Helmholtz Wave Research Articles

Strong diffraction effects accompany the transmission of a laser beam through inhomogeneous plasma microstructures.

In the study we thoroughly analyze diffraction effects accompanying the laser beam transmission through inhomogeneous plasma microstructures and simulate their diffraction patterns at the object output and in the near field. For this we solve the scalar Helmholtz wave equationin the first Rytov approximation and compute the diffraction spreading of the transmitted beam in free space. Diffraction effects are found to arise within the beam passage through inhomogeneous plasma microstructures even in the simplest approximations of the laser beam interaction with plasma. These effects become strong in the near-field region and significantly distort the patterns of plasma formations, as well as facilitate the appearance of various optical artifacts in the plasma images. By performing numerical simulations, we characterize in detail the features of the visualization of plasma formations in the field of a coherent laser beam registered by a lens system. The calculations are in good agreement with the experimental data. The study can find broad applications in the processing of the laser images of plasma microstructures registered by lens systems in the presence of strong diffraction effects.

RHDLPP: A multigroup radiation hydrodynamics code for laser-produced plasmas

In this paper, we introduce the RHDLPP, a flux-limited multigroup radiation hydrodynamics numerical code designed for simulating laser-produced plasmas in diverse environments. The code bifurcates into two packages: RHDLPP-LTP for low-temperature plasmas generated by moderate-intensity nanosecond lasers, and RHDLPP-HTP for high-temperature, high-density plasmas formed by high-intensity laser pulses. The core radiation hydrodynamic equations are resolved in the Eulerian frame, employing an operator-split method. This method decomposes the solution into two substeps: first, the explicit resolution of the hyperbolic subsystems integrating radiation and fluid dynamics; second, the implicit treatment of the parabolic part comprising stiff radiation diffusion, heat conduction, and energy exchange. Laser propagation and energy deposition are modeled through a hybrid approach, combining geometrical-optics ray-tracing in sub-critical plasma regions with a one-dimensional solution of the Helmholtz wave equation in super-critical areas. The thermodynamic states are ascertained using an equation of state, based on either the real gas approximation or the quotidian equation of state (QEOS). For ionization calculations, the code employs a steady-state collisional-radiation (CR) model using the screened-hydrogenic approximation. Additionally, RHDLPP includes RHDLPP-SpeIma3D, a three-dimensional spectral simulation post-processing module, for generating both temporally-spatially resolved and time-integrated spectra and imaging, facilitating direct comparisons with experimental data. The paper showcases a series of verification tests to establish the code's accuracy and efficiency, followed by application cases, including simulations of laser-produced aluminium (Al) plasmas, pre-pulse-induced target deformation of tin (Sn) microdroplets relevant to extreme ultraviolet lithography light sources, and varied imaging and spectroscopic simulations. These simulations highlight RHDLPP's effectiveness and applicability in fields such as laser-induced breakdown spectroscopy, extreme ultraviolet lithography sources, and high-energy-density physics.

A monolithic numerical model to predict the EMI shielding performance of lossy dielectric polymer nanocomposite shields in a rectangular waveguide: Design of an absorption-based sawtooth-shaped layer

A three-dimensional numerical model is constructed to predict the EMI shielding performance of a polymer nanocomposite shield in a rectangular waveguide. The Helmholtz wave equation for the electric field is implemented in component form and the set of coupled equations is solved via the finite element approach. Mesh convergence and model verification is performed by comparing free space model predictions for a flat, uniform layer to benchmark solutions calculated via transfer matrix theory. The capability of the model is showcased by exploring the role of geometry on the shielding performance of a sawtooth-shaped composite layer in a rectangular waveguide. Increasing the inclusion angle of the sawtooth, which is proportional to the ratio of the sawtooth amplitude and repeat unit width, reduces the transmitted power through the shield and increases the ratio of absorption to reflection of wave power by the shield. Thus, a rational design of this sawtooth geometry allows to overcome the typical trade-off between total shielding effectiveness and wave absorption contribution, thereby resulting in highly performant absorption-dominated shields.

Investigation on close-coupled gas atomization for Fe-based amorphous powder production via simulation and industrial trials: Part I. Melt breakup behaviors during primary atomization

The primary atomization process is simulated by Volume of Fluid (VOF) model and dynamic adaptive mesh method. The influence of melt mass flow rate and atomization pressure on the breakup process is investigated, and the effect of hot gas atomization is also evaluated. The results show that the breakup is insufficient when the melt mass flow rate is larger than 0.075 kg s−1, while the liquid film breakup inducing the nozzle clogging occurs when the melt mass flow rate is too low (0.025 kg s−1). The backflow of droplets occurs at low atomization pressure (1.0 MPa), and various defects (satellite powders, hollow powders and needle-shaped powders) appear when the atomization pressure is larger than 3.0 MPa. Although the breakup efficiency can be significantly improved by increasing the gas temperature, severe deformation of the gas-liquid interface is induced due to the Kelvin–Helmholtz wave and gas-liquid interaction, easily leading to the formation of hollow powders. Besides, the gas-to-melt ratio (GMR) is identified as a simple criterion for predicting primary atomization breakup modes, with liquid film breakup occurring when GMR≥4.4 and "fountain" breakup occurring when GMR≤4.3. In this work, not only the gas-liquid interaction is systematically analyzed by establishing a flow-heat transfer-VOF coupling model, but also the GMR is proposed to predict the breakup mode in the industrial production, which can provide theoretical and methodological guidance for the optimization of atomization operational parameters.

Numerical vibroacoustic analysis of multi-layered composite structure under hygrothermal loading using coupled FEM-IBEM micromechanical model

The vibroacoustic responses of multi-layer composite shallow shell panels subjected to harmonic mechanical excitation in hygrothermal environment is numerically investigated. A homogenized micromechanical finite element (FE) based on the higher-order mid-plane kinematics replicating quadratic function as well as the through the thickness stretching effect together with the indirect boundary element (IBE) scheme has been first time employed. The isoparametric Lagrangian element with zero-order Hermitian interpolation function (10 degrees of freedom per node) is used for structural model discretization to attain the hygro-thermo-elastic natural frequencies via Hamilton’s principle. The uniqueness of the model is the incorporation of the effective material properties under combined hygrothermal loading via a micromechanics approach. An IBE method is then implemented to attain structure-surrounding coupling and the Helmholtz wave equation is solved to compute the sound radiation responses. The effectiveness of the model is tested by converging it with the similar analytical/numerical results as well as the experimentally acquired data for vibrational as well as acoustic responses. The present scheme is further hold out for solving diverse numerical illustrations. The geometrical parameters, volume fraction of fiber, layup, and support conditions alongside the hygrothermal load is found to have significant influence on the hygro-thermo-acoustic characteristics. The functional inferences will assist in designing the imminent multi-layered structures that serve under the influence of severe hygrothermal loading during high-performance engineering applications.

Evaluation of the Accuracy of the Remote Determination of the Brewster Angle When Measuring Physicochemical Parameters of Soil

In precision farming technology, the moisture of the soil, its granulometric composition, specific conductivity and a number of other physical and chemical parameters are determined using remote radar sensing. The most important parameters are those measured in the area of the plant root system located well below the “air-surface” boundary. In order to create conditions for the penetration of electromagnetic waves through the “air-surface” interface with a minimum reflection coefficient, the irradiation of the Earth’s surface is carried out obliquely with an angle of incidence close to the Brewster angle. The reflection coefficient, and, consequently, the Brewster angle, depend on the complex dielectric permittivity of the surface soil layer and are not known a priori. To determine the Brewster angle, the usual method is to search for the minimum amplitude of the vertically polarized signal reflected from the surface. Another approach is when the first derivative of the dependence of the modulus of the complex amplitude of a vertically polarized interference wave, taken with respect to the angle of incidence, is set equal to zero. In turn, in real dielectrics such as agricultural soils, the amplitude of the vertically polarized signal reflected from the surface is directly proportional to the reflection coefficient and does not have a pronounced minimum, which reduces the accuracy of the measurements. Based on the solution of the Helmholtz wave equation for a three-layered structure of the propagation medium (air, upper fertile soil layer, soil layer below the groundwater level), a model of the process of forming an interference wave under oblique irradiation of a planar layered dielectric with losses has been developed. Using the developed model, factors influencing the accuracy of determining the Brewster angle have been identified. For the first time, it is proposed to use the phase shift between the oscillations of the interference waves with vertical and horizontal polarization to measure the Brewster angle. A comparative assessment of the accuracy of determining the Brewster angle using known amplitude methods and the proposed phase method has been carried out. The adequacy of the method was experimentally confirmed. Recommendations have been developed for the practical application of the phase method of finding the Brewster angle for assessing the dielectric permittivity of soil and its moisture content.

Shock-induced atomisation of a liquid metal droplet

The present study uses Galinstan as a test fluid to investigate the shock-induced atomisation of a liquid metal droplet in a high-Weber-number regime $(We \sim 400\unicode{x2013}8000)$ . Atomisation dynamics is examined for three test environments: oxidizing (Galinstan–air), inert (Galinstan–nitrogen) and conventional fluids (deionised water–air). Due to the readily oxidizing nature of liquid metals, their atomisation in an industrial scale system is generally carried out in inert atmosphere conditions. However, no previous study has considered gas-induced secondary atomisation of liquid metals in inert conditions. Due to experimental challenges associated with molten metals, laboratory scale models are generally tested for conventional fluids like deionised water, liquid fuels, etc. The translation of results obtained from conventional fluid to liquid metal atomisation is rarely explored. Here a direct multiscale spatial and temporal comparison is provided between the atomisation dynamics of conventional fluid and liquid metals under oxidizing and inert conditions. The liquid metal droplet undergoes breakup through the shear-induced entrainment mode for the studied range of Weber number values. The prevailing mechanism is explained based on the relative dominance of droplet deformation and Kelvin–Helmholtz wave formation. The study provides quantitative and qualitative similarities for the three test cases and explains the differences in morphology of fragmenting secondary droplets in the oxidizing test case (Galinstan–air) due to rapid oxidation of the fragmenting ligaments. A phenomenological framework is postulated for predicting the morphology of secondary droplets. The formation of flake-like secondary droplets in the Galinstan air test case is based on the oxidation rate of liquid metals and the properties of the oxide layer formed on the atomizing ligament surface.

An asymmetric graded indexed helical structured optical fibre for enhanced optical delays

Optical fibres are generally used to provide a guided path to optical signals. Optical fibres are classified as multi-mode or single-mode fibres. Multi-mode graded index (GRIN) optical fibres have been prevalent in short-range optical networks, while single-mode GRIN fibres are used for long-range links. Another potential application of optical fibre is the optical delays in numerous applications, such as quantum computing, network buffering, data synchronization, pattern correlation, etc. This paper conceptualizes a novel application of an asymmetric GRIN optical fibre for possible enhanced optical delays. An asymmetric GRIN parabolic profile with helical structure geometry has been proposed for this task. A detailed analysis has been presented to investigate the effects of various parameters of the proposed geometry. Further, a guided propagation has also been demonstrated with the help of ray optics and the Helmholtz wave equation to assess the feasibility of the proposal. The presented investigations revealed that the geometrical length of the proposed structure is significantly compact compared with the traditional GRIN fibre, proving it beneficial for optical delays.

Identifying optimal architectures of physics-informed neural networks by evolutionary strategy

Physics-Informed Neural Networks (PINNs) are artificial neural networks that encode Partial Differential Equations (PDEs) as an integral component of the ML model. PINNs are successfully used nowadays to solve PDEs, fractional equations, and integral–differential equations, including direct and inverse problems. Just as in the case of other kinds of artificial neural networks, the architecture, including the number and sizes of layers, activation functions, and other hyperparameters can significantly influence the network performance. Despite the serious work in this field, there are still no clear directions on how to choose an optimal network architecture in a consistent manner. In practice, expertise is required, with a significant number of manual trial and error cycles. In this paper, we propose PINN/GA (PINN/Genetic Algorithm), a fully automatic design of a PINN by an evolutionary strategy with specially tailored operators of selection, crossover, and mutation, adapted for deep neural network architecture and hyperparameter search. The PINN/GA strategy starts from the population of simple PINNs, adding new layers only if it brings clear accuracy benefits, keeping PINNs in the population as simple as possible. Since the examination of dozens of neural networks through the evolutionary process implies enormous computational costs, it employs a scalable computational design based on containers and Kubernetes batching orchestration. To demonstrate the potential of the proposed approach, we chose two non-trivial direct problems. The first is 1D Stefan transient model with time-dependent Dirichlet boundary conditions, describing the melting process, and the second is the Helmholtz wave equation over a 2D square domain. The authors found that PINNs accuracy gradually improves throughout the evolutionary process, exhibiting better performance and stability than parallel random search and Hyperopt Tree of Parzen Estimators, while keeping the network design reasonably simple.

Acoustic modal analysis with heat release fluctuations using nonlinear eigensolvers

Closed combustion devices like gas turbines and rockets are prone to thermoacoustic instabilities. Design engineers in the industry need tools to accurately identify and remove instabilities early in the design cycle. Many different approaches have been developed by the researchers over the years. In this work we focus on the Helmholtz wave equation based solver which is found to be relatively fast and accurate for most applications. This solver has been a subject of study in many previous works. The Helmholtz wave equation in frequency space reduces to a nonlinear eigenvalue problem which needs to be solved to compute the acoustic modes. Most previous implementations of this solver have relied on linearized solvers and iterative methods which as shown in this work are not very efficient and sometimes inaccurate. In this work we make use of specialized algorithms implemented in SLEPc that are accurate and efficient for computing eigenvalues of nonlinear eigenvalue problems. We make use of the n-tau model to compute the reacting source terms in the Helmholtz equation and describe the steps involved in deriving the Helmholtz eigenvalue equation and obtaining its solution using the SLEPc library.

An effective semilocal model for wave turbulence in two-dimensional nonlinear optics

The statistical evolution of ensembles of random, weakly interacting waves is governed by wave kinetic equations (WKEs). To simplify the analysis, one frequently works with reduced differential models of the wave kinetics. However, the conditions for deriving such reduced models are seldom justified self-consistently. Here, we derive a reduced model for the wave kinetics of the Schrödinger–Helmholtz equations in two spatial dimensions, which constitute a model for the dynamics of light in a spatially nonlocal, nonlinear optical medium. This model has the property of sharply localizing the frequencies of the interacting waves into two pairs, allowing for a rigorous and self-consistent derivation of what we term the semilocal approximation model (SLAM) of the WKE. Using the SLAM, we study the stationary spectra of Schrödinger–Helmholtz wave turbulence, and characterize the spectra that carry energy downscale, and waveaction upscale, in a forced-dissipated setup. The latter involves a nonlocal transfer of waveaction, in which waves at the forcing scale mediate the interactions of waves at every larger scale. This is in contrast to the energy cascade, which involves local scale-by-scale interactions, familiar from other wave turbulent systems and from classical hydrodynamical turbulence.

Acoustic radiation efficiencies of laminated composite structure under mechanical excitation and hygrothermal loading: Numerical analysis

In this research, the sound radiation efficiencies of multi-layer composite flat as well as curved structure under harmonic mechanical excitation in hygrothermal environment is numerically investigated using a coupled micromechanical finite element and boundary element scheme. The higher-order mid-plane kinematics including the stretching effect is equipped for the structural modeling and the Hamilton’s principle is utilized to evaluate the hygro-thermo-elastic natural frequencies alongside corresponding modes of the structures lying on an infinite rigid-baffle. An isoparametric Lagrangian element with zero order Hermitian interpolation function (ten degrees of freedom per node) is utilized for transforming the continuous structural equations into discrete counterparts. The effective composite material properties due to combined hygrothermal loading are considered via a micromechanical model. Then, an indirect boundary element method is employed to attain the structure-surrounding coupling and the sound radiation efficiencies are computed by solving the Helmholtz wave equation. The effectiveness of the model is established by converging it with the similar results available in the open literature for vibrational as well as sound radiation efficiencies and the results reveal the confirmation of the model accuracy. Upon solving diverse numerical illustrations, the sound radiation efficiency found to decrease with intensity of hygrothermal loads and the cross-ply spherical panels with less volume fraction of fiber depicted the least RE values over the entire frequency range considered.

Reflection and transmission of a Kelvin–Helmholtz wave incident on a shock in a jet

Screech tones in supersonic jets are underpinned by resonance between downstream-travelling Kelvin–Helmholtz waves and upstream-travelling acoustic waves. Specifically, recent works suggest that the relevant acoustic waves are guided within the jet and are described by a discrete mode of the linearised Euler equations. However, the reflection mechanism that converts downstream-travelling waves into upstream-travelling waves, and vice versa, has not been thoroughly addressed, leading to missing physics within most resonance models. In this work, we investigate the reflection and transmission of waves generated by the interaction between a Kelvin–Helmholtz wave and a normal shock in an under-expanded jet using a mode-matching approach. Both vortex-sheet and finite-thickness shear-layer models are explored, quantifying the impact of the shear layer in the reflection process. This approach could enable more quantitative predictions of resonance phenomena in jets and other fluid systems.

A diffraction problem for the biharmonic wave equation in one-dimensional periodic structures

This work concerns the propagation of flexural waves through one-dimensional periodic structures embedded in thin elastic plates. We show that the out-of-plane displacement of the plate only contains the Helmholtz wave component and the modified Helmholtz wave component is not supported when the Navier boundary condition is imposed. An adaptive finite element method with transparent boundary condition is developed for solving the associated boundary value problem. Numerical results show that the method is effective to solve the diffraction grating problem of the biharmonic wave equation.

Scalable multi-level deflation preconditioning for highly indefinite time-harmonic waves

Recent research efforts aimed at iteratively solving time-harmonic waves have focused on a broad range of techniques to accelerate convergence. In particular, for the famous Helmholtz equation, deflation techniques have been studied to accelerate the convergence of Krylov subspace methods. In this work, we extend the two-level deflation method to a multilevel deflation method for (heterogeneous) Helmholtz and elastic wave problems. By using higher-order deflation vectors, we show that up to the level where the coarse-grid linear systems remain indefinite, the near-zero eigenvalues of the these coarse-grid operators remain aligned with the near-zero eigenvalues of the fine-grid operator, keeping the spectrum of the preconditioned system away from the origin. Combining this with the well-known CSLP-preconditioner, we obtain a scalable solver for the highly indefinite linear systems. This can be attributed to a close to wave number independent convergence and an optimal use of the CSLP-preconditioner on the indefinite levels. There, we approximate the CSLP-preconditioner, while allowing the complex shift to be small, by using inner Bi-CGSTAB iterations instead of a multigrid F-cycle. The proposed method shows very promising results for the more challenging two- and three-dimensional heterogeneous time-harmonic wave problems.

Femtosecond laser-induced sub-wavelength plasma inside dielectrics. II. Second-harmonic generation

Second-harmonic emission at a frequency that is twice the laser frequency is an important diagnostic for nonlinear laser–plasma interaction. It is forbidden for centrosymmetric materials such as the bulk of sapphire. The symmetry, however, can be broken by dielectric discontinuities as a result of plasma generation inside a solid dielectric. In the present work, we explore the basic characteristics of experimentally observed second-harmonic emission during focusing a femtosecond Bessel beam inside sapphire. We employ three-dimensional particle-in-cell simulations and the Helmholtz wave equation for theoretical investigations. We analyze how the efficiency of second-harmonic generation and its polarization depend on the plasma parameters. We find that the second-harmonic is generated either due to the coalescence of two-surface electromagnetic waves or nonlinear interaction between the transverse electromagnetic wave and the longitudinal electron plasma wave driven by linear mode conversion. Experimental results agree with the theoretical predictions and confirm the existence of over-critical plasma inside the sapphire that is essential for the resonance of plasma waves or excitation of surface plasmons.

An efficient semi-analytical procedure to calculate acoustic radiation force and torque for axisymmetric irregular bodies

With the attributes of being biocompatible, label-free, and contact-free, sound has been widely used to manipulate micro-objects, such as cells, soft matters, and living things, by the acoustic radiation force and torque arising from wave scattering. The analytical theory exists for the acoustic radiation force and torque on separable geometries, including sphere and ellipsoid. However, it is still a challenge to handle irregular surfaces subject to non-orthogonal and inseparable boundary conditions. Here, we present a calculation method for an axisymmetric geometry with irregular cross-section excited by a time-harmonic plane wave with arbitrary incidence in the inviscid fluid. The method is based on conformal mapping from the irregular surface to a sphere in the new coordinate system. In this way, the separation of variables can be used to solve the corresponding Helmholtz wave equation subjecting to the spherical boundary conditions in the new coordinate system. The radiation force and the torque are asymptotically obtained using the far-field, closed-form solutions. The method is validated by comparisons with full 3D numerical solutions over a wide range of frequencies and incident angles. With a typical discrepancy of less than 5%, the proposed method is much more efficient than the full numerical simulations (via finite element method). Furthermore, it is found that the radiation force acting on different geometries follows the same tendency. The difference is that the presence of the radiation torque ensures that the symmetry axis of the irregular bodies is coincident with or orthogonal to the wave propagating direction.

Structural–Acoustic Response Analysis of Variable Stiffness Laminates with Inherent Material Damping

Sound radiation and transmission loss characteristics of variable stiffness composite plate reinforced with the curvilinear fibers are investigated numerically. The formulation is developed using higher-order shear flexible finite element model combined with Helmholtz wave equation. The governing equations obtained using Hamilton’s approach are further solved through the modal super position method to analyze the vibration response under steady state excitation. The inherent material damping of the laminate is accounted through the modal damping calculated using the modal strain energy approach. The acoustic pressure of the variable stiffness laminates is estimated using the Raleigh integral. Subsequently, acoustic response characteristics such as acoustic power level, radiation efficiency, directivity pattern, and transmission loss from the laminates are predicted using the estimated sound pressure for various forcing frequencies. A parametric study covering a wide range of design variables including center and edge fiber angles, lamination scheme, thickness ratio, and boundary conditions on the acoustic sound behavior arising from the vibration of curvilinear fiber composite plate is detailed. This study reveals that the acoustic response of the curvilinear fiber composite plate is significantly influenced by the curvilinear fiber angles at the center/edge fiber angle of the layers. It is hoped that the results obtained here will be useful for designers in developing structures with desired acoustic response characteristics.

Closure mechanism of the A1 and A2 modes in jet screech

This paper explores the screech closure mechanism for different axisymmetric modes in shock-containing jets. While many of the discontinuities in tonal frequency exhibited by screeching jets can be associated with a change in the azimuthal mode, there has to date been no satisfactory explanation for the existence of multiple axisymmetric modes at different frequencies. This paper provides just such an explanation. As shown in previous works, specific wavenumbers arise from the interaction of waves in the flow with the shocks. This provides new paths for driving upstream-travelling waves that can potentially close the resonance loop. Predictions using locally parallel and spatially periodic linear stability analyses and the wavenumber spectrum of the shock-cell structure suggest that the A1 mode resonance is closed by a wave generated when the Kelvin–Helmholtz mode interacts with the leading wavenumber of the shock-cell structure. The A2 mode is closed by a wave that arises owing to the interaction between the Kelvin–Helmholtz wave and a secondary wavenumber peak, which arises from the spatial variation of the shock-cell wavelength. The predictions are shown to closely match experimental data, and possible justifications for the dominance of each mode are provided based on the growth rates of the absolute instability.

Wavefield reconstruction and wave equation inversion for seismic surface waves

SUMMARY Surface waves are a particular type of seismic wave that propagate around the surface of the Earth, but which oscillate over depth ranges beneath the surface that depend on their frequency of oscillation. This causes them to travel with a speed that depends on their frequency, a property called dispersion. Estimating surface wave dispersion is of interest for many geophysical applications using both active and passive seismic sources, not least because the speed–frequency relationship can be used to infer the subsurface velocity structure at depth beneath the surface. We present an inversion scheme that exploits spatial and temporal relationships in the scalar Helmholtz (wave) equation to estimate dispersion relations of the elastic surface wave data in both active and passive surveys, while also reconstructing the wavefield continuously in space (i.e. between the receivers at which the wavefield was recorded). We verify the retrieved dispersive phase velocity by comparing the results to dispersion analysis in the frequency-slowness domain, and to the local calculation of dispersion using modal analysis. Synthetic elastic examples demonstrate the method under a variety of recording scenarios. The results show that despite the scalar approximation made to represent these intrinsically elastic waves, the proposed method reconstructs both the wavefield and the phase dispersion structure even in the case of strong aliasing and irregular sampling.