Numerical Simulation Of Wave Propagation Research Articles

Numerical simulation of wave propagation in blocky-layered medium with thin elastic and viscoelastic interlayers

Numerical simulation of wave propagation in blocky-layered medium with thin elastic and viscoelastic interlayers

Numerical simulation of wave propagation by using a hybrid method with an arbitrary order accuracy in both spatial and temporal approximations

This paper introduces an innovative numerical methodology designed to achieve high precision solution of acoustic wave propagation problem in isotropic material. During the temporal discretization process, the Krylov deferred correction (KDC) technique is employed, wherein a new variable is introduced to handle the second-order time derivative in the governing equations. An improved strategy is adopted for precisely implementing boundary conditions. Following this, the arbitrary-order generalized finite difference method (GFDM) is employed to simulate the transformed boundary value problem at each time step, enabling our framework to select the Taylor series expansion order arbitrarily. Ultimately, a hybrid numerical approach for wave problems is developed to achieve arbitrary-order accuracy in both spatial and temporal approximations. The developed scheme undergoes validation through four different numerical experiments in two- or three-dimension, wherein the numerical solutions obtained are compared with either analytical solutions or results from COMSOL software.

A non-invasive short range acoustic technique for liquid level measurement in containers

Techniques to measure liquid level in containers and vessels non-invasively are of great interest for the manufacturing, chemical industries, and energy sectors. Most commercial solutions use invasive techniques, which are not always practical and can lead to unsafe conditions when measuring devices encounter high pressures or hazardous chemical contents. In this study, a portable, non-invasive acoustic liquid level identification technique is developed. The technique relies on comprehensive analysis of acoustic guided wave modes in the container walls and their coupling to liquid contents to determine the optimal sensor separation distance and excitation signal characteristics, resulting in liquid level determination with high accuracy and precision. First, dispersion characteristics of guided wave propagation in container walls in contact with a liquid boundary are studied in detail. Based on the dispersion analysis a general-purpose short-range liquid level detection methodology is developed. The developed methodology is then validated using numerical wave propagation simulations and experiments on various metal containers with different physical characteristics, illustrating the versatility of the method. Compared to existing techniques, which are optimized for specific applications, the proposed method, due to its short range and non-invasive nature, has the potential for universal applicability on virtually any container, irrespective of its shape, size, wall-thickness, and whether it has any internal components.

Equation of State of Autoclaved Aerated Concrete-Oedometric Testing.

This paper aims at investigating the triaxial behavior of Autoclaved Aerated Concrete (AAC) under extremely high pressures, and experimentally determine Equation of State (EOS) for several different AAC densities. Oedometric tests were carried out using a home-made high-pressure triaxial apparatus, and pressures up to ~500 MPa were applied. The complete pressure-bulk strain relationships were measured, and new findings and insights were obtained. The paper presents the testing set-up and the measurement system. The data processing method accounting for the AAC pronounced shortening during the ongoing test is described using a weighted functions procedure for the circumferential strains' calculation, with which the confining pressure was determined. The boundary conditions effects on the test results were investigated, and a new technique for specimen insulation was suggested to ensure loading without friction and the prevention of local shear failure. The experimental EOS for different AAC densities were obtained. EOS curves for different specimens with the same density demonstrated good to very good repeatability of the EOS curves over the entire pressure range. Based on the tests results and the density's span, three classes of AAC are proposed. A preliminary attempt to apply the newly obtained EOS curves has been carried out to examine the energy dissipation for three different dynamic load levels. Although this is a preliminary stage that is beyond the objective of this paper, early interesting results were observed where an optimal AAC density, for which the highest energy has been absorbed, was identified. This finding encourages inclusion of that preliminary study as a closure section. Numerical simulations of wave propagation through ACC layers of different densities, laid on rigid supporting slabs, was carried out. The minimum total impulse imparted to the rigid slab was found for the optimal AAC density that has been determined above.

2-D seismic wave propagation using the distributional finite-difference method: further developments and potential for global seismology

SUMMARY We present a time-domain distributional finite-difference scheme based on the Lebedev staggered grid for the numerical simulation of wave propagation in acoustic and elastic media. The central aspect of the proposed method is the representation of the stresses and displacements with different sets of B-splines functions organized according to the staggered grid. The distributional finite-difference approach allows domain-decomposition, heterogeneity of the medium, curvilinear mesh, anisotropy, non-conformal interfaces, discontinuous grid and fluid–solid interfaces. Numerical examples show that the proposed scheme is suitable to model wave propagation through the Earth, where sharp interfaces separate large, relatively homogeneous layers. A few domains or elements are sufficient to represent the Earth’s internal structure without relying on advanced meshing techniques. We compare seismograms obtained with the proposed scheme and the spectral element method, and we show that our approach offers superior accuracy, reduced memory usage, and comparable efficiency.

Numerical simulation of wave propagation through a spherical particle within the framework of generalised Lorenz-Mie theory

Holography has been widely used for measuring and visualising transients in multiphase flows. Earlier, one of the drawbacks of this approach was the need to register on various photosensitive elements and its subsequent transfer to digital format and restoration. With the development of digital technologies, direct registration of interference patterns (holographic images) into a CCD matrix became possible. However, even in digital holography there are a number of problems that need to be solved. These problems pertain to recovery algorithms, efficient data processing and resolution, among others. Currently, the numerical implementation of the restoration and processing of holographic images can be done within the framework of classical diffraction theory or with the help of generalised Lorenz-Mie theory. The first implies an indirect solution of Maxwell's equations, i.e., application of the Huygens-Fresnel principle. The second approach involves a direct solution of Maxwell's equations for the holographic problem. In the framework of this work, a numerical simulation of holographic imaging of fields from spherical particles based on the generalised Lorenz-Mie theory is proposed. Within the framework of this work, a numerical implementation of modelling of holographic images of a homogeneous sphere based on the generalised Lorenz-Mie theory is presented. The implementation code in the python programming language is presented. The results of the study demonstrate the possibility of effective use of digital holography for visualisation and analysis of spherical objects.

A Deep Neural Networks-Based Sound Speed Reconstruction with Enhanced Generalization by Training on a Natural Image Dataset

Sound speed reconstruction has been investigated for quantitative evaluation of tissue properties in breast examination. Full waveform inversion (FWI), a mainstream method for conventional sound speed reconstruction, is an iterative method that includes numerical simulation of wave propagation, resulting in high computational cost. In contrast, high-speed reconstruction of sound speed using a deep neural network (DNN) has been proposed in recent years. Although the generalization performance is highly dependent on the training data, how to generate data for sufficient generalization performance is still unclear. In this study, the quality and generalization performance of DNN-based sound speed reconstruction with a ring array transducer were evaluated on a natural image-derived dataset and a breast phantom dataset. The DNN trained on breast phantom data (BP-DNN) could not reconstruct the structures on natural image data with diverse structures. On the other hand, the DNN trained on natural image data (NI-DNN) successfully reconstructed the structures on both natural image and breast phantom test data. Furthermore, the NI-DNN successfully reconstructed tumour structures in the breast, while the BP-DNN overlooked them. From these results, it was demonstrated that natural image data enables DNNs to learn sound speed reconstruction with high generalization performance and high resolution.

Задача о колебаниях в цепи связанных линейных осцилляторов с демпфирующей и антидемпфирующей границами

The problem of oscillations in a finite homogeneous chain of coupled harmonic oscillators is considered under special boundary conditions that ensure a stable flow of energy from one end of the chain to the other. The problem covers, as special cases, the classical problem of oscillations in a chain with free and fixed ends, as well as the problem of oscillations in a chain with absorbing and anti-absorbing boundaries. Absorbing boundary conditions are used in numerical simulation of wave propagation to minimize the influence of non-physical boundaries. An exact analytical solution to the considered problem is obtained. The dynamical system of the problem is studied. In particular, a description of the invariant subspaces of the system is given. The oscillatory properties of the system are investigated. The phenomenon of a significant increase in the amplitude of low-frequency oscillations has been discovered and studied. The problem is solved in Schrodinger variables. The solution is presented both in terms of infinite series of Bessel functions and in terms of natural oscillations (eigenmodes) of the system.

Numerical simulation of wave propagation through rigid vegetation and a predictive model of drag coefficient using an artificial neural network

Coastal vegetation has been widely recognized as an effective nature-based measure for coastal protection. Numerous studies based on macroscopic models have been performed to simulate vegetation-induced wave attenuation. The modeling accuracy is closely related to a key input parameter, the drag coefficient (CD). To date, however, no reliable prediction method is available for determining CD, limiting the practical application of the macroscopic models. In this study, vegetation-induced wave attenuation was investigated using a macroscopic model that solved Reynolds-averaged Navier-Stokes equations by introducing a vegetation resistance force to account for momentum loss. A stabilized k-ε turbulence model considering vegetation effects was developed for turbulence closure. The volume-of-fluid technique was used to capture the free surface. Using a calibration method to determine CD, the numerical model was validated based on several laboratory experiments. A correlation analysis was performed to reveal the potential contributions of dimensionless parameters about waves and vegetation to the calibrated CD. Multivariate non-linear regression (MNLR) and artificial neural network (ANN) methods were adopted to develop a CD predictive model. Comparisons indicated that the prediction performance of ANN model is superior to that of the MNLR model. The ANN model has the potential as a promising predictive tool for obtaining CD when simulating wave propagation through rigid vegetation.

Finite-difference modeling of Maxwell viscoelastic media developed from perfectly matched layer

In numerical simulation of wave propagation, both viscoelastic materials and perfectly matched layers (PMLs) attenuate waves. The wave equations for both the viscoelastic model and the PML contain convolution operators. However, convolution operator is intractable in finite-difference time-domain (FDTD) method. A great deal of progress has been made in using time stepping instead of convolution in FDTD. To incorporate PML into viscoelastic media, more memory variables need to be introduced, which increases the code complexity and computation costs. By modifying the nonsplitting PML formulation, I propose a viscoelastic model, which can be used as a viscoelastic material and/or a PML just by adjusting the parameters. The proposed viscoelastic model is essentially equivalent to a Maxwell model. Compared with existing PML methods, the proposed method requires less memory and its implementation in existing finite-difference codes is much easier. The attenuation and phase velocity of P- and S-waves are frequency independent in the viscoelastic model if the related quality factors (Q) are greater than 10. The numerical examples show that the method is stable for materials with high absorption (Q = 1), and for heterogeneous media with large contrast of acoustic impedance and large contrast of viscosity.

Numerical simulation of wave propagation in ice-covered ocean environments based on the equivalent-source method

Accurate modeling of sound propagation in ice-covered ocean environments can help with interpreting discrepancies between predictions and experimental observations in the changing Arctic Ocean; this is advantageous for environmental conservation, resource exploration, and naval applications. Building on the recent development of the equivalent-source (ES) method (ESM), herein, an ESM-based sub-ice model (ESM-SUBICE) is presented for wave propagation in an ice-covered ocean acoustic environment. The presented model solves exact governing equations for acoustic–elastic propagation in an ice-covered waveguide by expressing the wave solution in terms of a field superposition produced by several sets of ESs. Their unknown amplitudes are solved by strictly enforcing additional ice-layer boundary conditions. ESM-SUBICE achieves high efficiency using a water–seabed Green's function to automatically satisfy the boundary conditions at this interface. By further dividing the ocean environment into layers, ESM-SUBICE is extended for more general situations including stratified sound-speed structures and seabed range dependencies. ESM-SUBICE is benchmarked against a finite-element model, and it is found to produce high-quality solutions with high efficiency. Transmission-loss predictions for elastic, fluid, and free-surface ice representations in different ocean environments are compared to examine the effect of ice elasticity on propagation and scattering. The results suggest that the fluid representation is adequate for deep-water environments where the seabed is soft and the surface duct effect is insignificant; otherwise, for accurate predictions, the ice elasticity should be considered.

Impact of opacity effects on chromospheric oscillations inferred from NLTE inversions

Context. Spectropolarimetric inversions are a fundamental tool for diagnosing the solar atmosphere. Chromospheric inferences rely on the interpretation of spectral lines that are formed under nonlocal thermodynamic equilibrium (NLTE) conditions. In the presence of oscillations, changes in the opacity impact the response height of the spectral lines and hinder the determination of the real properties of the fluctuations. Aims. We aim to explore the relationship between the chromospheric oscillations inferred by NLTE inversion codes and the intrinsic fluctuations in velocity and temperature produced by the waves. Methods. We computed numerical simulations of wave propagation in a sunspot umbra with the code MANCHA. We used the NLTE synthesis and inversion code NICOLE to compute spectropolarimetric Ca II 8542 Å line profiles for the atmospheric models obtained as the output from the simulations. We then inverted the synthetic profiles and compared the inferences from the inversions with the known atmospheres from the simulations. Results. NLTE inversions of the Ca II 8542 Å line capture low-frequency oscillations, including those in the main band of chromospheric oscillations around 6 mHz. In contrast, waves with frequencies above 9 mHz are poorly characterized by the inversion results. Velocity oscillations at those higher frequencies exhibit clear signs of opacity fluctuations; namely the power of the signal at constant optical depth greatly departs from the power of the oscillations at constant geometrical height. The main response of the line to velocity fluctuations comes from low chromospheric heights, whereas the response to temperature shows sudden jumps between the high photosphere and the low chromosphere. This strong variation in the height where the line is sensitive to temperature is revealed as a strong oscillatory power in the inferred fluctuations, which is much stronger than the actual power from the intrinsic temperature oscillations. Conclusions. Our results validate the use of NLTE inversions to study chromospheric oscillations with frequencies below ∼9 mHz. However, the interpretation of higher-frequency oscillations and the power of temperature oscillations must be addressed with care, as these exhibit signatures of opacity oscillations.

Numerical Simulation of Wave Propagation in a Fractured Porous Medium

Numerical Simulation of Wave Propagation in a Fractured Porous Medium

Amplitude behavior in low-rank acoustic modeling

The numerical simulation of wave propagation can be represented by a propagator matrix applied to previous instances of the wavefield. Using the sparsity of the propagator matrix to approximate it by a low-rank representation, one can increment the wavefield’s phase from one time instance to another. Though this procedure does not pay attention to the amplitudes of the seismic waves, it is important to understand its dynamic properties. Here, we evaluate the amplitudes obtained by the low-rank method in the simulation of 2D acoustic wave propagation. In homogeneous media, where theoretical expressions for the wavefield are available, the method provides not only an excellent kinematic approximation, but also reliable amplitudes. For a single horizontal reflector below a homogeneous overburden, the reflection coefficients approximated by the low-rank method are of the same quality or slightly superior to those obtained by a second-order finite-difference (FD) method (implementation from SU). However, in more generally inhomogeneous media, our tests showed larger discrepancies between FD and low-rank modeling results. While comparing unfavorably with FD regarding computation time for small models, its quasi-linear scaling with model size makes the low-rank method superior for large models. Moreover, a generalization to more complex, e.g., anisotropic, media is straightforward.

Nonlinear hierarchical unit cell for passive, amplitude-dependent filtering of acoustic waves

In this letter we propose a nonlinear hierarchical unit cell for use in passive, amplitude-dependent filtering of acoustic energy transmission. Analogous to unit cell designs which filter on frequency using bandgaps, the nonlinear hierarchical unit cell filters on amplitude with minimal waveform distortion. Numerical simulations of wave propagation in a mechanical chain employing the proposed unit cell predict nearly zero transmission of energy at low amplitudes, and nearly perfect energy transmission at large amplitudes. We hypothesize that the amplitude-dependent transmission behavior results from the nonlinear unit cell locking at high amplitudes, and thus acting as a single mass. When this single mass has the same weight as the other masses in the chain, near-perfect transmission is possible. We investigate this behavior further through a nonlinear analysis using a harmonic balance method to predict wave transmission through the nonlinear unit cell. The analysis confirms the transmission behavior and the rigid body hypothesis at large amplitudes. To validate the simulated and analysis results, we design and construct a monatomic chain using steel masses and additively-manufactured serpentine springs, to include a specially-designed nonlinear spring in the hierarchical unit cell. We then document amplitude-dependent filtering in the experiment for multiple frequencies, with strong agreement documented between measured results and simulation. In addition, high-speed camera images of the hierarchical cell verify the hypothesized locking behavior at large amplitudes. We believe the ability to passively select transmission based on a signal’s amplitude, with minimal resulting distortion, may open new opportunities in wave control and filtering, and new approaches for conceiving wave-based devices.

Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions

Abstract In this paper, we introduce a stochastic partial differential equation model for the spatial dynamic of tumor–immune interactions. We perform numerical simulations in order to investigate the propagation of traveling waves in model system under the influence of random space-time fluctuations. One of methods is to solve a stochastic partial differential equation system for tumor–immune cell densities. The second method is to solve a stochastic partial differential algebraic equation system in order to assess the wave behavior of the solution in comparison with the deterministic approach. Finally, we discuss the implications of the model results.

An open-source implementation of GPU-accelerated Finite-Difference Time Domain (FDTD) simulations in the Unity game engine to visualize elastic wave propagation and scattering for pedagogical or training scenarios

Simulations of elastic wave propagation through solid materials can be realized through industry-standard software or custom implementations in a variety of programming languages. These codes are generally optimized for numerical accuracy and can be difficulty to interact with, slow to run, and often require external processing of results to create 3D visualizations. This is especially true for parallelized or Graphical Processing Unit (GPU)-based simulations which require additional libraries. These factors tend to limit the use of numerical simulations of wave propagation in pedagogical applications to simple 2D implementations of acoustic propagation through fluids. Here we present a fully-3D open-source Finite-Difference Time Domain (FDTD) simulation of elastic wave propagation that is implemented in the Unity game development engine. Due to the nature of the engine the simulation code runs entirely on the GPU without additional coding. Additionally the solutions are rendered as the simulation runs which removes the need for saving and post-processing large data files, as well as allowing the user to rotate the viewpoint in real-time to better visualize the wave propagation and scattering in complex geometries.

Realistic modelling of microstructural features in numerical simulations of wave propagation in metals

Research in numerical modelling in recent years has enabled realistic simulations of wave propagation in three dimensions in large volumes of material, while including features at small scale. A critical enabler has been the growth of computer power, especially Finite Element computations on GPUs. The presentation will start with the development of modelling capability for polycrystalline materials, for which extensive recent work has created realistic simulations for wave speed and grain-scattering attenuation, representing the microstructure at grain scale. This topic has received much attention in theoretical work over several decades, so the simulations have been helpful to evaluate and understand the theoretical models and the physics of the behaviour. Subsequently, the modelling has addressed several other microstructural interests, including simulations for creep damage and fatigue damage, each of which cause reductions of wave speed; models of equivalent macro material properties for these are developed based on studies of the microstructural information. Finally, recent work will be presented on wave propagation and scattering in Titanium alloys containing Macro Textured Regions (MTRs/Macrozones); these comprise microstructures of mixed scale, for which there is interest to use ultrasound to characterise the MTRs (large scale) against a background of a smaller scale regular microstructure.

Reduced basis methods for numerical room acoustic simulations with parametrized boundaries.

The use of model-based numerical simulations of wave propagation in rooms for engineering applications requires that acoustic conditions for multiple parameters are evaluated iteratively, which is computationally expensive. We present a reduced basis method (RBM) to achieve a computational cost reduction relative to a traditional full-order model (FOM) for wave-based room acoustic simulations with parametrized boundaries. The FOM solver is based on the spectral-element method; however, other numerical methods could be applied. The RBM reduces the computational burden by solving the problem in a low-dimensional subspace for parametrized frequency-independent and frequency-dependent boundary conditions. The problem is formulated in the Laplace domain, which ensures the stability of the reduced-order model (ROM). We study the potential of the proposed RBM in terms of computational efficiency, accuracy, and storage requirements, and we show that the RBM leads to 100-fold speedups for a two-dimensional case and 1000-fold speedups for a three-dimensional case with an upper frequency of 2 and 1 kHz, respectively. While the FOM simulations needed to construct the ROM are expensive, we demonstrate that the ROM has the potential of being 3 orders of magnitude faster than the FOM when four different boundary conditions are simulated per room surface.

Three-dimensional Gauss–Newton constant-Q viscoelastic full-waveform inversion of near-surface seismic wavefields

SUMMARY Full-waveform inversion (FWI) methods rely on accurate numerical simulation of wave propagation in the analysed medium. Acoustic or elastic wave equations are often used to model seismic wave propagation. These types of simulations do not account for intrinsic attenuation effects due to material anelasticity, and thus correction techniques have been utilized in practice to partially compensate the anelasticity. These techniques often only consider the waveform amplitude correction based on averaging of overall amplitude response over the entire data set, and ignore the phase correction. Viscoelastic wave equations account for the anelastic response in both waveform amplitude and phase, and are therefore a more suitable alternative. In this study, we present a novel 3-D Gauss–Newton viscoelastic FWI (3-D GN-VFWI) method. To address the main challenge of the Gauss–Newton optimization, we develop formulas to compute the Jacobian efficiently by the convolution of virtual sources and backward wavefields. The virtual sources are obtained by directly differentiating the viscoelastic wave equations with respect to model parameters. In order to resolve complex 3-D structures with reasonable computational effort, a homogeneous attenuation (Q factor) is used throughout the analysis to model the anelastic effects. Synthetic and field experiments are performed to demonstrate the utility of the method. The synthetic results clearly demonstrate the ability of the method in characterizing a challenging velocity profile, including voids and reverse velocity layers. The field experimental results show that method successfully characterizes the complex substructure with two voids and undulating limestone bedrock, which are confirmed by invasive tests. Compared to 3-D elastic FWI results, the presented viscoelastic method produces more accurate results regarding depths of the voids and bedrock. This study suggests that the improvement of imaging accuracy would warrant the widespread use of viscoelastic wave equations in FWI problems. To our best knowledge, this is the first reported study on 3-D GN-VFWI at any scale. This study provides the new theory and formulation for the use of Gauss–Newton optimization on the 3-D viscoelastic problem.