Three-dimensional Wave Propagation Research Articles

Three-dimensional thermomechanical wave propagation analysis of sandwich nanoplate with graphene-reinforced foam core and magneto-electro-elastic face layers using nonlocal strain gradient elasticity theory

This article investigates the propagation of bending, longitudinal, and shear waves in a smart sandwich nanoplate with a graphene platelet (GPL)-reinforced foam core and magneto-electro-elastic (MEE) surface layers using sinusoidal higher-order shear deformation theory (SHSDT). The suggested nanoplate is comprised of a Ti–6Al–4V foam core placed between MEE surface layers. The MEE surface layers are composed of a volumetric combination of cobalt-ferrite (CoFe2O4) and barium-titanate (BaTiO3). The foam core and MEE face layers’ material characteristics are temperature dependent. In this study, three different core types are considered: metallic solid core (Type-I), GPL-reinforced solid core (Type-II) and GPL-reinforced foam core (Type-III), as well as three different foam distributions: symmetrical foam I (S-Foam I), symmetrical foam II (S-Foam II) and uniform foam (U-Foam). To derive the nanoplate's equations of motion and determine the system response, Hamilton's principle and Navier's method are employed. The effects of various parameters such as the wave number, nonlocal parameter, foam void coefficient and distribution pattern, GPL volume fraction, and thermal, electric, and magnetic charges, on the phase velocity and wave frequency are investigated via analytical calculations. The findings of the research indicate that the 3-D wave propagation characteristics of the sandwich nanoplate can be considerably modified or tuned with respect to external loads and material parameters. Thus, the proposed sandwich structure is expected to provide important contributions to radar stealth applications, protection of nanoelectromechanical devices from high frequency and temperature environments, advancement of smart nanoelectromechanical sensors characterized by lightweight and temperature sensitivity and wearable health equipment applications.

A comparison of ground motions predicted through one-dimensional site response analyses and three-dimensional wave propagation simulations at regional scales

One-dimensional (1D) site response analysis (SRA), which considers vertically propagating seismic waves from the bedrock to the surface, has been a common technique among geotechnical engineers to examine site-specific ground shaking. However, observations from past earthquakes and analytical studies indicate that idealizations ingrained in 1D SRA may be too severe to capture the ground truth, such as the omissions of spatial variability of soil properties, surface topography, and basin and directivity effects. Physics-based three-dimensional ground motion simulations (GMSs) can incorporate these factors and yield more reliable predictions. In this study, we utilize ground motions from 57 physics-based broadband (from 0 to 8–12 Hz) GMS for a region of Istanbul. A total of 2912 sites with experimentally measured soil profiles that are distributed over the 30 km-by-12.5 km area are also modeled as soil columns and analyzed through 1D SRA. The ground responses from 1D SRA and three-dimensional (3D) GMS are then compared for all 57 earthquake scenarios. These systematic comparisons are then used for examining model features that are correlated with variations in the ratios of various ground motion intensity measures (IMs) and for developing regression-based formulas that can be used for determining simple factors for the considered region to correctly scale (up or down) the site-specific ground motion intensities obtained from 1D SRA, including peak ground acceleration (PGA), peak ground velocity (PGV), and spectral acceleration ( Sa) values.

Basin boundary seismic effects in Mexico City southern region

AbstractRapid changes in geotechnical and geological ground conditions lead to significant ground motion variability. This condition mainly occurs at the so-called basin edges, where there is an abrupt transition between soft highly compressible soils and stiffer materials. This problem becomes more relevant in areas where ground subsidence drastically changes the dynamic response of high plasticity clay deposits, such as those found in Mexico City, due to fundamental site period evolution with time. This paper presents site response analyses at an abrupt transition area in the southeast Mexico City region, along the edges of the Xochimilco-Chalco lakes. Considerable damage associated with three-dimensional wave propagation effects was observed in this zone during the September 2017 Puebla-Mexico earthquake. A series of three-dimensional finite difference numerical models of the basin edge were developed to evaluate ground motion variability, considering topographic effects and soil non-linearities. Good agreement between the computed response and the observed damage during the 2017 Puebla-Mexico earthquake reconnaissance was found. In addition, several normal and subduction events with a return period of 250 years were considered to evaluate the effect that frequency content, and strong ground motion duration have on the soil response variability. From the results gathered here, it was established the relevance of accounting for three-dimensional wave propagation fields to assess site effects at basin-edge zones properly and to be able to implement proper risk mitigation measurements at these zones.

A borehole array data–based approach for conducting 1D site response analyses II: Accounting for modeling errors

Site response estimates from one-dimensional (1D) site response analyses (SRAs) carry inaccuracies due to modeling and parametric errors. Modeling errors are due to the condensation of the three-dimensional (3D) wave propagation phenomenon to the vertical propagation of a horizontally polarized wave through a soil column, and parametric errors are due to the incomplete knowledge of the distributions of soil parameters, leading to the selection of nonoptimal input parameters for a site of interest. While parametric errors are traditionally handled using different soil parameters (e.g. alternative shear-wave velocity profiles), modeling errors are generally neglected. This paper proposes an approach for conducting linear elastic 1D SRAs to improve site response predictions and account for modeling errors. First, ground-motion data from borehole array sites are collected, processed, and screened for appropriateness (e.g. expected shear strains lower than 0.01%, signal-to-noise ratio higher than 3). Second, 1D SRA predictions in terms of transfer functions and amplification factors are compared against observations, and the discrepancies are quantified as residuals. Finally, the residuals are partitioned into a model bias term [Formula: see text], a site term [Formula: see text] with standard deviation [Formula: see text], and a event- and site-specific remaining residual [Formula: see text] with standard deviation [Formula: see text]. Values for [Formula: see text] and [Formula: see text] for forward predictions are recommended. The sensitivity of the site response residuals to region, site type (1D- or 3D-like), and the applicability of findings to outcropping applications are discussed, and an example application for a hypothetical project site is presented.

Dispersion Analysis of Three-Dimensional Elastic Wave Propagation in Transversely Isotropic Media Using Optimally Blended Spectral-Element Method

The accuracy of the optimally blended spectral-element method for wave propagation in a homogeneous transversely isotropic elastic medium was investigated. A nonstandard quadrature rule obtained by combining Gauss quadrature rule and Gauss–Lobatto–Legendre quadrature rule was used to compute elementary matrixes. The mass and stiffness matrixes were represented as the triple tensor-product of elementary matrixes as second-order tensors. The solution of the eigenvalue problem representing the semidiscretized version of elastic wave equation for plane harmonic wave propagation was obtained using the Rayleigh quotient approximation technique. The resulting eigenvalues subsequently were used to estimate the phase and group velocity of three bulk waves. The variations of the errors in these velocities of the bulk waves with the number of grid points per wavelength were depicted graphically. These variations were shown for different polynomial orders and various angles of deviation from the symmetry axis. The effect of a change in the order of time discretization on error variation also was shown. The solution obtained by the optimally blended spectral-element method was found to be more accurate than that from the classical spectral-element method for low-order polynomials. The improvement in solution accuracy was demonstrated through dispersion analysis.

The impact of soil variability on uncertainty in predictions of induced vibrations

Soil is by nature a variable, non-homogeneous material, which has implications for engineering problems involving wave propagation. This paper investigates the impact of the spatial heterogeneity of the stiffness of the soil on the three-dimensional wave propagation. A dynamic Random Finite Element Model is presented in which the soil variability is modelled by means of random fields, applied to the Young’s modulus of the soil, following a Monte-Carlo approach. The results show the importance of accounting for soil variability when making predictions on the maximum vibration level. Deterministic analysis is demonstrated to be insufficient when quantifying the maximum vibration level, because no information on the expected variability of the maximum vibration level is obtained. Furthermore, the scale of fluctuation and anisotropy of the random field strongly impact the estimation of the maximum vibration level, and the time of occurrence.

Morphological and Spectral Features of Ionospheric Structures at E- and F-Region Altitudes over Poker Flat Analyzed Using Modeling and Observations

Electron density irregularities in the ionosphere modify the phase and amplitude of trans-ionospheric radio signals. We aim to characterize the spectral and morphological features of E- and F-region ionospheric irregularities likely to produce these fluctuations or "scintillations". To characterize them, we use a three-dimensional radio wave propagation model-"Satellite-beacon Ionospheric scintillation Global Model of upper Atmosphere" (SIGMA), along with the scintillation measurements observed by a cluster of six Global Positioning System (GPS) receivers called Scintillation Auroral GPS Array (SAGA) at Poker Flat, AK. An inverse method is used to derive the parameters that describe the irregularities by estimating the best fit of model outputs to GPS observations. We analyze in detail one E-region and two F-region events during geomagnetically active times and determine the E- and F-region irregularity characteristics using two different spectral models as input to SIGMA. Our results from the spectral analysis show that the E-region irregularities are more elongated along the magnetic field lines with rod-shaped structures, while the F-region irregularities have wing-like structures with irregularities extending both along and across the magnetic field lines. We also found that the spectral index of the E-region event is less than the spectral index of the F-region events. Additionally, the spectral slope on the ground at higher frequencies is less than the spectral slope at irregularity height. This study describes distinctive morphological and spectral features of irregularities at E- and F-regions for a handful of cases performed using a full 3D propagation model coupled with GPS observations and inversion.

Nonlinear least-squares estimation of shear elasticity and shear viscosity in viscoelastic media

Rapid simulations of three-dimensional (3D) shear wave propagation in viscoelastic media with a Kelvin-Voigt model are enabled by Green’s functions accelerated with graphics processing unit parallelization and high-performance computing resources. Cross-correlation analysis of the wave motion yields an estimate of shear elasticity, where the errors in the estimated values increase with rising shear viscosity. To reduce errors in the shear elasticity estimates, a nonlinear least-squares routine that accounts for the effects of propagation, attenuation, and dispersion is applied to 3D simulated shear wave data in viscoelastic media. The nonlinear least-squares approach also yields an estimate of the shear viscosity when applied to simulated particle velocity waveforms. Post-processing, cross-correlation analysis, and the nonlinear least-squares routine are evaluated on a desktop computer using MATLAB. Estimation of viscoelastic parameters is performed at the focal depth for several combinations of shear elasticity and shear viscosity. The errors in the estimated shear elasticities obtained from the nonlinear least-squares routine and cross-correlation are determined as a function of the input shear elasticity and shear viscosity values. The estimates provided by the nonlinear least-squares routine for shear elasticity and shear viscosity are shown to approach the simulation input values for each combination of viscoelastic parameters.

An open-source GPU-accelerated application of the elastodynamic finite integration technique (EFIT) to three-dimensional wave propagation and scattering in anisotropic materials of arbitrary geometries

As ultrasonic testing (UT) becomes increasingly widespread, accurate numerical simulations of the complex wave propagation behavior in solids become increasingly useful. This is especially true for complex geometries and material properties created by new manufacturing processes such as those use to create Additively Manufactured Metals (AMMs). Here, we present an open-source fully-anisotropic Elastodynamic Finite Integration (EFIT) implementation written in the Julia language which can be deployed to high-performance computers for large-scale parallel simulations. Results of benchmarks against existing isotropic implementations, published data, and collected measurements will be presented.

Radial Seismic Metamaterials Based on Layering Theory: Broadband Shielding of Ultra-Low Frequency Seismic Surface Waves

In this paper, a novel radial seismic metamaterial (LRSM) based on layering theory is proposed. Compared with traditional seismic metamaterials, the structure of LRSM is a periodic array of multi-layer rings distributed along the radial direction. By using the finite element method, the dispersion relationship and displacement vector field of LRSM with different layers are studied, and the influence of structural geometric parameters and circumferential continuity on the band gap characteristics of LRSM is discussed. The frequency domain analysis of finite periodic structure and the three-dimensional transient wave propagation analysis are carried out. The results show that the LRSM has ultra-low frequency broadband characteristics, which is produced by the coupling between the local resonance of the LRSM and the surface wave mode. Comparing three LRSMs with different layers, the initial frequency and bandwidth do not change monotonically with the increase of the number of layers. There is an optimal bandgap characteristic in two layers, and the relative bandwidth can reach 83.9%. The increase of the number causes the change of the structural stiffness, which is caused by the change of the local resonance strength. The position and width of the band gap in the LRSM are very sensitive to the height of the structure. The increase of the height of the LRSM can move the first band gap to the low frequency, and the total bandwidth increases, which is mainly caused by the increase of the equivalent mass of the system with the increase of the height of the structure. Further, it is verified that LRSM can effectively attenuate seismic surface waves of 0.1–20 Hz, and its maximum amplitude attenuation can exceed 85%. The novel periodic structure proposed in this paper can provide new options for the fields of earthquake and low-frequency vibration reduction.

A flexible fully nonlinear potential flow model for wave propagation over the complex topography of the Norwegian coast

Coastal wave propagation and transformation are complicated due to the significant variations of water depth and irregular coastlines, which are typically present at the Norwegian fjords. A potential flow model provides phase-resolved solutions with low demands on computational resources. Many potential flow models are developed for offshore waves and lack of numerical treatments of coastal conditions. In the presented work, several modifications are introduced to a fully nonlinear potential flow model with a σ-grid for the purpose of coastal wave modelling: Shallow water breaking criteria are included in addition to deepwater breaking algorithms to approximate breaking waves over a complete range of water depth. A new coastline algorithm is introduced to detect complex coastlines and ensure robust simulations near the coast. The algorithm is compatible with structured grid arrangement in the horizontal plane and allows for high-order discretisation schemes for the free surface boundary conditions for an accurate representation of complex free surfaces. A parallelised solver for the Laplace equation is utilised to ensure fast simulations for large domains with multi-core infrastructures.The proposed model is validated against theories and experiments for various two- and three-dimensional nonlinear wave propagation and transformation cases that represent typical coastal conditions. The simulations show a good representation of nonlinear waves, and the results compare well with experiments. Furthermore, two large-scale engineering scenarios are simulated, where the applicability of the coastline algorithm and the parallel commutation capability of the model are demonstrated.

Investigation of electromagnetic wave propagation in the bicomplex 3D-FEM using a wavenumber Whitney Hodge operator

PurposeThe purpose of this paper is to show the applicability of a discrete Hodge operator in the context of the De Rham cohomology to bicomplex-valued electromagnetic wave propagation problems. It was applied in the finite element method (FEM) to get a higher accuracy through conformal discretization. Therewith, merely the primal mesh is needed to discretize the full system of Maxwell equations.Design/methodology/approachAt the beginning, the theoretical background is presented. The bicomplex number system is used as a geometrical algebra to describe three-dimensional electromagnetic problems. Because we treat rotational field problems, Whitney edge elements are chosen in the FEM to realize a conformal discretization. Next, numerical simulations regarding practical wave propagation problems are performed and compared with the common FEM approach using the Helmholtz equation.FindingsDifferent field problems of three-dimensional electromagnetic wave propagation are treated to present the merits and shortcomings of the method, which calculates the electric and magnetic field at the same spatial location on a primal mesh. A significant improvement in accuracy is achieved, whereas fewer essential boundary conditions are necessary. Furthermore, no numerical dispersion is observed.Originality/valueA novel Hodge operator, which acts on bicomplex-valued cotangential spaces, is constructed and discretized as an edge-based finite element matrix. The interpretation of the proposed geometrical algebra in the language of the De Rham cohomology leads to a more comprehensive viewpoint than the classical treatment in FEM. The presented paper may motivate researchers to interpret the form of number system as a degree of freedom when modeling physical effects. Several relationships between physical quantities might be inherently implemented in such an algebra.

Structure–Soil–Structure Interaction Analysis for Lateral Seismic Earth Pressure of Deeply Buried Structure in Layered Ground

Abstract Based on the theory of three-dimensional wave propagation, this paper proposed a method to evaluate the amplified seismic responses of two adjacent structures in horizontally stratified gr...

A time-domain approach for the simulation of three-dimensional seismic wave propagation using the scaled boundary finite element method

A direct time-domain approach to simulate seismic wave propagation in three-dimensional unbounded media is proposed based on the Scaled Boundary Finite Element Method (SBFEM). A domain of interest is commonly partitioned into a far field and a near field. The far field is modelled by the semi-analytical SBFEM satisfying rigorously the radiation conditions at infinity. Separate scaled boundary finite elements are employed to reach a balance between computational efficiency and accuracy. The near field is discretized into arbitrarily-shaped scaled boundary finite elements without the occurrence of hanging nodes. This advantage of the SBFEM in mesh generation is leveraged by incorporating the automatic octree-based meshing technique. By exploiting the geometrical similarity of both bounded and unbounded SBFE subdomains the computational cost is reduced. Inspired by the Domain Reduction Method (DRM), seismic waves are introduced to the system via a single layer of elements in the near field. This formulation of seismic input is mathematically convenient as it avoids the direct participation of the formulation of the far field. The proposed approach is attractive in a reliable simulation of the far field, flexible mesh generation of the near field and simple formulation of the seismic excitations. These merits are demonstrated through numerical simulations of seismic wave propagation in a free field and different examples featuring complex geometries in the near fields.

Higher order and continuity [formula omitted] projections with piece-wise constant test functions

We focus on the isogeometric L2 projection problem employing higher-order C1 continuity basis functions that preserve the partition of unity. We show that the rows of the system of linear equations can be combined, and the test functions can be sum up to 1 using the partition of unity property at the quadrature points. Thus, the test functions in higher continuity IGA can be set to piece-wise constants. This formulation is equivalent to testing with piece-wise constant basis functions, with supports span over some parts of the domain. The resulting method is a Petrov–Galerkin formulation with piece-wise constant test functions. This observation has the following consequences. The numerical integration cost can be reduced because we do not need to evaluate the test functions since they are equal to 1. This observation is valid for any basis functions preserving the partition of unity property. It is independent of the problem dimension and geometry of the computational domain. The resulting method is equivalent to a linear combination of the collocations at points and with weights resulting from applied quadrature over the spans defined by supports of the piece-wise constant test functions. We show the algorithm for finding optimal supports of the piece-wise constant test functions using the inf–sup condition argument. We also discuss how the piece-wise constants’ replacement of the test functions can be utilized in modern integration methods, with weighted quadrature, sum factorization, and row by row assembly. We test our method on the isogeometric L2 projection, the explicit dynamics simulations in two and three dimensions, the heat transfer problem, and the three-dimensional elastic wave propagation simulations.

Time-domain modeling of wave-based room acoustics including viscothermal and relaxation effects in air.

Air absorption can be a significant source of attenuation, which should be considered in long-duration wideband acoustics simulations. In this short contribution, a time-domain model for three-dimensional wave propagation including viscothermal and relaxation effects (air absorption) is developed and coupled with locally reactive impedance wall conditions through a conservative energy framework. The model is discretised with a finite-difference time-domain method, and numerical stability is established with a discrete energy balance. Numerical examples are presented to demonstrate the proposed method.

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.