On the viscoelastic-electromagnetic-gravitational analogy

Constraint energy minimizing generalized multiscale discontinuous Galerkin method

A modified mixed-domain method is presented in this paper to model wave propagation in heterogeneous media. The previous mixed domain method is only accurate for weakly heterogeneous media. Both the phase and amplitude corrections are introduced to the mixed-domain method in order to model wave propagation in more strongly heterogeneous media. Two cases are studied to validate the modified method. A two-layered medium with density and speed of sound variation is first studied. A human skull with density and speed of sound variation is then used to further evaluate the accuracy of this method. Result from the angular spectrum approach is used as the benchmark for the layered medium and result from the MATLAB toolbox k-Wave is used as the benchmark for the case of skull. Simulation results show that the modified mixed-domain method is more accurate for strongly heterogeneous media.

The attenuation vector plays an important role in investigating waves propagating in dissipative, isotropic or anisotropic media. It is defined as the imaginary part of the complex-valued traveltime gradient. The real part of the complex-valued traveltime gradient is referred to as the propagation vector. In this paper, simple expressions for the attenuation vector are derived for seismic body waves propagating in heterogeneous, weakly dissipative, anisotropic media. The ray-theory perturbation method is used, in which the dissipative medium is considered to be a small perturbation of a perfectly elastic medium. The general perturbation procedure proposed by Klimeš (2002) is very suitable for this purpose, as it does not require ray tracing in dissipative media; the computation of rays in the reference perfectly elastic medium, followed by quadratures along them, is quite sufficient. It is shown that the waves propagating in a heterogeneous, weakly dissipative, anisotropic or isotropic medium are, in general, inhomogeneous. This means that their attenuation vector is not parallel to the propagation vector. This holds even for waves generated by a point-source in a homogeneous anisotropic weakly dissipative medium. An exception is a wave generated by a point-source in a homogeneous isotropic dissipative medium, where the generated wave is homogeneous. Thus, the commonly used concept of homogeneous waves can be applied neither to heterogeneous nor to anisotropic dissipative media. The situation is different for plane waves propagating in homogeneous dissipative isotropic or anisotropic media. In this case, the homogeneity or inhomogeneity of the plane wave may be chosen freely. Besides the attenuation vector, we also study the complex-valued ray-velocity vector. In heterogeneous media, the complex-valued ray-velocity vector is generally inhomogeneous, that is, its real and imaginary parts are not parallel. It is also shown that twice the scalar product of the attenuation vector with the ray-velocity vector in the reference medium yields 1/Q, where Q is the position- and direction-dependent quality factor. Quality factor Q does not depend on the inhomogeneity of the wave under consideration and offers a convenient measure of intrinsic material dissipation.

We have extended a new highly accurate numerical scheme for unstructured 2D and 3D meshes based on the discontinuous Galerkin approach to simulate seismic wave propagation in heterogeneous media containing fluid-solid interfaces. Because of the formulation of the wave equations as a unified first-order hyperbolic system in velocity stress, the fluid can be in movement along the interface. The governing equations within the moving fluid are derived from well-known first principles in fluid mechanics. The discontinuous Galerkin approach allows for jumps of the material parameters and the solution across element interfaces, which are handled by Riemann solvers or numerical fluxes. The use of Riemann solvers at the element interfaces makesthe treatment of the fluid particularly simple bysetting the shearmodulus in the fluid region to zero. No additional compatibility relations, such as vanishing shear stress or continuity of normal stresses, are necessary to couple the solid and fluid along an interface. The Riemann solver automatically recognizes the jump of the material coefficients at the interface and provides the correct numerical fluxes for fluid-solid contacts. Therefore, wave propagation in the entire computational domain containing heterogeneous media, namely moving fluids and elastic solids, can be described by a uniform set of acoustic and elastic wave equations. The accuracy of the proposed scheme is confirmed by comparing numerical results against analytic solutions. The potential of the new method was demonstrated in a 3D model problem typical for marine seismic exploration with a fluid-solid interface determined by a complicated bathymetry.

Numerical simulation of anti-plane wave propagation in heterogeneous media

Based on a robust numerical method for calculation of band structure for stress-waves in heterogeneous media, a layered composite was designed and fabricated. It is shown that in the second pass band of this simple 2-phase layered structure, the direction of the energy flow is opposite that of the phase velocity. The ultrasonic stress-wave transmission through multiple thicknesses of this structure was studied experimentally and various quantities, such as band structure, phase velocity, group velocity, and a representation of the velocity of energy propagation were measured and compared with numerical predictions. The numerical calculations are based on hybrid (mixed min-max) variational principles presented previously by Nemat-Nasser and coworkers. In the cases where a closed form solution exists, the numerical results of this method agree very accurately with the results of the exact solution. Furthermore, the computational efficiency of the mixed variational method renders it an excellent tool for design and analysis of stress-wave propagation in heterogeneous media.

ABSTRACTTime-domain constant-Q (CQ) viscoelastic wave equations have been derived to efficiently model Q, but are known to break down in accuracy in describing CQ attenuation at low Q. In view of this, a new time-domain viscoelastic wave equation for modeling wave propagation in anelastic medium is evaluated based on Kjartansson’s CQ model to improve the accuracy in describing CQ attenuation at low Q. We use an approximate frequency-domain viscoelastic wave equation to replace the accurate frequency-domain viscoelastic wave equation. Then, a new time-domain wave equation is derived by converting the approximate viscoelastic wave equation from the frequency domain to the time domain. The newly derived viscoelastic wave equation consists of several Laplacian differential operators with variable fractional order. We use an arbitrary-order Taylor series expansion (TSE) to approximate the derived mixed domain fractional Laplacian operators, and realize the decoupling of the wavenumber and fractional order. Then, the proposed viscoelastic wave equation can be solved directly using the staggered-grid pseudospectral method (SGPSM). We evaluate the precision of the new viscoelastic wave equation by comparing the numerical solutions with the analytical solutions in homogeneous medium. Theoretical curve analysis and numerical results indicate that the proposed fractional viscoelastic wave equation has higher precision in describing CQ attenuation than that of the traditional fractional viscoelastic wave equation, especially for cases that P-wave quality factor QP is less than 10, and S-wave quality factor QS is less than 8. Furthermore, we use two numerical examples to verify the effectiveness of the TSE SGPSM in heterogeneous media. The discussion shows that the advantage of using our fractional viscoelastic wave equation over the traditional fractional viscoelastic wave equation is the higher precision in describing CQ attenuation at different frequency.

Atherosclerosis is a chronic vascular disease affecting large and medium sized arteries. Several treatment options are already available for treatment of this disease. Targeting atherosclerotic plaques by high intensity focused ultrasound (HIFU) using dual mode ultrasound arrays (DMUA) was recently introduced in literature. We present a finite difference time domain (FDTD) simulation modeling of the wave propagation in heterogeneous medium from the surface of a 3.5 MHz array prototype with 32-elements. After segmentation of the ultrasound image obtained for the treatment region in-vivo, we integrated this anatomical information into our simulation to account for different parameters that may be caused by these multi-region anatomical complexities. The simulation program showed that HIFU was able to induce damage in the prefocal region instead of the target area. The HIFU lesions, as predicted by our simulation, were well correlated with the actual damage detected in histology.

The dynamics of a single crack moving through a heterogeneous medium is studied in the quasistatic approximation, which may be directly applicable to cracks grown under fatigue. In a model scalar system and for mode III (tearing) loading, the crack surface is found to be self-affine with a roughness exponent of zeta = 1/2. Mode I (tensile) loading, however, leads to a crack surface that is only logarithmically rough, quite unlike those seen in most experiments. Residual stresses are found, potentially, to lead to increased crack surface roughness. But elastic wave propagation effects may be needed to explain the very rough crack surfaces observed experimentally.

Piezoelectric ultrasonic transducers typically employ composite structures to improve their transmission and reception sensitivities. The geometry of the composite is regular with one dominant length scale and, since these are resonant devices, this dictates the central operating frequency of the device. In order to construct a wide bandwidth device it would seem natural therefore to utilise resonators that span a range of length scales. In this paper we consider such a device and build a theoretical model to predict its performance. A fractal medium is used as this contains a wide range of length scales and yields to a renormalisation approach. The propagation of an ultrasonic wave in this heterogeneous medium is then analysed and used to construct expressions for the electrical impedance, and the transmission and reception sensitivities of this device as a function of the driving frequency.

Seismic data are routinely used to infer in situ properties of earth materials on many scales, ranging from global studies to investigations of surficial geological formations. While inversion and imaging algorithms utilizing these data have improved steadily, there are remaining challenges that make detailed measurements of the properties of some geologic materials very difficult. For example, the determination of the concentration and orientation of fracture systems is prohibitively expensive to simulate on the fine grid and, thus, some type of coarse-grid simulations are needed. In this paper, we describe a new multiscale finite element algorithm for simulating seismic wave propagation in heterogeneous media. This method solves the wave equation on a coarse grid using multiscale basis functions and a global coupling mechanism to relate information between fine and coarse grids. Using a mixed formulation of the wave equation and staggered discontinuous basis functions, the proposed multiscale methods have the following properties. • The total wave energy is conserved. • Mass matrix is diagonal on a coarse grid and explicit energy-preserving time discretization does not require solving a linear system at each time step. • Multiscale basis functions can accurately capture the subgrid variations of the solution and the time stepping is performed on a coarse grid. We discuss various subgrid capturing mechanisms and present some preliminary numerical results.

We present a scattering attenuation model based on the statistical wave propagation theory in random media. It is suitable for the weak wavefield fluctuation regime and has practically no restriction in the frequency domain. The presented formulas allow to quantify scattering attenuation in complex geological regions using simple statistical estimates from well-log data. This knowledge is important for further petrophysical interpretations of reservoir rocks. To test our theory we perform numerical simulations of seismic wave propagation in 3-D elastic random media using a finite-difference solution of the elastodynamic wave equation. From the synthetic seismograms we determine the scattering attenuation (Q −1) with help of spectral decay methods. We find good agreement of the frequency-dependent Q values and the theoretical predictions.

We analyze a model for the evolution of shock fronts in polycrystalline materials. This model is based on the idea of Meyers and Carvalho [Mater. Sci. Eng. 24, 131 (1976)] that the shock velocity anisotropy within the polycrystal is the most important factor in shock front broadening. Our analysis predicts that the shock front width increases as the $1∕2$ power of the front penetration distance into the crystal. Our theoretical prediction is in plausible agreement with previous experimental results for the elastic precursor rise time, and it should therefore provide a useful shock width estimate. Furthermore, our theoretical framework is also applicable to other problems involving front propagation in heterogeneous media.

Hydraulic fracture trajectories in rocks and other materials are highly affected by spatial heterogeneity in their mechanical properties. Understanding the complexity and structure of fluid-driven fractures and their deviation from the predictions of homogenized theories is a practical problem in engineering and geoscience. We conduct a Monte Carlo simulation study to characterize the influence of heterogeneous mechanical properties on the trajectories of hydraulic fractures propagating in elastic media. We generate a large number of random fields of mechanical properties and simulate pressure-driven fracture propagation using a phase-field model. We model the mechanical response of the material as that of an elastic isotropic material with heterogeneous Young modulus and Griffith energy release rate, assuming that fractures propagate in the toughness-dominated regime. Our study shows that the variance and the spatial covariance of the mechanical properties are controlling factors in the tortuousness of the fracture paths. We characterize the deviation of fracture paths from the homogenous case statistically, and conclude that the maximum deviation grows linearly with the distance from the injection point. Additionally, fracture path deviations seem to be normally distributed, suggesting that fracture propagation in the toughness-dominated regime may be described as a random walk.

In the present study seismic wave propagation in heterogeneous media is numerically simulated by using the pseudospectral method with the staggered grid RFFT differentiation in order to clarify the cause for the complicated distribution characteristics of strong ground motion in regions with basin structure. The results show that the maximum amplitudes of simulated ground acceleration waveforms are closely related to the basin structure. Interference of seismic waves in the basin strongly affects the distribution of maximum seismic waveforms, which may result in peak disasters during earthquakes. Peak disasters might be away from basin boundaries or earthquake faults. Seismic energy transmitted into the basin from the bedrock can hardly penetrate the bottom of the basin and then travel back into the bedrock region. The seismic energy is absorbed by basin media, and transferred into the kinematical energy of seismic waves with great amplitude in the basin. Seismic waves between basins may result in serious damage to buildings over the basin. This is significant for aseismatic research. Geological surveys in and around urban areas would benefit aseismatic research and mitigation of seismic disasters of a city. Such geological surveys should involve seismic velocity structure in the media above the bedrock besides such subjects as active faults and geological structure.