Three-dimensional Wave Propagation Research Articles

Seismic Earth Pressure Due to Rayleigh Waves in Viscoelastic Media

ABSTRACT This paper presents an analytical solution to calculate the seismic earth pressure induced by Rayleigh wave in viscoelastic media using the three-dimensional wave propagation theory. Two soil-structure systems are considered: a substructure inserted into a homogeneous soil layer and a substructure surrounded by a multilayered soil profile. Closed-form solutions are proposed as functions of damping ratio, Poisson’s ratio, and the height of the substructure. Furthermore, two special cases were investigated to examine the influences of a loose and a stiff thin layer in the layered soil profiles. The results obtained for different scenarios were compared systematically in a comprehensive numerical example.

Three-Dimensional Trampolinelike Behavior in an Ultralight Elastic Metamaterial

Elastic metamaterials possess band gaps, or frequency ranges that are forbidden to wave propagation. Existing solutions for impeding three-dimensional (3D) wave propagation largely rest on high-volume fractions of mass inclusions that induce and tailor negative effective density-based local resonances. This study introduces a class of elastic metamaterials that achieve low-frequency band gaps with a volume fraction as low as 3% (mass density as low as $0.034\phantom{\rule{0.1em}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$). The working of the proposed design hinges on a 3D trampolinelike mode behavior that gives rise to wide, omnidirectional, and low-frequency band gaps for elastic waves despite very low-mass densities. Such a 3D trampoline effect is derived from a network of overhanging nodal microarchitectures that act as locally resonating elements, which give rise to band gaps at low frequencies. The dynamic effective properties of the metamaterial are numerically examined, which reveal that the band gap associated with the trampoline effect is resulted from a negative effective modulus coupled with a near-zero yet positive effective density. The experimental characterization is then made possible by fabricating the metamaterial via a light-based printing system that is capable of realizing microarchitectures with overhanging microfeatures. This design strategy could be useful to applications where simultaneous light weight and vibration control is desired.

A Discontinuous Galerkin Method for Three-Dimensional Poroelastic Wave Propagation: Forward and Adjoint Problems

We develop a numerical solver for three-dimensional wave propagation in coupled poroelastic-elastic media, based on a high-order discontinuous Galerkin (DG) method, with the Biot poroelastic wave equation formulated as a first order conservative velocity/strain hyperbolic system. To derive an upwind numerical flux, we find an exact solution to the Riemann problem, including the poroelastic-elastic interface; we also consider attenuation mechanisms both in Biot's low- and high-frequency regimes. Using either a low-storage explicit or implicit-explicit (IMEX) Runge-Kutta scheme, according to the stiffness of the problem, we study the convergence properties of the proposed DG scheme and verify its numerical accuracy. In the Biot low frequency case, the wave can be highly dissipative for small permeabilities; here, numerical errors associated with the dissipation terms appear to dominate those arising from discretisation of the main hyperbolic system. We then implement the adjoint method for this formulation of Biot's equation. In contrast with the usual second order formulation of the Biot equation, we are not dealing with a self-adjoint system but, with an appropriate inner product, the adjoint may be identified with a non-conservative velocity/stress formulation of the Biot equation. We derive dual fluxes for the adjoint and present a simple but illuminating example of the application of the adjoint method.

Analyzing three-dimensional wave propagation with the hybrid reproducing kernel particle method based on the dimension splitting method

By introducing the dimension splitting method into the reproducing kernel particle method (RKPM), a hybrid reproducing kernel particle method (HRKPM) for solving three-dimensional (3D) wave propagation problems is presented in this paper. Compared with the RKPM of 3D problems, the HRKPM needs only solving a set of two-dimensional (2D) problems in some subdomains, rather than solving a 3D problem in the 3D problem domain. The shape functions of 2D problems are much simpler than those of 3D problems, which results in that the HRKPM can save the CPU time greatly. Four numerical examples are selected to verify the validity and advantages of the proposed method. In addition, the error analysis and convergence of the proposed method are investigated. From the numerical results we can know that the HRKPM has higher computational efficiency than the RKPM and the element-free Galerkin method.

Cross-sectional photoacoustic tomography image reconstruction with a multi-curve integration model

Background and objectiveIn acoustic inversion of photoacoustic tomography (PAT), an imaging model that precisely describes both the ultrasonic wave propagation and the detector properties is of crucial importance. Inspired by the multi-stripe integration model in clinical X-ray computed tomography systems, in this work, we introduce the Multi-Curve-Integration-based acoustic inversion for cross-sectional Photoacoustic Tomography (MCI-PAT). MethodsWe assumed that in cross-sectional PAT system, the three-dimensional (3-D) wave propagation problem could be reduced to a two-dimensional (2-D) problem in a limited, yet sufficient field of view. Under such condition, the MCI-PAT imaging model is generated by integrating several circular acoustic curves, the centers of which are points evenly distributed on the finite-size ultrasonic transducer surface. In this way, the spatial detector response is taken into account, while its computational burden does not largely increase because the integration process is performed only on a 2-D plane. ResultsAs proven by simulation, phantom and in vivo small animal experiments, the MCI-PAT method leads to promising improvement in PAT image quality. Comparing to traditional imaging models that considered only a single acoustic curve, our proposed method successfully improved the visibility of small structures and achieved evident enhancement on signal-to-noise ratio. ConclusionsThe performance of the MCI-PAT method demonstrates that for cross-sectional PAT, a 2-D simplification of the propagation of multiple photoacoustic waves is feasible. Due to its simplicity, our method can be used as an add-on to current system models considering only a single acoustic curve.

An Analytical Solution for Longitudinal Impedance of a Large-Diameter Floating Pile in Soil with Radial Heterogeneity and Viscous-Type Damping

An analytical model is presented for solving the longitudinal complex impedance of a large-diameter floating pile in viscoelastic surrounding soil with radial heterogeneity and viscous-type damping, taking the effect of three-dimensional wave propagation of soil and lateral inertia of the pile shaft into account. The corresponding analytical solution for longitudinal impedance is also derived and validated via comparisons with existing solutions. The influences of the pile length, Poisson’s ratio of the pile shaft and the viscous damping coefficient, as well as the degree and radius of disturbed surrounding soil, on the longitudinal impedance of the pile shaft are examined by performing parametric analyses. It is demonstrated that the proposed analytical model and solution are suitable for the longitudinal vibration problem of a large-diameter pile and radially inhomogeneous surrounding soil, especially when the pile slenderness is low. In addition, the present solution can be easily degenerated to describe the longitudinal vibration problem relating to a large-diameter floating pile in radially homogenous soil or a pile with fixed-end supports.

One-Way Wave Equation Derived from Impedance Theorem

The wave equations for longitudinal and transverse waves being used in seismic calculations are based on the classical force/moment balance. Mathematically, these equations are 2nd order partial differential equations (PDE) and contain two solutions with a forward and a backward propagating wave, therefore also called “Two-way wave equation”. In order to solve this inherent ambiguity many auxiliary equations were developed being summarized under “One-way wave equation”. In this article the impedance theorem is interpreted as a wave equation with a unique solution. This 1st order PDE is mathematically more convenient than the 2nd order PDE. Furthermore the 1st order wave equation being valid for three-dimensional wave propagation in an inhomogeneous continuum is derived.

Dependence of skull surface wave propagation on stimulation sites and direction under bone conduction.

In order to better understand bone conduction sound propagation across the skull, three-dimensional (3D) wave propagation on the skull surface was studied, along with its dependence on stimulation direction and location of a bone conduction hearing aid (BCHA) actuator. Experiments were conducted on five Thiel embalmed whole head cadaver specimens. Stimulation, in the 0.1-10 kHz range, was sequentially applied at the forehead and mastoid via electromagnetic actuators from commercial BCHAs, supported by a 5-N steel band. The head response was quantified by sequentially measuring the 3D motion of ∼200 points (∼15-20 mm pitch) across the ipsilateral, top, and contralateral skull surface via a 3D laser Doppler vibrometer (LDV) system, guided by a robotic positioner. Low-frequency stimulation (<1 kHz) resulted in a spatially complex rigid-body-like motion of the skull that depended on both the stimulation condition and head support. The predominant motion direction was only 5-10 dB higher than other components below 1 kHz, with no predominance at higher frequencies. Sound propagation direction across the parietal plates did not coincide with stimulation location, potentially due to the head base and forehead remaining rigid-like at higher frequencies and acting as a large source for the deformation patterns across the parietal sections.

A full-field simulation methodology for sonic boom modeling on adaptive Cartesian cut-cell meshes

This paper develops a full-field direct numerical simulation methodology of sonic boom in a stratified atmosphere. The entire flow field, ranging from the near field around a supersonic body to the far field extending to the ground, is modeled by the three-dimensional Euler equations with a gravitational source term. Thus far, it has been solved using a structured grid, and an application of previous simulation to complex geometries has been limited. In this study, we realize a full-field simulation by employing the following four numerical approaches: (i) a hierarchical structured adaptive mesh refinement (AMR) method, (ii) a Cartesian cut cell method, (iii) a well-balanced finite volume method, and (iv) a segmentation method of the computational domain. A new well-balanced, MUSCL-Hancock scheme applied over Cartesian AMR and cut cell grids for a stratified atmosphere is formulated. The computational results of an oblique shock wave in a stratified atmosphere agree well with the exact solution. A full-field simulation successfully reproduces the Drop test for Simplified Evaluation of Non-symmetrically Distributed sonic boom (D-SEND) #1, conducted by the Japan Aerospace Exploration Agency (JAXA). The results of this simulation are in good agreement with those of the previous computational study, the waveform parameter method, and flight test measurements. The grid convergence study shows that the mesh size is fine enough to assess pressure signatures over the entire flow field. These results demonstrate that a full-field simulation with AMR and cut cell grids is a powerful tool for extensively analyzing three-dimensional shock wave propagation in a stratified atmosphere.

Unified Approach to Solving the Trajectory Problem by Usage of Neural Maps

In modern robotics, there are many approaches to planning the trajectory of the portal manipulator. Methods based on the use of graphs, methods of potential fields, with the use of cellular decomposition, intelligent and others are used. Many varieties are based on similar mathematical apparatus. An important aspect of the trajectory search method applicability is convenience of chosen workspace and obstacles representation method. The approach proposed for manipulator path modelling implies a multi-level process, which includes periodic calculation and updating of invalid configurations set, considering the held cargo dimensions, a neural map creation and activation by the wave front propagation method etc. During experimental simulations with different activation functions, numbers of artificial neurons and synaptic connection implementation types, few methods of neural map activation have been considered, including flat and/or three-dimensional raster methods and wave propagation with static obstacles.

Behaviour of reinforcement in drop tower beam tests

Drop tower tests help to gain understanding about the general behaviour of reinforced concrete members under impact loading and to analyse strains and strain rates occurring within their reinforcement. For this purpose, beam and slab specimens are usually employed. The main advantage of beams compared to slabs is that they are less complex due to the almost two-dimensional instead of three-dimensional wave propagation within them. To investigate the steel strains and strain rates, ten impact tests on beam specimens with various impact energies were performed. The impactor sizes and velocities were varied. The reinforcement bars of the beams were instrumented with semiconductor strain gauges. The measured data suggest that the occurring strains in beam tests are independent of the loading velocity. The same was found for the strain rates. The reason is that higher impact energies mostly influence the concrete damage due to spalling on the impact-facing side which happens after the maximum strains occurred. The strains in the reinforcement bars generally result from the overall deflection because of the impact, the spreading of longitudinal waves in the horizontal direction, and the localized cracking of the concrete due to the formation of a punching cone.

Three-Dimensional Magnetostatic Waves Caused by Electromagnetic Fields in Metallized Ferrite Plates

By using a well-known technique for finding the characteristics of magnetic fields based on the method of decomposition of unknown quantities in a series in small parameter (chosen as the ratio of amplitude of the tangential harmonic field to the amplitude of normal constant field) and restricting ourselves to the first two terms of decomposition, we establish the conditions of excitation and study the regularities of propagation of three-dimensional magnetostatic waves in metallized normally magnetized ferrite plates depending on the character of the external electromagnetic field and electrophysical characteristics of the material.

A weight-adjusted discontinuous Galerkin method for the poroelastic wave equation: Penalty fluxes and micro-heterogeneities

We introduce a high-order weight-adjusted discontinuous Galerkin (WADG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in anisotropic porous media. We use a coupled first-order symmetric stress-velocity formulation [1,2]. Careful attention is directed at (a) the derivation of an energy-stable penalty-based numerical flux, which offers high-order accuracy in presence of material discontinuities, and (b) proper treatment of micro-heterogeneities (sub-element variations) in the numerical scheme. The use of a penalty-based numerical flux avoids the diagonalization of Jacobian matrices into polarized wave constituents necessary when solving element-wise Riemann problems. Micro-heterogeneities are accurately and stably incorporated in the numerical scheme using easily-invertible weight-adjusted mass matrices [3]. The convergence of the proposed numerical scheme is proven and verified by using convergence studies against analytical plane wave solutions. The proposed method is also compared against an existing implementation using the spectral element method to solve the poroelastic wave equation [4].

Reflected wave manipulation via acoustic metamaterials with decoupled amplitude and phase

We present the analytical derivation and numerical simulation for the wavefront manipulation of reflected waves at a lossy surface. It is found that the incident plane wave can be reflected in the desired direction or conversed to the trapped surface mode without parasitic scatterings by introducing acoustic energy loss at the surface. To realize the lossy inhomogeneous surface, holey-structured acoustic metamaterial arrays with decoupled reflection amplitude and phase are used. Anomalous reflection phenomena, including non-specular reflection, negative reflection, and conversion of the propagating wave into the surface wave, are achieved by modulating the decoupled amplitude and phase. Our results provide a practical and straightforward way to manipulate the reflected waves, which can also be extended to the three-dimensional wave propagation.

Application of Elastodynamic Finite Integration Technique (EFIT) to three-dimensional wave propagation and scattering in arbitrary geometries

Over several decades, railroad ultrasonic examination (UE) industry techniques have primarily been developed through simple analytical modeling and experimental approaches. However, with present-day computational capabilities, we can use numerical techniques such as the Elastodynamic Finite Integration Technique (EFIT) to fine-tune systems for complex applications before the fabrication process begins. EFIT is well-established as a useful method in numerical analysis of ultrasonic wave propagation with distinct advantages over the Finite Difference Time Domain method. Several software packages exist that use EFIT as the primary method for simulating the behavior of ultrasonic waves over time in two or three dimensions, but none of them are well-suited for railroad UE R&amp;D. This paper explores the development of a tool developed for this purpose which was designed to: (1) allow for the input of various profile geometries, boundary conditions, and material inclusion geometries (such as a bolt hole in a railroad track); (2) allow for the input of specific ultrasonic impulses from varying emitter designs; and (3) produce verifiable results, as confirmed by experimental measurements.

Magnetic resonance elastography of the human brain using a multiphase DENSE acquisition.

PurposeIn magnetic resonance elastography (MRE), a series of time‐shifted images is acquired at specific phase offsets in relation to an induced mechanical excitation. To efficiently gather the set of phase offset images and to overcome limitations due to prolonged TEs and related susceptibility artifacts at low‐frequency MRE, we developed an improved displacement encoding with a stimulated echoes (DENSE) method.MethodsThe proposed multiphase DENSE‐MRE acquisition scheme allows full sampling of the wave propagation in 1 encoding direction during each TR using multiple readouts at specific phase offsets. With this approach, all phase offsets can be imaged in 1 TR without the need for whole sequence repetitions at time‐shifted offsets relative to the excitation motion. We tested this technique in phantom experiments with 60 Hz and in the brain of 4 volunteers using 20‐Hz harmonic excitation.ResultsThree‐dimensional wave propagation could be acquired in 7 minutes 30 seconds. Following background phase elimination, clear wave images were obtained, showing the propagation of the waves over time. Calculated shear modulus maps of the phantom matched well to the maps obtained by conventional gradient‐echo MRE. In the brain, low‐frequency DENSE‐MRE images were free of susceptibility‐induced artifacts and the calculated maps showed a median global complex shear modulus magnitude of 0.72 kPa and phase angle of 1.03 rad across volunteers.ConclusionThe proposed multiphase DENSE approach allows efficient low‐frequency MRE with short TEs and is well‐suited for low‐frequency MRE of the human brain.

Effects of Central Tube on Shape of Modal Duct of a Helicoid in a Simple Expansion Chamber

The shape of the modal duct of an acoustic wave propagating in a muffling system varies with the internal geometry. This shape can be either as a result of plane wave propagation or three-dimensional wave propagation. These shapes depict the distribution of acoustic pressure that may be used in the design or modification of mufflers to create resonance at cut-off frequencies and hence achieve noise attenuation or special effects on the output of the noise. This research compares the shapes of acoustic duct modes of two sets of four pitch configurations of a helicoid in a simple expansion chamber with and without a central tube. Models are generated using Autodesk Inventor modeling software and imported into ANSYS 18.2, where a fluid volume from the complex computer-aided-design (CAD) geometry is extracted for three-dimensional (3D) analysis. Mesh is generated to capture the details of the fluid cavity for frequency range between 0 and 2000Hz. After defining acoustic properties, acoustic boundary conditions and loads were defined at inlet and outlet ports before computation. Postprocessed acoustic results of the modal shapes and transmission loss (TL) characteristics of the two configurations were obtained and compared for geometries of the same helical pitch. It was established that whereas plane wave propagation in a simple expansion chamber (SEC) resulted in a clearly defined acoustic pressure pattern across the propagation path, the distribution in the configurations with and without the central tube depicted three-dimensional acoustic wave propagation characteristics, with patterns scattering or consolidating to regions of either very low or very high acoustic pressure differentials. A difference of about 80 decibels between the highest and lowest acoustic pressure levels was observed for the modal duct of the geometry with four turns and with a central tube. On the other hand, the shape of the TL curve shifts from a sinusoidal-shaped profile with well-defined peaks and valleys in definite multiples of π for the simple expansion chamber, while that of the other two configurations depended on the variation in wavelength that affects the location of occurrence of cut-on or cut-off frequency. The geometry with four turns and a central tube had a maximum value of TL of about 90 decibels at approximately 1900Hz.

On perfectly matched layers for discontinuous Petrov–Galerkin methods

In this article, several discontinuous Petrov–Galerkin (DPG) methods with perfectly matched layers (PMLs) are derived along with their quasi-optimal graph test norms. Ultimately, two different complex coordinate stretching strategies are considered in these derivations. Unlike with classical formulations used by Bubnov–Galerkin methods, with so-called ultraweak variational formulations, these two strategies in fact deliver different formulations in the PML region. One of the strategies, which is argued to be more physically natural, is employed for numerically solving two- and three-dimensional time-harmonic acoustic, elastic, and electromagnetic wave propagation problems, defined in unbounded domains. Through these numerical experiments, efficacy of the new DPG methods with PMLs is verified.

Stability analysis and applications of traveling wave solutions of three-dimensional nonlinear modified Zakharov–Kuznetsov equation in a magnetized plasma

Propagation of three-dimensional nonlinear solitary waves in a magnetized electron–positron plasma is analyzed. Modified extended mapping method is further modified and applied to three-dimensional nonlinear modified Zakharov–Kuznetsov equation to find traveling solitary wave solutions. As a result, electrostatic field potential, electric field, magnetic field and quantum statistical pressure are obtained with the aid of Mathematica. The new exact solitary wave solutions are obtained in different forms such as periodic, kink and anti-kink, dark soliton, bright soliton, bright and dark solitary waves, etc. The results are expressed in the forms of trigonometric, hyperbolic, rational and exponential functions. The electrostatic field potential and electric and magnetic fields are shown graphically. The soliton stability of these solitary wave solutions is analyzed. These results demonstrate the efficiency and precision of the method that can be applied to many other mathematical physical problems.

Solutions of nonplanar KP-equations for dusty plasma system with GE-method

The propagation of three-dimensional nonlinear dust acoustic waves in a dusty plasma consisting of positive and negative dust grains as well as Boltzmann distributed electrons and ions is investigated. Using a reductive perturbation method, Cylindrical Kadomtsev-Petviashvili and Spherical Kadomtsev-Petviashvili equations, appropriate for describing the evolution of the system, are derived. The Generalized Expansion method is used to find the various solutions of the obtained nonlinear Kadomtsev-Petviashvili equations. New classes of triangular, hyperbolic solitary, rational, and Jacobi elliptical type solutions are obtained and graphically presented. It is found that the proposed dusty plasma model can support solitary waves with compressive and rarefactive potential pulses. Moreover, the effects of plasma parameters on this solitary wave structure are investigated. The current findings are applied to a cosmic-type plasma in different regions of space, viz. cometary tails, mesosphere, and Jupiter's magnetosphere as well as laboratory-type plasma such as low temperature experiments, where a dusty plasma with opposite polarity is dominant.