Wave Propagation In Heterogeneous Media Research Articles

Non‐local dispersive model for wave propagation in heterogeneous media: one‐dimensional case

AbstractNon‐local dispersive model for wave propagation in heterogeneous media is derived from the higher‐order mathematical homogenization theory with multiple spatial and temporal scales. In addition to the usual space–time co‐ordinates, a fast spatial scale and a slow temporal scale are introduced to account for rapid spatial fluctuations of material properties as well as to capture the long‐term behaviour of the homogenized solution. By combining various order homogenized equations of motion the slow time dependence is eliminated giving rise to the fourth‐order differential equation, also known as a ‘bad’ Boussinesq problem. Regularization procedures are then introduced to construct the so‐called ‘good’ Boussinesq problem, where the need for C1 continuity is eliminated. Numerical examples are presented to validate the present formulation. Copyright © 2002 John Wiley & Sons, Ltd.

Non‐local dispersive model for wave propagation in heterogeneous media: multi‐dimensional case

AbstractThree non‐dispersive models in multi‐dimensions have been developed. The first model consists of a leading‐order homogenized equation of motion subjected to the secularity constraints imposing uniform validity of asymptotic expansions. The second, non‐local model, contains a fourth‐order spatial derivative and thus requires C1 continuous finite element formulation. The third model, which is limited to the constant mass density and a macroscopically orthotropic heterogeneous medium, requires C0 continuity only and its finite element formulation is almost identical to the classical local approach with the exception of the mass matrix. The modified mass matrix consists of the classical mass matrix (lumped or consistent) perturbed with a stiffness matrix whose constitutive matrix depends on the unit cell solution. Numerical results are presented to validate the present formulations. Copyright © 2002 John Wiley & Sons, Ltd.

The Effect of Material Heterogeneity on the Shock Response of Layered Systems in Plate Impact Tests

In the study of shock wave propagation in solids, scattering, dispersion, and attenuation play a critical role in determining the thermomechanical response of the media. These phenomena can be attributed to a number of nonlinearities arising from the wave characteristics, loading conditions, material heterogeneity (measured at various spatial scales ranging from nanometers to a few millimeters). The nonlinear effects in general can be ascribed to impedance (and geometric) mismatch present at various length scales as often encountered in composite material systems, apart from material nonlinearities arising from inelastic effects, void nucleation and growth, cracking, and delamination. However, in nearly brittle material systems, elastic effects dominate and is the only effect considered here. Uniaxial strain experiments on the S-2 glass/polymer composite system display markedly different behavior than that observed in monolithic metallic systems [6] and this motivated the present work. Stress profiles measured at various locations along the direction of wave propagation in the plate impact experiments showed that the shock wave rise time increased with propagation in addition to the reduction of peak stress. In this work, we specifically address the issue of shock wave rise time in a simplified multi-layered system. A careful analysis of wave propagation in heterogeneous medium is performed by considering the elastic/acoustic properties of individual lamina in a layered system. An analytical model has been developed to describe the scattering process (reflection/transmission) at various layer interfaces of multilayer composite system. FEM results are then used to compare with the analytical predictions. These results show that the rise time can be a consequence of multiple internal reflections and transmissions occurring at the heterogeneous interfaces, it is further shown that the rise time depends on the magnitude of impedance mismatch and the number of layers.

Uniformly Valid Multiple Spatial-Temporal Scale Modeling for Wave Propagation in Heterogeneous Media

A novel dispersive model for wave propagation in heterogeneous media is developed. The method is based on a higher-order mathematical homogenization theory with multiple spatial and temporal scales. By this approach a fast spatial scale and a series of slow temporal scales are introduced to account for rapid spatial fluctuations of material properties as well as for the long-term behavior of the homogenized solution. The problem of secularity arising from the classical multiple-spatial-scale homogenization theory for wave propagation problems is resolved, giving rise to a uniformly valid dispersive model. The proposed dispersive model is solved analytically and its solution is found to be in good agreement with the numerical solution of the source problem in a heterogeneous medium.

Uniformly Valid Multiple Spatial-Temporal Scale Modeling for Wave Propagation in Heterogeneous Media

Uniformly Valid Multiple Spatial-Temporal Scale Modeling for Wave Propagation in Heterogeneous Media

A New Interface Method for Hyperbolic Problems with Discontinuous Coefficients: One-Dimensional Acoustic Example

A new numerical method, called the explicit simplified interface method (ESIM), is developed in the context of acoustic wave propagation in heterogeneous media. Equations of acoustics are written as a first-order linear hyperbolic system. Apart from interfaces, a standard scheme (Lax–Wendroff, TVD, and WENO) is used in a classical way. Near interfaces, the same scheme is used, but it is applied on a set of modified values deduced from numerical values and jump conditions at interfaces. It amounts to modifying the scheme so that its order of accuracy is maintained at irregular points, despite the nonsmoothness of the solution. This easy-to-implement interface method requires only a few additional computational resources, and it can be applied to other partial differential equations.

Absorbing boundary and free-surface conditions in the phononic lattice solid by interpolation

We have recently developed a new lattice-Boltzmann-based approach for modelling compressional wave propagation in heterogeneous media, which we call the phononic lattice solid by interpolation (PLSI). In this paper, we propose an absorbing boundary condition for the PLSI method in which the microscopic reflection coefficients at the boundaries of a model are set to zero and viscous layers are added to the boundaries. Numerical simulation examples using the PLSI method and comparisons with exact solutions demonstrate that artificial boundary reflections can be almost completely eliminated when the incidence angle is less than approximately 70°, Beyond this angle, remanent artificial boundary reflections become visible. We propose four methods for modelling free-surface reflections in PLSI simu-lations. In the first three methods, special collision rules at a free surface are specified to take into account the effect of a free surface on quasi-particle movements (i.e. wave propagation) They are termed the specular bouncing, backward bouncing I, and combined bouncing methods. They involve quasi-particle reflections with a coefficient of − 1 and require the free surface to be located exactly along lattice nodes. For the fourth method, we modify the backward bouncing I model for the case when a free surface is located at any position along lattice links and thus term it the backward bouncing II model. It uses the reflection coefficient at the free surface to calculate the reflected number densities during PLSI simulations. Hence, the free surface is handled in the same way as an interface within a model. Numerical examples and comparisons with exact solutions show that these four methods used at the microscopic scale are all appropriate for modelling macroscopic waves reflected from free surfaces.

Dynamic Domain Decomposition in Approximate and Exact Boundary Control in Problems of Transmission for Wave Equations

This paper is concerned with dynamic domain decomposition for optimal boundary control and for approximate and exact boundary controllability of wave propagation in heterogeneous media. We consider a cost functional which penalizes the deviation of the final state of the solution of the global problem from a specified target state. For any fixed value of the penalty parameter, optimality conditions are derived for both the global optimal control problem and for local optimal control problems obtained by a domain decomposition and a saddle-point--type iteration. Convergence of the iterations to the solution of the global optimality system is established. We then pass to the limit in the iterations as the penalty parameter increases without bound and show that the limiting local iterations converge to the solution of the optimality system associated with the problem of finding the minimum norm control that drives the solution of the global problem to a specified target state.

Mechanisms of reaction wave propagation during combustion synthesis of advanced materials

Using a digital high-speed microscopic video recording technique developed in our laboratory, we have investigated hightemperature combustion synthesis waves in the Ni+Al system. Based on the results obtained, we identify characteristic features of rapid high-temperature reaction wave propagation in heterogeneous media. It is shown that, while at time scales ∼10 -1 s the reaction wave propagation appears to be steady, at time scales ∼10 -3 s it exhibits features of hesitations followed by jumps, and at even faster time scales ∼10 -4 s is shown to follow a novel scintillating reaction wave mechanism.

Use of a transient wave propagation code for 3D simulation of cw radiated transducer fields

In order to compute continuous wave (cw) ultrasonic wave fields in complex media, where analytical approaches are extremely difficult, numerical simulations on large computational grids must be employed. The use of a time domain code, normally used for transient wave propagation in heterogeneous media, is used here as a tool to simulate continuous wave fields. As a starting point, the case of the three-dimensional pressure field radiated from a circular aperture in water is computed. These numerical simulations are performed on a massively parallel computer and compared with experiment and known theory. The computations are performed on very large three-dimensional grids that span the near to the far field. As a trial case, a numerical computation of the radiated field from a continuous-wave excited transducer in a baffle is compared with an analytical evaluation using the Rayleigh surface integral and experiment. In addition, results are presented that show the effect of a small defect placed in the beam. To do this, a small cylindrical copper scatterer was placed in the near field, in both the computation and an accompanying experiment. These cases are done in preparation for using the same approach for computing cw fields radiated from a transducer into complex heterogeneous media.

Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers

This paper reports on the development of a parallel numerical methodology for simulating large-scale earthquake-induced ground motion in highly heterogeneous basins. We target large sedimentary basins with contrasts in wavelengths of over an order of magnitude. Regular grid methods prove intractable for such problems. We overcome the problem of multiple physical scales by using unstructured finite elements on locally-resolved Delaunay triangulations derived from octree-based grids. The extremely large mesh sizes require special mesh generation techniques. Despite the method's multiresolution capability, large problem sizes necessitate the use of distributed memory parallel supercomputers to solve the elastic wave propagation problem. We have developed a system that helps automate the task of writing efficient portable unstructured mesh solvers for distributed memory parallel supercomputers. The numerical methodology and software system have been used to simulate the seismic response of the San Fernando Valley in Southern California to an aftershock of the 1994 Northridge Earthquake. We report on parallel performance on the Cray T3D for several models of the basin ranging in size from 35 000 to 77 million tetrahedra. The results indicate that, despite the highly irregular structure of the problem, excellent performance and scalability are achieved.

Distributed three-dimensional finite-difference modeling of wave propagation in acoustic media

Finite-difference modeling of wave propagation in heterogeneous media is a useful technique in a number of disciplines, including earthquake and oil exploration seismology, laboratory ultrasonics, ocean acoustics, radar imaging, nondestructive evaluation, and others. However, the size of the models that can be treated by finite-difference methods in three spatial dimensions has limited their application to supercomputers. We describe a finite-difference domain-decomposition method for the three-dimensional acoustic wave equation which is well suited to distributed parallelization. We have implemented this algorithm using the PVM message-passing library, and show here benchmarks on two different distributed memory architectures, the IBM SP2 and a network of low-cost PCs running the Linux operating system. We present performance measurements of this algorithm on both the low-bandwidth PC network (10-Mbits/s Ethernet) and the high-bandwidth SP2 cluster (40-Mbits/s switch). These results demonstrate the feasibility of doing distributed finite-difference acoustic modeling on networks of workstations, but point to the substantial efficiencies that can be expected as higher bandwidth networks become available. © 1997 American Institute of Physics.

Physics and Modeling of Shock-Wave Dispersion in Heterogeneous Composites

Wave dispersion in heterogeneous solids due to scattering within the microstructure is examined. Underlying physics is explored. A theory based on a quasi-harmonic normal model description of the scattering wave energy is developed and compared with phenomenological nonlinear anelasticity models of wave propagation in heterogeneous media.

Two-phase flow behind a shock wave with phase transitions and chemical reactions

A theoretical and experimental study was undertaken of the problems of shock wave propagation in heterogeneous media containing an oxidant in the gaseous phase and a fuel in the condensed phase. The fuel was in the form of dispersed droplets in the flow of oxidant or in the form of a thin film on the wall of a tube. A system of governing equations with boundary conditions is composed that makes it possible to simulate numerically the initiation of detonation and the acceleration and slowing down of an unsteady wave to a self-sustaining regime. To verify the mathematical model, experiments were carried out in a shock tube for pure gas, for inert model, experiments were carried out in a shock tube for pure gas, for inert dispersed droplets, and for combustible dispersed droplets. The comparison of the results shows good agreement of theoretical and experimental data.

Scale effects on dynamic wave propagation in heterogeneous media

To study scale‐effects on wave propagation in 3‐D heterogeneous media, we conducted dynamic and static laboratory experiments on seven samples of glass beads cast with epoxy. We measured P and S wave velocities and frequency dispersion by the pulse‐transmission method, and static elastic Young's modulus by a uniaxial stress test. We changed the scale of the heterogeneity by varying the diameter of the glass beads in each sample, and obtained wavelength to scale ratios varying from 0.2 to 20. We observed about 22% P‐wave velocity dispsersion and 15% S‐wave velocity dispersion within this range of wavelength to scale ratios. We observed no scale effects on the static Young's modulus of the same seven samples. It is clear that strong wave velocity dispersion in the experiment is due to the dynamic wavelength‐scale effects caused by scattering.

Rayleigh scattering in elastic composite materials

This paper is devoted to long wave propagation in heterogeneous media. More specifically, we deal with Rayleigh diffraction in elastic materials with a periodic microstructure whose heterogeneities are in finite concentration or show great contrasts in properties. This study is based on the homogenization method but contrary to the usual procedure in which only the first significant terms are used, the developments are established up to the third order. We demonstrate that the terms of a superior order successively introduce effects of polarization, of celerity dispersion and of attenuation and we thus bring to the fore a characteristic distance of mode conversion. Finally we demonstrate that the effect of dispersion alone appears in macroscopically isotropic materials.

Dynamics of nonlinear waves in chemical active media

Chemical waves are prototypes of self-organized spatio-temporal structures observed in a large class of distributed active media of physical, chemical or biological origin far from thermodynamic equilibrium. We discuss recent progress in the understanding of complex dynamics of spiral waves which was obtained by exploiting the possibility of external control of pattern formation in a light-sensitive modification of the Belousov-Zhabotinsky reaction. Also we comment on some aspects of wave propagation in heterogeneous media.

A Novel VSP Method for Fault Proximity Detection using Low Velocity Waveguides

A potentially useful seismic technique for fault detection has been tested by finite-difference modelling of elastic wave propagation in heterogeneous media. It employs a surface-to-borehole vertical seismic profiling (VSP) geometry. Geophones are lowered into the borehole and seismic sources are fired at or near the ground surface. Small disruptions to a low-velocity zone (LVZ) bounded by higher velocity media can be detected by the technique with the analysis of seismic arrival patterns recorded at geophones in a nearby borehole. A disruption in the LVZ will act as a secondary source when an incident wave strikes it. Some of the energy radiated from the secondary source can be captured by the LVZ and will propagate as guided waves. Both offset VSP and walkaway VSP recording geometries are considered for the numerical simulation. In the offset geometry, a spread of geophones is deployed in the borehole around the coal seam while a single source is fired. Both disrupted and continuous LVZ models have been considered for numerical simulations. Explosive sources (P-wave energy only) are used. Synthetic seismograms show that channel waves can be excited by the scattered energy at a LVZ disruption. The maximum amplitude of the channel waves is comparable to those of direct body waves, which clearly indicates that the disruption in the LVZ can act as an efficient source to generate guided waves. In the walkaway VSP geometry, a single geophone is fixed in the LVZ and a spread of sources is fired on the surface away from the borehole. Synthetic seismograms demonstrate that the apex of the channel-wave arrivals pinpoints the location of the disruption of the LVZ for a layered model. Although this technique has parallels with in-seam seismic coal exploration, it does not require that the source be placed within the low-velocity channel. This method is equally applicable to any exploration target which relates to a LVZ.

Interface conditions for acoustic and elastic wave propagation

The divergence theorem is used to handle the physics required at interfaces for acoustic and elastic wave propagation in heterogeneous media. The physics required at regular and irregular interfaces is incorporated into numerical schemes by integrating across the interface. The technique, which can be used with many numerical schemes, is applied to finite differences. A derivation of the acoustic wave equation, which is readily handled by this integration scheme, is outlined. Since this form of the equation is equivalent to the scalar SH wave equation, the scheme can be applied to this equation also. Each component of the elastic P‐SV equation is presented in divergence form to apply this integration scheme, naturally incorporating the continuity of the normal and tangential stresses required at regular and irregular interfaces.

Three-dimensional numerical modeling of geoacoustic scattering from seafloor topography

A three-dimensional (3-D), second-order finite-difference method was used to create synthetic seismograms for wave propagation in heterogeneous media in order to investigate the scattering of elastic and acoustic energy due to topography on the seafloor. The method uses a fully staggered grid in Cartesian coordinates as developed by Virieux [Geophysics 51, 889-901 (1986)]. Numerical results were generated for two models: a linear fault scarp on the seafloor, and a flat seafloor containing a rectangular channel. Wave-front snapshots allow the scattering and focusing of different wave modes with direction to be visualized. Compressional and shear wave backscattering from the sides of the features can be seen together with the trapped compressional wave energy propagating inside the channel. The results illustrate the effects of out of the plane scattering due to simple seafloor topographic features.