratic terms and a perfectly matching layer condition

Frequency-domain simulation of seismic waves plays an important role in seismic inversion, but it remains challenging in large models. The recently proposed physics-informed neural network (PINN), as an effective deep-learning method, has achieved successful applications in solving a wide range of partial differential equations (PDEs), and there is still room for improvement on this front. For example, PINN can lead to inaccurate solutions when PDE coefficients are nonsmooth and describe structurally complex media. Thus, we solve the acoustic and visco-acoustic scattered-field (Lippmann-Schwinger) wave equation in the frequency domain with PINN instead of the wave equation to remove the source singularity. We first illustrate that nonsmooth velocity models lead to inaccurate wavefields when no boundary conditions are implemented in the loss function. Then, we add the perfectly matched layer (PML) conditions in the loss function to better couple the real and imaginary parts of the wavefield. Moreover, we design new neurons by replacing the classical affine function with a quadratic function in the argument of the activation function to better capture the nonsmooth features of the wavefields. We find that the PML condition and the quadratic functions improve the results including handling attenuation and discuss the reason for this improvement. We also illustrate that a network trained to predict a wavefield for a specific medium can be used as an initial model of the neural network for predicting other wavefields corresponding to PDE-coefficient alterations and improving the convergence speed accordingly. This pretraining strategy should find applications in iterative full-waveform inversion and time-lag target-oriented imaging when the model perturbation between two consecutive iterations or two consecutive experiments is small.

A Numerical Mode Matching (NMM) Method for Optical Fibers With Kerr Nonlinearity

Recently, the reshaping of the transverse light field profile in nonlinear multimode fibers (MMFs) has attracted great attentions. However, the electromagnetic (EM) fields in such nonlinear media cannot be easily and accurately calculated by using the traditional computational electromagnetics methods due to the nonlinearity of Kerr optical materials in graded-index (GRIN) MMFs. The iterative division method based on the 2.5-D spectral element numerical mode matching (SNMM) method is developed to address this problem. The SNMM method is a semi-analytical method, which can effectively reduce the computational costs to obtain highly accurate solutions. Starting from the numerical results of SNMM, the iterative division method solves electromagnetic fields of the linear-to-nonlinear transition process together iteratively. In this work, several improvements are proposed to extend the SNMM for Kerr optical nonlinear MMFs. The governing equations with anisotropic permittivity with Kerr optical nonlinear media are established in the cylindrical coordinate system. The absorbing boundary condition (ABC) is introduced to replace the perfectly matched layer (PML) so that the eigenmodes of a waveguide can be obtained accurately. Numerical results show the validity, accuracy and advantages of the proposed method in simulated MMFs.

Enhanced mixed boundary for modeling infinite domain in 2.5D soil vibration analysis

In this paper, an enhanced mixed 2.5D boundary for treating the infinite domains of the half-space is proposed. Unlike the perfectly matched layer (PML) that has a fixed outer boundary, the proposed method replaces the fixed outer boundary by infinite elements (IFE) to allow waves to propagate uninterruptedly to infinity. Such an approach not only improves the absorption capacity of the PML, but reduces its dependence on the attenuation function. The proposed boundary in the stretched coordinates is formulated in detail, for which the criteria for selecting the reference wavenumbers for multi types of waves are deliberately analyzed. A comprehensive parametric study for assessing the accuracy of the proposed boundary is presented, where the superiority of the proposed boundary relative to the pure PML and IFE is demonstrated. Particularly, the optimal ranges of parameters of the proposed boundary are identified. For illustration, the proposed boundary is applied to analyzing the frequency and time domain responses of a surface train moving at sub- and super-critical speeds. The results indicate that the proposed method can accurately predict the wave propagation characteristics, including the Mach cone at super-critical speeds.

Time‐domain scaled boundary perfectly matched layer for elastic wave propagation

AbstractA novel artificial boundary method called the scaled boundary perfectly matched layer (SBPML) is proposed for the transient analysis of elastic wave propagation in unbounded domains. This method constructs a generalized perfectly matched layer (PML) around the finite interior subdomain based on a coordinate transformation inspired by the scaled boundary finite element method (SBFEM). It can model the infinite exterior subdomains of heterogeneous materials, featuring both parallel and radial straight‐line physical surfaces and interfaces. In addition, generally‐shaped (even nonconvex‐shape) artificial boundaries can be used to provide high flexibility in choosing the geometry of the finite interior subdomain. Two coordinate mapping technologies, namely the modified scaled boundary coordinate transformation from the SBFEM and the complex coordinate stretching from the PML, are successively utilized to construct the proposed method. By introducing the integral of stress as an auxiliary variable, the resulting SBPML subdomain is represented in the time domain using a mixed displacement‐stress unsplit‐field formulation. This has the added benefit of allowing for easy incorporation into standard dynamic finite element methods. The performance of the SBPML is demonstrated through several numerical examples, including P‐SV and SH wave propagation problems occurring in unbounded domains with complex geometries and heterogeneous material properties. The numerical results are evaluated in comparison to both reference solution and existing artificial boundary methods, ultimately demonstrating the flexibility, robustness, accuracy, and stability of the proposed approach.