Random Mass Density Field Research Articles

Impact force and moment problems on random mass density fields with fractal and Hurst effects.

This paper reports the application of cellular automata to study the dynamic responses of Lamb-type problems for a tangential point load and a concentrated moment applied on the free surface of a half-plane. The medium is homogeneous, isotropic and linear elastic while having a random mass density field with fractal and Hurst characteristics. Both Cauchy and Dagum random field models are used to capture these effects. First, the cellular automata approach is tested on progressively finer meshes to verify the code against the continuum elastodynamic solution in a homogeneous continuum. Then, the sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst parameters. Overall, the mean response amplitude is lowered by the mass density field's randomness, while the Hurst parameter (especially, for β < 0.2) is found to have a stronger influence than the fractal dimension on the response. The resulting Rayleigh wave is modified more than the pressure wave for the same random field parameters. Additionally, comparisons with previously studied Lamb-type problems under normal in-plane and anti-plane loadings are given. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Anti-plane shear Lamb's problem on random mass density fields with fractal and Hurst effects

This paper reports a study of transient dynamic responses of the anti-plane shear Lamb's problem on random mass density field with fractal and Hurst effects. Cellular automata (CA) is used to simulate the shear wave propagation. Both Cauchy and Dagum random field models are used to capture fractal dimension and Hurst effects in the mass density field. First, the dynamic responses of random mass density are evaluated through a comparison with the homogenerous computational results and the classical theoretical solution. Then, a comprehensive study is carried out for different combinations of fractal and Hurst coefficients. Overall, this investigation determines to what extent fractal and Hurst effects are significant enough to change the dynamic responses by comparing the signal-to-noise ratio of the response versus the signal-to-noise ratio of the random field.

Lamb's problem on random mass density fields with fractal and Hurst effects.

This paper reports on a generalization of Lamb's problem to a linear elastic, infinite half-space with random fields (RFs) of mass density, subject to a normal line load. Both, uncorrelated and correlated (with fractal and Hurst characteristics) RFs without any weak noise restrictions, are proposed. Cellular automata (CA) is used to simulate the wave propagation. CA is a local computational method which, for rectangular discretization of spatial domain, is equivalent to applying the finite difference method to the governing equations of classical elasticity. We first evaluate the response of CA to an uncorrelated mass density field, more commonly known as white-noise, of varying coarseness as compared to CA's node density. We then evaluate the response of CA to multiscale mass density RFs of Cauchy and Dagum type; these fields are unique in that they are able to model and decouple the field's fractal dimension and Hurst parameter. We focus on stochastic imperfection sensitivity; we determine to what extent the fractal or the Hurst parameter is a significant factor in altering the solution to the planar stochastic Lamb's problem by evaluating the coefficient of variation of the response when compared with the coefficient of variation of the RF.

Acoustic waves in random fields

The effect of space- and time-dependent random mass density, velocity, and pressure fields on frequencies and amplitudes of acoustic waves is considered by means of the analytical perturbative method. The analytical results, which are valid for weak fluctuations and long wavelength sound waves, reveal frequency and amplitude alteration, the effect of which depends on the type of random field. In particular, the effect of a random mass density field is to increase wave frequencies. Space-dependent random velocity and pressure fields reduce wave frequencies. While space-dependent random fields attenuate wave amplitudes, their time-dependent counterparts lead to wave amplification. In another example, sound waves that are trapped in the vertical direction but are free to propagate horizontally are affected by a space-dependent random mass density field. This effect depends on the direction along which the field is varying. A random field, which varies along the horizontal direction, does not couple vertically standing modes but increases their frequencies and attenuates amplitudes. These modes are coupled by a random field which depends on the vertical coordinate, but the dispersion relation remains the same as in the case of the deterministic medium.

Frequency and amplitude alterations of sound modes in a space-dependent random mass density field

We investigate the effect of a space-dependent random mass density field on small amplitude acoustic modes that are settled in a semi-infinite medium of a temperature growing linearly with depth. Using a perturbation method, the dispersion relation is derived in the form of Hill's determinant. Numerical solutions of this equation lead to the following conclusions: (a) a weak random field (with σeff = 0.05) essentially affects long waves which experience attenuation and a frequency reduction; (b) for a stronger random field (with σeff = 0.1), high-order sound modes behave as sound waves as they are attenuated and their frequencies are increased; (c) for a sufficiently strong random field (with σeff = 0.2), mode coupling occurs, as a result of which the dispersive curves cross each other, the sound modes loose their identities, and some modes are amplified. Here σeff denotes the effective strength of a random field.

Random sound waves in a weakly stratified atmosphere

The influence of random mass density and velocity fields on the frequencies and amplitudes of the sound waves that propagate along a constant gravity field is examined in the limit of weak random fields, small amplitude oscillations and a weakly stratified medium. Using a perturbative method, we derive dispersion relations from which we conclude that the effect of a space-dependent random mass density field is to attenuate sound waves. Frequencies of these waves are higher than in the case of a coherent medium. A time-dependent random mass density field increases frequencies and amplifies the sounds waves. On the other hand, a space-dependent random flow reduces the wave frequencies and attenuates the sound waves. The time-dependent random flow raises the frequencies of the sound waves and amplifies their amplitudes. In the limit of the gravity-free medium the above results are in an agreement with the former findings.

The effect of time-dependent random mass density field on frequencies of solar sound waves

The eect of a time-dependent random mass density eld on the frequencies and amplitudes of solar p-modes approximated as sound waves is considered by analytical perturbative means and numerical simulations for one-dimensional hydrodynamic equations. The analytical results, which are worked out for a Gaussian spec- trum of the random mass fluctuations, show frequency increase and amplitude amplication, in agreement with numerical simulations.

Solar acoustic oscillations in a random density field

The influence of a space-dependent random mass density field on the development of solar p -modes is investigated using analytical and numerical means. Using a perturbative approach, which is valid for a weak random field and small amplitude waves, we derive a linear dispersion relation whose solutions correspond to attenuated oscillations. The real part of the frequency of these oscillations exceeds the one of waves propagating in a medium without random density. We give an interpretation of the unphysical nature of the frequency shift and of the amplitude attenuation which is similar to Landau damping. The analytical findings are compared with the results of the numerical solution of a model wave equation. We find that, for weak random fields and for wavelengths which are a few times the correlation length of the random density fluctuations, numerical results agree with the analytical theory. Two practical formulas for deriving the correlation spectrum of the random density field from observations are also given.

The Intermittent Behavior of the Cosmic Mass Field Revealed by a L[CLC]y[/CLC]α Forest QSO

The intermittent behavior of the space-scale distribution of Ly$\alpha$ transmitted flux of QSO HS1700+64 has been analyzed via a discrete wavelet transform. We found that there are strong indications of intermittency on scales down to about 10 $h^{-1}$ kpc. These are: 1.) the probability distribution function of the local fluctuations of the flux is significantly long-tailed on small scales, and 2.) the local power spectrum of the flux shows prominent spiky structures on small scales. Moreover, the local power spectrum averaged on regions with different sizes shows similar spiky structures. Therefore, the random mass density field traced by the Ly$\alpha$ forests is rougher on smaller scales, consistent with singular clustering.