Perfectly Matched Layer Research Articles

A stable model order reduction scheme for an acoustic finite element model with perfectly matched layers

Perfectly matched layers (PMLs) are often used in an exterior acoustic finite element model to simulate the Sommerfeld radiation condition. However, the time-domain model is often too large to efficiently handle the transient simulation due to the frequency-dependent stretching function and the number of elements needed per wavelength. Krylov subspace-based model order reduction can alleviate this problem by only matching the important moments of the original model. However, stability-preservation is not guaranteed, which is the key to perform time-domain simulation. This work proposes a stability-preserving model order reduction scheme for the time-domain PMLs. By assessing the stability conditions of time-domain PMLs, a two-step strategy is applied. Firstly, a one-sided split basis is used to preserve the form of Lyapunov function of the original model such that a stable intermediate reduced order model (ROM) results. Secondly, an unsplit basis is applied on the modal transformation of this intermediate ROM to reduce the size further. The proposed method is verified by numerical simulations.

A stable decoupled perfectly matched layer for the 3D wave equation using the nodal discontinuous Galerkin method

In outdoor acoustics, the calculations of sound propagating in air can be computationally heavy if the domain is chosen large enough for the sound waves to decay. The computational cost is lowered by strategically truncating the computational domain with an efficient boundary treatment. One commonly used boundary treatment is the perfectly matched layer (PML), which dampens outgoing waves without polluting the computed solution in the inner domain. The purpose of this study is to propose and assess a new perfectly matched layer formulation for the 3D acoustic wave equation, using the nodal discontinuous Galerkin finite element method. The formulation is based on an efficient PML formulation that can be decoupled to further increase the computational efficiency and guarantee stability without sacrificing accuracy. This decoupled PML formulation is demonstrated to be long-time stable, and an optimization procedure for the damping functions is proposed to enhance the performance of the formulation.

Adaptive-coefficient finite difference frequency domain method for time fractional diffusive-viscous wave equation arising in geophysics

The diffusive-viscous wave (DVW) equation is a widely used model to describe frequency dependent attenuation of seismic wave in fluid-saturated porous medium. In this paper taking power law frequency dependent attenuation into account, we first introduce a modified DVW equation (time fractional DVW equation) in which the first order temporal derivative of viscous term is replaced with a fractional order temporal derivative. In consideration of that most of the existing numerical simulations for seismic wave equations are based on time domain methods and truncation with some specific boundary conditions, we incorporate the absorbing boundary condition as complex-frequency-shifted (CFS) perfectly matched layer (PML) into the time fractional DVW equation, and then develop an adaptive-coefficient (AC) finite difference frequency domain (FDFD) method for numerical simulation. The corresponding analytical solution for homogeneous time fractional DVW equation is provided for model validation, and the effectiveness of the developed AC FDFD method is verified by some numerical examples including the homogeneous model and the layered model. Numerical results show that AC FDFD method is more accurate than the traditional 2nd-order FDFD method for numerical modeling of time fractional DVW equation with CFS PML absorbing boundary condition, while requiring similar computational costs.

Single hollow-core microstructure fiber biosensor for pancreatic cancer diagnosis in terahertz regime

Abstract Pancreatic cancer is a kind of malignant tumor that is difficult to detect in its early stages, developing rapidly and with a 5-year survival rate of only 5% to 10%. Therefore early diagnosis and discovery of pancreatic cancer are very important for the successful treatment of the disease. Here, we report a single hollow-core microstructural fiber (SHC-MSF) biosensor based on a ZEONEX substrate, which has been optimized for the early detection of pancreatic cancer biomarkers. The proposed SHC-MSF biosensor adopts a single-aperture structure to increase the contact range with assay analytes to improve the detection sensitivity. Its biosensing performance was numerically analyzed using a finite element method with a perfect matching layer. Numerical results demonstrated that the proposed MSF-biosensor presented ultra-high sensitivity (bilirubin: 105.55%, glucose: 105.34%, creatinine: 105.67%) and negligible confinement loss (bilirubin: 5.52×10-14 cm-1, glucose: 1.65×10-14 cm-1, creatinine: 5.57×10-14 cm-1) in the range of 0.3~2.0 THz. Moreover, the SHC-MSF biosensor could selectively detect and distinguish cancer markers of different concentrations in the blood to achieve a more accurate diagnosis of pancreatic cancer. Finally, fabrication tolerance analysis of the proposed MSF-biosensor is provided to ensure the feasibility of rapid preparation.

High birefringent slotted core slotted cladding porous core PCF for THz waveguide

A novel design featuring a porous core photonic crystal fiber (PC-PCF) is proposed in this paper for terahertz (THz) waveguide application. This work uses finite element method (FEM) and a boundary condition related to perfectly matched layer (PML) for analyzing the guiding properties. This methodology is supported by a fiber type featuring a slotted core and slotted cladding, utilizing Zeonex as the material. It offers high birefringence valued at 0.0978, small confinement loss of 10−14 dB/cm, a meager amount of effective material loss (EML) valued at 0.011697 dB/cm and a very high core power fraction (CPF) of 67.156%. Additionally, it shows a flattened pattern of dispersion measuring 0.4 ± 0.2 ps/THz/cm at an extensive frequency range of 0.5 THz to 2 THz. Other significant waveguide properties: effective refractive index, numerical aperture (NA), V-parameter, effective area, spot size, and beam divergence of the suggested fiber are numerically analyzed and conferred thoroughly. Regarding its mode of operation, the proposed fiber primarily functions in multimode, but it can also operate in single mode for higher porosity levels and lower core diameters. This unique characteristic renders the proposed PC-PCF structure well-suited for a broad range of applications, offering versatility as to its mode of operation.

FDTD simulation of beamforming shape of a woolly horseshoe bat (Rhinolophus luctus)

Abstract The practicality of the Finite-Difference Time-Domain (FDTD) method is employed to calculate the bat biosonar beamforming shape obtained from the three dimensional (3D) digital model of a woolly horseshoe bat (Rhinolophus luctus), as well as to acquire the nearfield sound pressure distribution of the transient sound field. The multiple sound pressure amplitude can be obtained by Fourier transform of the instantaneous sound pressure in the nearfield FDTD region. According to Kirchhoff integral, the far field normalized directivity pattern is obtained. The single point source is positioned at the center of the computational region, which is surrounded by four different number of perfectly matched layer (PML). The FDTD method is utilized to calculate the absorption of waves by the PML. The trustworthiness of the algorithm is authenticated by the outcomes. It demonstrates that FDTD can produce satisfactory outcomes for biosonar calculations on the woolly horseshoe bat.

Model order reduction of time-domain acoustic finite element simulations with perfectly matched layers

This paper presents a stability-preserving model reduction method for an acoustic finite element model with perfectly matched layers (PMLs). PMLs are often introduced into an unbounded domain to simulate the Sommerfeld radiation condition. These layers act as anisotropic damping materials to absorb the scattered field, of which the material properties are frequency- and coordinate-dependent. The corresponding time-domain model size is often very large due to this frequency-dependent property and the number of elements needed per wavelength. Therefore, to enable efficient transient simulations, this paper proposes a two-step method to generate stable reduced order models (ROMs) of such systems. Firstly, the modified and stable version of PMLs is projected by a one-sided split basis, which gives a stable intermediate ROM. Secondly, the intermediate ROM is modified to satisfy the stability-preserving condition by applying the modal transformation. Applying any one-sided model order reduction method on this modified model leads to a stable and small ROM. This two-step method is further extended to account for the locally-conformal PML model by reformulating it in curvilinear coordinates, which works for arbitrary convex truncated domains. The proposed method is successfully verified by several numerical simulations.

Simulation of seismic waves in fluid-solid coupled thermoelastic media

This research advances seismic wave propagation analysis by incorporating temperature effects within fluid-solid coupled media. Leveraging the Lord-Shulman theory, we develop propagation equations that apply thermoacoustic and thermoelastic concepts to the intricacies of fluid-solid-thermal coupled media. This method facilitates the energy exchange between different regions by using distinct wave propagation equations in various media and implementing appropriate boundary conditions at the interfaces. Subsequently, we use high-order staggered grids and formulate corresponding perfectly matched layer absorbing boundaries to enhance simulation accuracy and reduce computational demands. This approach is validated through two numerical examples and rock-physics experiments, highlighting the crucial influence of thermal variations on subsurface wave dynamics and the method’s effectiveness in synthesizing seismic records and simulating converted wave information. The results indicate that our method provides a new approach for analyzing seismic wave propagation characteristics in fluid-solid coupled environments with temperature variations.

An Improved Uniaxial Perfectly Matched Layer Based on Finite Element Method for Hyperbolic Media

An Improved Uniaxial Perfectly Matched Layer Based on Finite Element Method for Hyperbolic Media

Highly sensitive photonic crystal fiber chemical sensor for detecting carbon disulfide and bromoform in the THz regime

In this study, an innovative photonic crystal fiber (PCF) designed specifically for the detection of carbon disulfide and bromoform liquid chemicals within the THz frequency range was introduced. The PCF’s structural design was achieved using the Finite Element Method (FEM) and Perfectly Matched Layer (PML) boundary conditions within COMSOL Multiphysics, ensuring precision through appropriate numerical parameters. Six distinct configurations were developed, incorporating circular, square, rectangular slotted, benzene-shaped circular, and two elliptical core designs, as well as an eight-elliptical core design. The PCF models were constructed utilizing the dielectric material TOPAS for accurate simulation and analysis. Various crucial parameters of the proposed PCF were examined across a wide THz spectrum spanning from 0.2 to 1.2 THz. The PCF model exhibited a peak output at the operating frequency of 0.8 THz for the square-shaped core design, achieving a relative sensitivity of 96.891% for bromoform and 95.603% for carbon disulfide. Remarkably low material losses of 0.0081104 cm−1 for bromoform and 0.006703 cm−1 for carbon disulfide were observed, along with a core power fraction of 93.107% for bromoform and 94.263% for carbon disulfide. The effective area was determined to be 1.77 × 10−07 μm2 for bromoform and 1.70 × 10−07 μm2 for carbon disulfide, while the confinement loss measured 2.25 × 10−17 dB/cm for bromoform and 4.76 × 10−17 dB/cm for carbon disulfide. These superior attributes strongly suggest that this model will be crucial in applications like supercontinuum generation, sensing, and biomedical imaging.

Meshless method for wave propagation in poroelastic transversely isotropic half‐space with the use of perfectly matched layer

AbstractNumerical investigation of wave propagation in transversely isotropic poroelastic half‐space with the use of a new stretched coordinate system through the Meshless Local Petrov–Galerkin (MLPG) formulation is presented in this paper. To this end, the u−p formulation of Biot is adopted as the framework of the porous media. One approach to numerically solve the infinite domain problems is the use of an absorber layer in which the whole half‐space is divided into two parts, that is (i) a finite part, in which the responses are interested, and (ii) the remaining semi‐infinite part, which is replaced by a Perfectly Matched Layer (PML). The stretched coordinates in the PML are introduced in such a way that the wave propagating in it does not generate spurious reflection to the finite part. Comparing the numerical results with some existing exact solutions and evaluating the norm of error demonstrate that the response functions in the finite part are achievable as precise as desired. Some new results are also presented which show the validity of the numerical approach in poroelastic transversely isotropic domain.

A compact open boundary treatment for a pure thermal plume by direct numerical simulation

This study introduces a novel open boundary treatment method for simulating thermal plumes in natural convection, utilizing a combination of the Perfectly Matched Layer (PML) and local one-dimensional inviscid (LODI) methods within a compressible flow solver for Direct Numerical Simulation (DNS). This innovative approach significantly reduces the size of the computational domain yet effectively captures the complex transition from laminar to turbulent flow. Validation of the proposed methodology includes comparisons with existing empirical formulations, experimental data, and well-resolved DNS results, incorporating both time-averaged and spectral analyses. Additionally, the buoyancy-induced laminar-turbulent transition is investigated to enhance understanding of the dynamics and behaviors of thermal plumes. The findings offer insights into optimizing computational efficiency without compromising the accuracy needed to analyze intricate fluid dynamics in thermal engineering applications.

Design and theoretical analysis of broadband bend-resistant low-loss hollow-core anti-resonant fiber

With the development of modern fiber-optic communication technology, bend-resistant low-loss fiber within the S+C+L bands is essential to meeting the requirements of dense wavelength division multiplexing (DWDM) systems, and a small-tube-added multi-nested hollow-core anti-resonant fiber (SM-HC-ARF) is proposed. In conventional double-nested HC-ARF, a pair of circular tubes are tangent to both the inner and outer tubes to form a multi-nested structure. A small tube is put in the gap between the nested structure to enhance the anti-resonance effect further. The influence of fiber structural parameters on transmission characteristics is analyzed by the finite element method combined with the perfect matching layer condition. The numerical results show that, in the range of 1460–1625 nm, the bending loss and the confinement loss are less than 1.17 (at a bending radius of 8 cm) and 0.80 dB/km, respectively. Specifically, at 1550 nm, the bending loss is 0.43 dB/km, and the confinement loss is 0.33 dB/km. The broadband anti-bending low-loss SM-HC-ARF can be expected to apply in the DWDM system and other optical communication technologies.

A novel model order reduction technique for solving horizontal refraction equations in the modeling of three-dimensional underwater acoustic propagation

Modeling three-dimensional (3D) underwater acoustic propagation is vital for underwater detection, localization, and navigation. This article introduces a novel model order reduction (MOR) technique to solve horizontal refraction equations (HREs) in the modeling of 3D underwater acoustic propagation. This approach relies on an adiabatic approximation of the 3D sound field, representing the field as a combination of local vertical modes with their modal coefficients governed by a system of two-dimensional (2D) HREs. Inspired by normal mode theory, the coefficients in the expansion over vertical modes are determined by projecting them onto a lower-dimensional, orthogonal space defined by their transverse eigenfunctions. By artificially truncating the horizontal domain in the transverse directions using two perfectly matched layers (PMLs), the eigenproblem associated with the transverse eigenfunctions of the modal coefficients is closed and thus can be solved through a modal projection method. The modal projection method enables fast computation of modal coefficients in a longitudinally invariant environment within seconds, offering a naturally wide-angle solution covering ±90°. The MOR method is extended to encompass fully 3D cases by introducing an admittance matrix, a memory-saving strategy that prevents numerical overflow when the longitudinal range is large. Moreover, the fact that the outer boundaries of the PMLs are range-independent allows the proposed MOR technique to perform well on a coarse grid when employing the Magnus scheme, significantly saving the numerical cost for the fully 3D simulation. Numerical simulations are provided for both longitudinally invariant and fully 3D scenarios, demonstrating the high accuracy and efficiency of the proposed MOR technique in solving HREs.

Simulation of ideal blackbody based on multi-space boundary condition in FDTD

Abstract Investigation shows that it is difficult to solve the electromagnetic field surrounding an ideal blackbody using the analytic method. To solve this problem, the simulation method should be adapted, and the Maxwell function should be expanded to multi space. In this paper, a multi space simulation scheme of finite difference time domain (FDTD) and a multi space boundary condition (MSBC) are proposed for the simulation of an ideal blackbody. Compared with the traditional absorbing boundary condition (ABC) or perfectly matched layer (PML), which can only be assigned in the outermost layers of the simulation space, the proposed MSBC is simpler and can be arranged inside the computing space. Numerical simulation shows that MSBC has good electromagnetic wave absorption performance and can simulate the ideal blackbody accurately. At the same time, the simulation results also show some interesting characteristics of electromagnetic waves around the ideal blackbody.

Ultra-high-sensitive plasmonic sensor based on asymmetric hexagonal nano-ring resonator for cancer detection

A highly sensitive sensor based on two metal-insulator-metal waveguides coupled to an asymmetric hexagonal nano-ring resonator detecting cancerous cells is proposed. This novel design is utilized to facilitate the sensing of human cells. The sensing mechanism of the presented optical structure can act as a refractive index measurement in biological, chemical, biomedical diagnosis, and bacteria detection, which leads to achieving high sensitivity in the structure. The main goal is to achieve the highest sensitivity concerning the optimum design. As a result, the sensitivity of the designed topology reaches a maximum value of about 1800 nm/RIU (nm/refractive index unit) by controlling the angle of the resonator. It is evident that the sensitivity parameter is improved, and the reason for the increase in sensitivity is due to the asymmetry of the resonator, which has an 81 % increase in sensitivity compared to the symmetrical resonator, especially for blood cancer cells. The maximum quality factor obtains 131.65 with a FOM of 90.4 (RIU−1). The sensing performance of this proposed structure is numerically investigated using the finite difference time domain (FDTD) method with the perfectly matched layer (PML). Accordingly, the suggested high sensitivity sensor makes this structure a promising therapeutic candidate for sensing applications that can be used in on-chip optical devices to produce highly complex integrated circuits.

Development of Dispersion Calculator SAFEDC of Guided Wave with Complex Material and Waveguide

The propagation characteristics of guided waves (GW) are related to both the material, loading and geometry parameters. Correct understanding of GW is the key step towards further GW-based structural health monitoring. A dispersion calculator of GW propagating in the complex geometry with viscoelastic and anisotropic material parameter in the open waveguide is developed based on the semi-analytical finite element (SAFE) method and perfectly matched layer (PML). By assuming the simple harmonic motion along the propagation direction and adopting the mesh discretization in the cross section, SAFE features the capability of GW calculation in the complex waveguide at only a cost of limited calculation resources. Hence, besides the capabilities of the most off-the-shelf tools focusing on the analytical solution of GW in simple plate or pipeline, the developed algorithm further offers the solution of GW in open waveguide with laminates of arbitrary ply stacking sequences and in arbitrarily complex cross section. Quadratic one-dimensional and triangular elements are adopted to automatically discretize the plate structure and complex cross section, respectively. The obtained results of GW in plates are in a good agreement with the results obtained with the off-the shelf-tools that adopt analytical derivations, and show stronger robustness of mode searching and sorting by avoiding the complex root-finding procedure required in the analytical derivation. The most GW features, including phase velocity, group velocity, attenuation, wave structure in terms of displacement, stress, and strain, and animation of wave propagation are directly calculated. All these functions are integrated and updated in the previously developed freeware SAFEDC, whose 1st version SAFEDC v1.0 was published in September 2023.

Modelling and simulation for the investigation on the ultrasonic propagation mechanism in advanced microelectronic packages

The miniaturisation, ultra-thinness and high-density multi-layer structure of advanced microelectronic packages complicate the propagation mechanism of ultrasonic waves. In this paper, a finite element model is used to simulate ultrasonic wave propagation in flip chip packages, investigating the laws of transmission and reflection at the lamination boundaries. The acoustic field of ultrasonic transducers is simulated using MATLAB and Abaqus software. The angular spectrum method (ASM) based on the Fourier transform is adopted to more precisely reveal the distribution characteristics and attenuation relationship of near‐field ultrasonic waves. The influence of the frequency and size of the ultrasonic transducer on the propagation characteristics of ultrasonic waves is analysed. Based on an acoustic field map generated by the detection model, the waveform conversions of acoustic waves in a multi-layer structure are analysed. The results show that ultrasonic waves are mainly presented in the form of reflected and transmitted waves at the layered interface and the model with a perfectly matched layer (PML) has higher accuracy. Therefore, this method is applied to ultrasonic testing in a flip chip package, which cannot only effectively exclude interference from boundary reflection but also greatly improve the reliability of waveform conversions analysis.

Simplified Tunnel–Soil Model Based on Thin-Layer Method–Volume Method–Perfectly Matched Layer Method

In order to analyze the ground vibration responses induced by the dynamic loads in a tunnel, this paper proposes a new simplified tunnel–soil model. Specifically, based on the basic theory of the thin-layer method (TLM), the basic solution of three-dimensional layered foundation soil displacement was derived in the cylindrical coordinate system. The perfectly matched layer (PML) boundary condition was applied to the TLM. Subsequently, a tunnel–soil dynamic interaction analysis model was established using the volume method (VM) in conjunction with the TLM-PML method. The displacement frequency response function of the foundation soil around the tunnel foundation was derived. Finally, a ground vibration test under an impact load in a tunnel was carried out. The test and calculated results were compared. The comparison results show that the ground vibration acceleration response values within 25 m from the load are similar. Compared with the test results, the theoretical calculation results exhibit a decreasing trend in the range of 40–80 Hz between 25 and 60 m, with the maximum reduction being approximately one order of magnitude. In addition, the experimental comparison demonstrates that the model can be used to analyze the ground vibrations caused by underground loads.

Dynamic stress response in reservoirs stimulated by fluctuating fluid pressure during pulsating hydraulic fracturing

Determining the most effective scheme for fracturing operations depends on accurate calculations of reservoir stress. Traditional fracturing theories usually focus on stable fluid injection modes, which employ static assumptions in modelling and analysis. However, pulsating hydraulic fracturing (PHF) introduces fluctuating fluid pressure to stimulate reservoirs, which requires considering dynamic factors and exploring the dynamic stress response within the reservoir. Furthermore, reservoirs are often classified as homogeneous continuous or porous media materials, but the boundaries between these classifications are sometimes apparent. This paper investigated the development of both an elastodynamics model and a poroelastodynamics model for calculating dynamic stress responses during PHF. The staggered grid finite difference method addressed these models, with the outer boundary employing the perfectly matched layer (PML) method to simulate an infinite reservoir. The numerical simulation results were extensively verified and compared with analytical solutions, and the differences between various models and their relevant conditions were thoroughly analyzed. Analyzing fluid pressure fluctuation parameters revealed noteworthy insights into reservoir stress response characteristics. Results indicate that as the Biot coefficient and permeability increase, the disparities between the elastodynamics and poroelastodynamics models become prominent. In practical terms, high Biot coefficient or high permeability reservoirs necessitate using the poroelastodynamics model for accurate reservoir stress analysis. As the frequency of fluid pressure fluctuation increases, the amplitude of response of reservoir circumferential stress also increases, while the response amplitude of radial stress and fluid pressure decreases. These findings offer valuable theoretical guidance for parameter design in engineering applications during PHF.