Waves In Heterogeneous Media Research Articles

An unsplit-field viscoelastic complex-frequency-shifted perfectly matched layer for analysis of transient waves in heterogeneous media based on an efficient voxel element method

Wave propagation analysis of heterogeneous unbounded media employing domain discretization methods necessitates truncation of the finite computational domain at its truncation boundaries. The Perfectly Matched Layer (PML) is a robust absorbing boundary condition, which is extensively employed to favorably absorb outgoing scattered waves from the interior computational domain, regardless of frequencies and directions of outgoing waves. Nonetheless, most available PMLs are developed for elastic domains, while their developments in domain discretization methods, which consider damping using the Rayleigh damping formulation, are insufficient. In this study, a viscoelastic PML formulation is developed that accommodates the Rayleigh-type damping. The PMLs are basically developed from a coordinate transformation concept using a complex stretching function. Suitable stretching functions can efficiently control the absorptive properties and the spectral character of the PML. Besides, the stretching functions play a crucial role in the long-time stability of transient wave simulations. The standard stretching function usually results in spurious reflections, especially when propagating waves incident to the truncation boundary at near-grazing incidence. On the other hand, the complex-frequency-shifted (CFS) stretching function has been shown to mitigate long-time instability. Therefore, the proposed viscoelastic PML in the time-domain and frequency-domain are derived using the CFS stretching function for the first time. Comprehensive formulations of the proposed viscoelastic CFS-PML are given, where the displacement field is represented as the sole unknown variable of the problem. Hence, the proposed CFS-PML can be incorporated into any domain discretization method. However, the proposed CFS-PML is further developed in the context of the voxel element method, where element-by-element iterative strategies are implemented to solve the governing equations without storing global matrices in the computer memory. Comprehensive numerical experiments are conducted to demonstrate the validity and efficiency of the proposed viscoelastic CFS-PML formulation in analyzing soil profiles with various damping ratios and properties. Moreover, long-time stability for problems involving waveguides and near-grazing wave incidence is studied, where no instabilities are detected.

P-SV-wave propagation in heterogeneous media: Velocity-stress distributional finite-difference method

We have developed a 2D distributional-finite-difference (DFD) algorithm for modeling the propagation of seismic waves in heterogeneous media in the time domain. We revisit the classic staggered finite-difference algorithm by substituting the standard finite-difference operators with the recently introduced DFD operators with the methodological differences we underline. We find that the DFD operators improve the simulation accuracy while maintaining the simple structure of the finite-difference algorithm. Thanks to its weak formalism, the newly developed algorithm accurately and naturally accounts for the free surface, which is a substantial improvement for finite-difference approaches. Moreover, we develop an efficient factorization for the DFD operators. It limits the computational cost of the proposed algorithm to twice that of the finite-difference algorithm. Numerical examples demonstrate that the extra computational burden is well compensated by the superior accuracy of the DFD operators.

A refined spectral element model for wave propagation in multiscale hybrid epoxy/carbon fiber/graphene platelet composite shells

The propagation of elastic waves in heterogeneous media is of interest for impact dynamics and non-destructive detection. This work presents a refined spectral element model (RSEM) to study the wave propagation in multiscale hybrid composite (MHC) shells subjected to impulsive loadings. The doubly-curved MHC shell consists of epoxy, carbon fibers, and graphene platelets (GPLs). The GPLs are functionally distributed along the thickness of the shell. For the three-phase MHC, the Halpin-Tsai micromechanical model in conjunction with the Mori-Tanaka approach is exploited to determine the effective material properties. In the framework of four-variable shear deformation theory, the governing equations along with the natural boundary conditions are derived using Hamilton's principle. A two-node spectral shell element is developed according to the closed-from solutions. The accuracy of the RSEM is verified by comparison with published results in aspects of the natural frequency and transient responses. The wave dispersion characteristics, including the wave number, phase velocity, and group velocity are examined. In the context of high frequency and short wavelength, the proposed RSEM achieves high computational efficiency benefiting from its independence of mesh structure. The time domain responses clearly indicate the wave-boundary interactions, e.g., wave reflection, dispersion, and interference. It is revealed that the present model can well capture the fundamental wave modes of the MHC shell. Moreover, the inclusion of GPLs plays a significant role in improving transverse moduli and mitigating the discontinuities of inter-laminar shear stress.

Distributional finite-difference modelling of seismic waves

SUMMARYThis study introduces a distributional finite-difference method (DFDM) for modelling the propagation of elastic waves in heterogeneous media in the time domain. DFDM decomposes the modelling domain into multiple elements that can have arbitrary sizes. When large elements are used, the proposed method closely resembles the finite-difference method because the wavefield is updated using operations involving band diagonal matrices only. Thus DFDM is computationally efficient. When smaller elements are used, DFDM looks closer to the finite-element or the spectral element methods and permits to mesh complicated structures. A complete multidomain algorithm for modelling elastic wave propagation in arbitrarily heterogeneous media is presented. The algorithm’s stability is discussed, and the usual Courant condition governs the stability of the proposed scheme. Numerical examples show that the proposed algorithm accurately accounts for free surfaces, solid–fluid interfaces and accommodates non-conformal meshes in their basic form. Seismograms obtained using the proposed method are compared to those computed using analytical solutions and the spectral element method. To achieve comparable accuracy, DFDM requires fewer points per wavelength than the spectral element method, for example.

СРАВНИТЕЛЬНЫЙ АНАЛИЗ 1: НАЧАЛЬНАЯ СТАДИЯ ДИНАМИЧЕСКОГО ДЕФОРМИРОВАНИЯ И РАЗРУШЕНИЯ АЛЮМИНИЕВЫХ СПЛАВОВ 1561 И 1565

Transition of dynamically deformed solids into structure-unstable state is analyzed theoretically and experimentally from the physical kinetics approach. Propagation of shock wave in heterogeneous medium is considered to be in the form of superposition of two modes: (i) propagation of plane front with the mean velocity u(x,t) and (ii) random motions of local regions of medium because of chaotic stress fields. resulting from interaction of shock wave with the heterogeneous medium. This interaction results in particle velocity distribution, the quantitative characteristic of which is the particle velocity dispersion. Transition from one scale of dynamic deformation to another is found to be the so-called defect (decrease) of the particle velocity which characterizes the intensity of energy and momentum exchange between scale levels. Both characteristics are registered in real time in the process of shock-wave tests by using precise interferometric technique. Two materials, 1561 and 1565 aluminum alloys, were tested under uniaxial strain conditions within impact velocity range of 250–750 m/s. Owing to fine focusing of laser beam of interferometer up to 50–70 µm, the registered dynamic strength (spall strength) of material corresponds to single mesoscale structural element, which allows the incipient stage of dynamic fracture to be revealed in detail. The incipient strength is found to be maximum when velocity variation becomes equal to the velocity defect.

Unfitted hybrid high-order methods for the wave equation

We design an unfitted hybrid high-order (HHO) method for the wave equation. The wave propagates in a domain where a curved interface separates subdomains with different material properties. The key feature of the discretization method is that the interface can cut more or less arbitrarily through the mesh cells. We address both the second-order formulation in time of the wave equation and its reformulation as a first-order system. We prove H1-error estimates for a space semi-discretization in space of the second-order formulation, leading to optimal convergence rates for smooth solutions. Numerical experiments illustrate the theoretical findings and show that the proposed numerical schemes can be used to simulate accurately the propagation of acoustic waves in heterogeneous media, with meshes that are not fitted to the geometry. For the second-order formulation, the implicit, second-order accurate Newmark scheme is used for the time discretization, whereas (diagonally-implicit or explicit) Runge–Kutta schemes up to fourth-order accuracy are used for the first-order formulation. In the explicit case, we study the CFL condition on the time step and observe that the unfitted approach combined with local cell agglomeration leads to a comparable condition as when using fitted meshes.

Convergence Analysis of Hybrid High-Order Methods for the Wave Equation

We prove error estimates for the wave equation semi-discretized in space by the hybrid high-order (HHO) method. These estimates lead to optimal convergence rates for smooth solutions. We consider first the second-order formulation in time, for which we establish \({\varvec{H}}^1\) and \({\varvec{L}}^2\)-error estimates, and then the first-order formulation, for which we establish \({\varvec{H}}^1\)-error estimates. For both formulations, the space semi-discrete HHO scheme has close links with hybridizable discontinuous Galerkin schemes from the literature. Numerical experiments using either the Newmark scheme or diagonally-implicit Runge–Kutta schemes for the time discretization illustrate the theoretical findings and show that the proposed numerical schemes can be used to simulate accurately the propagation of elastic waves in heterogeneous media.

System formulation of the nonlinear relaxing wave equation for high-order finite-difference approaches

Finite difference solutions of the wave equation can describe the propagation of waves in heterogeneous media. High-order finite-difference approaches have been used extensively for long-range propagation for seismological applications. For ultrasound imaging, therapy, and neuromodulation, the description of ultrasound propagation must also include attenuation, dispersion, and nonlinearity. Describing these physical phenomena with high-order, stencils has been a persistent challenge. Here we develop a formulation of the nonlinear, attenuating, dispersing, wave equation in system form that is ideally suited for a wide variety of high-order finite-difference stencils. Relaxation functions are modeled in the wave equation by stretching the velocity and pressure variables into a new relaxed coordinate system that describes attenuation, dispersion, and perfectly matched layers using a single formalism. The final system of equations, composed only of first-order derivatives, allows for the implementation of sophisticated stencils. Then, a high-order finite-difference solution of these equations is presented. Solutions of wave propagation are demonstrated for ultrasound imaging applications for deep human abdominal imaging and compared to experimental measurements showing that the numerical method accurately captures the speckle statistics, aberration strength, reverberation strength, and coherence properties. Applications are demonstrated for transcranial neuromodulation, array design, super-resolution, and machine learning.

Sensitivity of sonic boom propagation throughout a turbulent atmosphere

Sensitivity of sonic boom propagation throughout a turbulent atmosphere is investigated. Three types of boom of same amplitude but different initial shapes: an ideal N-wave, a measured boom (NASA data for F-18) and a “low” boom (C25D mock-up) with increased rise time are studied. The atmosphere is supposed to be a quiescent and isothermal medium with a superposed synthetic velocity field with homogeneous and isotropic statistical properties satisfying a von Kármán energy spectrum. Using the “random field generation method,” the flow velocity turbulent field is governed by three independent parameters: a random matrix, an intensity parameter and a scale parameter (turbulence integral scale). The flow velocity is then used as a base flow for a in-house software called FLHOWARD designed to compute the propagation of acoustic shock waves in heterogeneous media. In order to reduce the number of simulations compared to a Monte-Carlo approach, the study is performed within the generalized chaos polynomial (gPC) framework. Various convergence tests have been performed to define the optimal discretization and gPC order. Stochastic evolution of selected metrics along a 1 km distance are investigated.

Simulation of elastic wave propagation across fractures using a nodal discontinuous Galerkin method—theory, implementation and validation

SUMMARY An existing nodal discontinuous Galerkin (NDG) method for the simulation of seismic waves in heterogeneous media is extended to media containing fractures with various rheological behaviour. Fractures are treated as 2-D surfaces where Schoenberg’s linear slip or displacement discontinuity condition is applied as an additional boundary condition to the elastic wave equation which is in turn implemented as an additional numerical flux within the NDG formulation. Explicit expressions for the new numerical flux are derived by considering the Riemann problem for the elastic wave equation at fractures with varying rheologies. In all cases, we obtain further first order differential equations that fully describe the temporal evolution of the particle velocity jump at the fracture. Our flux formulation allows to separate the effect of a fracture from flux contributions due to simple welded interfaces enabling us to easily declare element faces as parts of a fracture. We make use of this fact by first generating the numerical mesh and then building up fractures by selecting appropriate element faces instead of adjusting the mesh to pre-defined fracture surfaces. The implementation of the new numerical fluxes into NDG is verified in 1-D by comparison to an analytical solution and in 2-D by comparing the results of a simulation valid in 1-D and 2-D. Further numerical examples address the effect of fracture systems on seismic wave propagation in 1-D and 2-D featuring effective anisotropy and coda generation. Finally, a study of the reflective and transmissive behaviour of fractures indicates that reflection and transmission coefficients are controlled by the ratio of signal frequency and relaxation frequency of the fracture.

Space-Time Discontinuous Petrov–Galerkin Methods for Linear Wave Equations in Heterogeneous Media

Abstract We establish an abstract space-time DPG framework for the approximation of linear waves in heterogeneous media. The estimates are based on a suitable variational setting in the energy space. The analysis combines the approaches for acoustic waves of Gopalakrishnan–Sepulveda [J. Gopalakrishnan and P. Sepulveda, A space-time DPG method for the wave equation in multiple dimensions, Space-Time Methods. Applications to Partial Differential Equations, Radon Ser. Comput. Appl. Math. 21, Walter de Gruyter, Berlin 2019, 129–154] and Ernesti–Wieners [J. Ernesti and C. Wieners, A space-time discontinuous Petrov–Galerkin method for acoustic waves, Space-Time Methods. Applications to Partial Differential Equations, Radon Ser. Comput. Appl. Math. 21, Walter de Gruyter, Berlin 2019, 99–127] and is based on the abstract definition of traces on the skeleton of the time-space substructuring. The method is evaluated by large-scale parallel computations motivated from applications in seismic imaging, where the computational domain can be restricted substantially to a subset of the full space-time cylinder.

Finite element heterogeneous multiscale method for elastic waves in heterogeneous media

A finite element heterogeneous multiscale method (FE-HMM) is proposed for the simulation of time-dependent elastic waves in a rapidly varying heterogeneous elastic medium. It is based on a standard finite element discretization of an effective wave equation at the macro scale, whose a priori unknown effective material coefficients are computed on sampling domains at the micro scale within each macro finite element. Hence the computational effort becomes independent of the highly heterogeneous elastic medium at the smallest scale. Optimal error estimates and convergence rates in the energy and the L2 norm are derived, which are explicit in the macro and micro discretization errors. Numerical experiments verify the sharpness of the error bounds and illustrate the versatility of the method for non-periodic, layered or stochastic media.

Piecewise parabolic method for simulating one-dimensional shear shock wave propagation in tissue-mimicking phantoms

The recent discovery of shear shock wave generation and propagation in the porcine brain suggests that this new shock phenomenology may be responsible for a broad range of traumatic injuries. Blast-induced head movement can indirectly lead to shear wave generation in the brain, which could be a primary mechanism for injury. Shear shock waves amplify the local acceleration deep in the brain by up to a factor of 8.5, which may tear and damage neurons. Currently, there are numerical methods that can model compressional shock waves, such as comparatively well-studied blast waves, but there are no numerical full-wave solvers that can simulate nonlinear shear shock waves in soft solids. Unlike simplified representations, e.g., retarded time, full-wave representations describe fundamental physical behavior such as reflection and heterogeneities. Here we present a piecewise parabolic method-based solver for one-dimensional linearly polarized nonlinear shear wave in a homogeneous medium and with empirical frequency-dependent attenuation. This method has the advantage of being higher order and more directly extendable to multiple dimensions and heterogeneous media. The proposed numerical scheme is validated analytically and experimentally and compared to other shock capturing methods. A Riemann step-shock problem is used to characterize the numerical dissipation. This dissipation is then tuned to be negligible with respect to the physical attenuation by choosing an appropriate grid spacing. The numerical results are compared to ultrasound-based experiments that measure planar polarized shear shock wave propagation in a tissue-mimicking gelatin phantom. Good agreement is found between numerical results and experiment across a 40 mm propagation distance. We anticipate that the proposed method will be a starting point for the development of a two- and three-dimensional full-wave code for the propagation of nonlinear shear waves in heterogeneous media.

Application of generalised finite differences method to reflection and transmission problems in seismic SH waves propagation

A matrix formulation of the generalised finite difference method is introduced. A necessary and sufficient condition for the uniqueness of the solution is demonstrated, and important practical consequences are obtained. A generalised finite differences scheme for SH wave is obtained, the stability of the scheme is analysed and the formula for the velocity of the wave due to the scheme is obtained in order to deal with the numerical dispersion. The method is applied to seismic waves propagation problems, specifically to the problem of reflection and transmission of plane waves in heterogeneous media. A heterogeneous approach without nodes at the interface is chosen to solve the problem in heterogeneous media. Copyright © 2017 John Wiley & Sons, Ltd.

Fractional biharmonic operator equation model for arbitrary frequency-dependent scattering attenuation in acoustic wave propagation.

This paper proposes a fractional biharmonic operator equation model in the time-space domain to describe scattering attenuation of acoustic waves in heterogeneous media. Compared with the existing models, the proposed fractional model is able to describe arbitrary frequency-dependent scattering attenuation, which typically obeys an empirical power law with an exponent ranging from 0 to 4. In stark contrast to an extensive and rapidly increasing application of the fractional derivative models for wave absorption attenuation in the literature, little has been reported on frequency-dependent scattering attenuation. This is largely because the order of the fractional Laplacian is from 0 to 2 and is infeasible for scattering attenuation. In this study, the definition of the fractional biharmonic operator in space with an order varying from 0 to 4 is proposed, as well as a fractional biharmonic operator equation model of scattering attenuation which is consistent with arbitrary frequency power-law dependency and obeys the causal relation under the smallness approximation. Finally, the correlation between the fractional order and the ratio of wavelength to the diameter of the scattering heterogeneity is investigated and an expression on exponential form is also provided.

Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation

The numerical simulation of time-harmonic waves in heterogeneous media is a tricky task which consists in reproducing oscillations. These oscillations become stronger as the frequency increases, and high-order finite element methods have demonstrated their capability to reproduce the oscilla-tory behavior. However, they keep coping with limitations in capturing fine scale heterogeneities. We propose a new approach which can be applied in highly heterogeneous propagation media. It consists in constructing an approximate medium in which we can perform computations for a large variety of frequencies. The construction of the approximate medium can be understood as applying a quadrature formula locally. We establish estimates which generalize existing estimates formerly obtained for homogeneous Helmholtz problems. We then provide numerical results which illustrate the good level of accuracy of our solution methodology.

Influence of initial stress and gravity on torsional surface wave in heterogeneous medium

The present paper investigates the effect of gravity and initial stress on the propagation of torsional surface waves in heterogeneous medium. The dispersion equation has been obtained for rigid and traction free boundaries in terms of Whittaker function. The present study reveals that torsional surface waves can propagate in both the cases. In this paper, we assume the expansion of Whittaker function up to linear term. It is observed that, in presence of initial stress and gravity for both rigid and traction free boundary, the phase velocity of torsional surface waves increases with the growth of rigidity. In both the cases, it has also been noticed that as the gravity increases, the phase velocity of the torsional surface waves decreases in presence of heterogeneity and initial stress. It is concluded that in presence of traction free boundary, the phase velocity of the torsional surface waves is more than the rigid boundary.

Cellular wave mode of the infiltration combustion of porous media

The paper summarizes the results of experimental studies of the structuring of the infiltration combustion front in a burning titanium powder layer. The unsteady processes of formation and propagation of cellular and nonuniform combustion waves in heterogeneous media in straight-through and semi-closed channels in conditions of restricted gas exchange and presence of inert gaseous impurities are analyzed. The structural characteristics of cellular combustion waves in a metal powder layer under conditions of natural gas infiltration are reported.

The subgrid modeling of propagation of acoustic waves in heterogeneous media with multiscale isotropic random elastic stiffness and density

The effective coefficients in the problem of the acoustic wave propagation have been calculated for a multiscale 3D isotropic medium using a subgrid modeling approach. The density and the elastic stiffness have been represented mathematically by the Kolmogorov multiplicative cascades, which, to date, appear to be the only mechanisms for generating a stationary multifractal fields with a log-stable probability distribution. The fields with the stable distribution are described with the help of linear combination random values , and weight coefficients , , which satisfy certain conditions in the nodes of spatial grid . The parameters of the stable distribution of the random values , are equal: , , , . The wavelength is assumed to be large as compared with the scale of heterogeneities of the medium. We consider the regime in which the waves propagate over a distance of the typical wave length in source. The theoretical results obtained in this paper are compared with the results of a direct 3D numerical simulation.

Dynamic earthquake rupture simulations on nonplanar faults embedded in 3D geometrically complex, heterogeneous elastic solids

Dynamic propagation of shear ruptures on a frictional interface in an elastic solid is a useful idealization of natural earthquakes. The conditions relating discontinuities in particle velocities across fault zones and tractions acting on the fault are often expressed as nonlinear friction laws. The corresponding initial boundary value problems are both numerically and computationally challenging. In addition, seismic waves generated by earthquake ruptures must be propagated for many wavelengths away from the fault. Therefore, reliable and efficient numerical simulations require both provably stable and high order accurate numerical methods.We present a high order accurate finite difference method for: a) enforcing nonlinear friction laws, in a consistent and provably stable manner, suitable for efficient explicit time integration; b) dynamic propagation of earthquake ruptures along nonplanar faults; and c) accurate propagation of seismic waves in heterogeneous media with free surface topography.We solve the first order form of the 3D elastic wave equation on a boundary-conforming curvilinear mesh, in terms of particle velocities and stresses that are collocated in space and time, using summation-by-parts (SBP) finite difference operators in space. Boundary and interface conditions are imposed weakly using penalties. By deriving semi-discrete energy estimates analogous to the continuous energy estimates we prove numerical stability. The finite difference stencils used in this paper are sixth order accurate in the interior and third order accurate close to the boundaries. However, the method is applicable to any spatial operator with a diagonal norm satisfying the SBP property. Time stepping is performed with a 4th order accurate explicit low storage Runge-Kutta scheme, thus yielding a globally fourth order accurate method in both space and time. We show numerical simulations on band limited self-similar fractal faults revealing the complexity of rupture dynamics on rough faults.