Fractal and Hurst Effects in Stochastic Mechanics

In this paper how to generate respiratory flow that has fractal signal feature is introduced. Physiological signal have fractal feature have been verified by many researchers, such as heart beat rate, interbreath interval. Mechanical ventilators are used to provide life support for patients with respiratory failure. But these machines can damage the lung, causing them to collapse. On the other hand, fractal feature can be used as an indication of health situation; as a result in patient simulation the physiological signal should also have fractal features. The fractal feature is generated by fractional Brownian motion simulation. The fractal dimension is decided by Hurst exponent in routine. The algorithm is realized by R language and result is input into LabVIEW which have friendly interface and easy for simulation control usage. The method can be used in design of mechanical ventilator and medical patient simulator.

Most of the recent research on random surface contact mechanics has been on self-affine surfaces. In such models, the fractal dimension (which represents the ‘roughness’) and the Hurst parameter (which represents the ‘spatial memory’) are linearly dependent. In this study, we investigate the non-adhesive, frictionless contact between elastic solids with non-self-affine manifolds. In particular, we use Cauchy and Dagum covariance functions, which can decouple the fractal and Hurst effects, to describe the height distribution of the random surfaces. A numerical model based on the Boussinesq point load fundamental solutions is employed along with the discrete convolution FFT method to perform the contact analysis. We investigate the true contact area evolution under increasing load for surfaces with a wide range of fractal and Hurst parameters. It is observed that the contact area evolution at light loads is almost independent of the Hurst parameter and non-monotonically dependent on the fractal dimension. By contrast, previous studies predicted the contact evolution to be weakly dependent on the Hurst parameter and the fractal dimension. The curvature of the plots of the slope of the contact area evolution is found to depend on the fractal dimension, contrary to previous studies, which predicted either convexity or concavity.

Acceleration waves on random fields with fractal and Hurst effects

Geophysical granular flows, such as snow avalanches and debris flows, represent a great hazard to man and infrastructures. These natural phenomena are characterized by the rapid flow of a granular solid phase, embedded in an ambient fluid. Since such events occur with little warning, in addition to investigating the triggering conditions, it is of crucial interest to predict their propagation and run-out distances. The main purpose of the present research is to better understand the flow dynamics of dry granular flows, in the presence of no-slip bottom boundary conditions, and predict their propagation by using a depth-averaged approach. In this case, a stratification of different flow regimes, such as quasi-static and dense collisional, occurs and makes the utilization of the classical single layer depth-averaged models insufficient, because of important uncertainties about the velocity and shear stress distributions along the flow depth. In the first part of the present work, an experimental-numerical study on dam-break flows of dry granular material is reported. It has been found that a modification to the formula, proposed in the Savage-Hutter model for calculating the earth-pressure coefficient, leads to an improved agreement with experimental data, in presence of no-slip bottom conditions. In the second part, an experimental study on steady state velocity profiles of dry granular flows is reported. The granular material, used in this experimental research, was Ottawa sand (ASTM C-778 20/30). The velocity profiles at the side walls and free surfaces have been obtained, through granular PIV techniques. The measurements are in accordance with other experimental works on different granular materials and suggest the occurrence of a rheological stratification along the flow depth, in case of no-slip bottom condition. In order to better describe such a complex flow dynamics, a two-layer depth-averaged model has been proposed. The dynamics of the two layers, ideally corresponding to dense-collisional and quasi-static regimes, have been considered independently. As well, mass exchanges between the layers have calculated through a physically based closure equation. The well-known mathematical issue of the hyperbolicity loss in two-layer models has been carefully addressed. In order to overcome it, a local modification of source terms of the original mathematical model has been proposed. Such a treatment consists of introducing an extra resistance at the interface, if necessary, to avoid the hyperbolicity loss. This approach has been found to be robust and yields reasonable results, as long as the asymptotic steady state solution and boundary conditions fulfil the hyperbolicity requirements of the original model. Through comparisons with experimental data, the two-layer approach turns out to be very promising to describe the complex flow dynamics in case of no-slip bottom boundary conditions, although a more detailed description of the basal shear stress seems to be required. In the third part of the present dissertation, the same two-layer approach has been used in order to rewrite the mathematical model in a curvilinear coordinates system, attached to the interface between the two layers. The model equations have been obtained, by using the same approach proposed in Tai and Kuo (2008). In the model derivation, the physically negligible quantities are identified in a more rational way than the previous model, by means of a scaling approximation.

Aim at the dynamic reliability of stochastic truss structures under stationary random excitation. In consideration of the randomness of structural physical parameters and the geometric parameters and the random excitations were first transformed into sinusoidal ones in terms of the pseudo excitation method ( PEM) , that turned the joint-random problem into a single random problem for which only structural parameters remain random, and thus the analysis is greatly simplified. The mean value and the variance of the mean square value of the structural displacement and velocity were computed by using the theory of spectral analysis. And then the expressions of stochastic truss structural dynamic reliability were educed from the Poisson formula. Finally, the influence of the randomness of structural parameters on the structural dynamic reliability was analyzed by an example; the method of this paper was proved feasible and available comparing with the Monte Carlo method.

We introduce the concept of spatial and temporal complexity with emphasis on how its fractal characterization for 1D, 2D or 3D hemodynamic brain signals can be carried out. Using high-resolution experimental data sets acquired in animal and human brain by noninvasive methods - such as laser Doppler flowmetry, laser speckle, near infrared, or functional magnetic resonance imaging - the spatiotemporal complexity of cerebral hemodynamics is demonstrated. It is characterized by spontaneous, seemingly random (that is disorderly) fluctuation of the hemodynamic signals. Fractal analysis, however, proved that these fluctuations are correlated according to the special order of self-similarity. The degree of correlation can be assessed quantitatively either in the temporal or the frequency domain respectively by the Hurst exponent (H) and the spectral index (beta). The values of H for parenchymal regions of white and gray matter of the rat brain cortex are distinctly different. In human studies, the values of beta were instrumental in identifying age-related stiffening of cerebral vasculature and their potential vulnerability in watershed areas of the brain cortex such as in borderline regions between frontal and temporal lobes. Biological complexity seems to be present within a restricted range of H or beta values which may have medical significance because outlying values can indicate a state of pathology.

Aiming at the design of a structure under random loading, e.g. a tail of airplane in flight through a three-dimensional turbulence, the mean square values of deflections and stresses are evaluated by a stochastic method on the basis of simplified representations of the structure with finite span and its spanwise lift distribution. In this analysis the typical power-spectrum of stationary isotropic three-dimensional turbulence is used, and it is assumed that the scale of turbulence is much smaller than the length of span, and that the intensity of turbulence is so small that a stall does not occur. It is well-known that the variance of stress is used in estimating the strength and fatigue life.

Fractal patterns have been shown to change in resting- and task-state blood oxygen level-dependent signals in bipolar disorder patients. However, fractal characteristics of brain blood oxygen level-dependent signals when responding to external emotional stimuli in pediatric bipolar disorder remain unclear. Blood oxygen level-dependent signals of 20 PBD-I patients and 17 age- and sex-matched healthy controls were extracted while performing an emotional Go-Nogo task. Neural responses relevant to the task and Hurst exponent of the blood oxygen level-dependent signals were assessed. Correlations between clinical indices and Hurst exponent were estimated. Significantly increased activations were found in regions covering the frontal lobe, parietal lobe, temporal lobe, insula, and subcortical nuclei in PBD-I patients compared to healthy controls in contrast of emotional versus neutral distractors. PBD-I patients exhibited higher Hurst exponent in regions that involved in action control, such as superior frontal gyrus, inferior frontal gyrus, inferior temporal gyrus, and insula, with Hurst exponent of frontal orbital gyrus correlated with onset age. The present study exhibited overactivation, increased self-similarity and decreased complexity in cortical regions during emotional Go-Nogo task in patients relative to healthy controls, which provides evidence of an altered emotional modulation of cognitive control in pediatric bipolar disorder patients. Hurst exponent may be a fractal biomarker of neural activity in pediatric bipolar disorder.

Current particle dispersion models do not accurately predict the particle clustering that occurs in turbulent flow due to interaction of the particles with turbulent eddies. This clustering arises due to the effects of centrifugal forces which act to throw heavy particles out of the turbulent eddies, causing the particles to collect in high-concentration sheets lying between the eddies. The current paper proposes a stochastic vortex structure (SVS) model for simulation of particle clustering and collisions in turbulent flows. A new measure for particle drift relative to the fluid velocity is proposed that is related to the cross product of the fluid acceleration and velocity fields. Tests were conducted comparing the predictions of the SVS model with direct numerical simulation (DNS) and with three different stochastic Lagrangian methods in statistically stationary homogeneous isotropic turbulence with particles having Stokes numbers based on integral length scaling of order unity, assuming one-way fluid-particle coupling. The tests examined different turbulent flow features that are important for particle dispersion and clustering, as well as for prediction of the particle collision rate and collision distribution. The results indicate that the SVS model performs reasonably well for predicting particle concentration heterogeneity and collision rate, and that differences between the SVS and DNS results can be attributed to the fact that the SVS model neglects the small-scale velocity fluctuations within the turbulent flow.

Simulation of primary atomization with an octree adaptive mesh refinement and VOF method

Small target detection in sea clutter using all-dimensional Hurst exponents of complex time sequence

The issue is about turbulence as a natural physical phenomenon as related to observations as distinct from the problem of turbulence. The dichotomic distinction between laminar and turbulent flows is problematic in several respects. First, most flows termed “turbulent” are in reality partly turbulent: some portions of the flow as turbulent and some as laminar—the coexistence of the two regimes in one flow is a common feature with continuous transition of laminar into turbulent state via the entrainment process through the boundary between the two. Moreover, the reality is not that simple as the laminar/turbulent dichotomy as, e.g. the behavior of passive objects in flows with small Reynolds number looks as perfectly turbulent reflecting the qualitative difference between the chaotic flow properties in Eulerian and Lagrangian settings. These examples illustrate the enormous difficulties in defining what is both (i) turbulence and (ii) the turbulence problem. As concerns (i) one can only provide a description of major qualitative universal (sic) features of turbulent flows as obtained almost exclusively from observations (rather than by any theoretical deliberations) which form most important part of the “essence” of turbulence. This is because these mostly widely known qualitative features of all turbulent flows are essentially the same, i.e., it is meaningful to speak about qualitative universality of turbulent flows. It has to be stressed that the term “phenomenon of turbulence” as used above is mostly associated with the observational aspects, which in turbulence play far more important role due the unsatisfactory state of “theory”: there seems to exist no such a thing based on first principles. Hence it is vital to put the emphasis on the physical aspects based in the first place on observations.

Statistical analysis of processes is a relatively fast and precise method for traffic differentiation and identification. Many statistical properties of the network traffic show similarities to economic or climate-related processes. One such common feature is that they all show certain fractal features – such as self-similarity. In this paper, the self-similar features of currency exchange rates, water levels, and network packet arrival rates are analyzed and compared. We analyze their autocorrelation function and estimate their Hurst exponent to present similarities and differences between these processes. Furthermore, for general and IoT-specific, live LTE core control traffic, we revisit the dimension-invariance of packet arrival rates and present two interesting functions – the index of dispersion of intervals and the index of dispersion for counts.

We have previously modeled the turbulent wave-front phase by using a fractional Brownian motion (J. Opt. Soc. Am. A, Vol. 21, N° 10, pp. 1962-1969, October 2004). Non-Kolmogorov turbulence is primarily considered within this approach. Now, in this work we study the relationship between seeing, usual measure of quality associated to a groundbased telescope, and Hurst exponent, characteristic parameter of a fractional Brownian motion. The theory behind the differential image motion monitor (DIMM), a standard and widely used instrument for seeing measurements, is reviewed by us. It is shown that there is a direct connection between both parameters. Thus, it is concluded that Hurst exponent is a quantifier of the atmospheric turbulent state.

Understanding granular flows has always been important for predicting natural phenomena such as rockslides and soil erosion, as well as industrial processes such as coal-based fossil fuel systems and solids processing. As such, it becomes important to understand granular flows from both a classical granular flow and tribological perspective. Inherently important in the study of granular flows is the study of the individual particle level interactions, which define the global behavior of the flow. The current work examines both the coefficient of restitution (COR) and coefficient of friction (COF) for various material combinations. COR and tribological experiments are performed on various sphere and plate (disk) materials, such as low carbon steel, tungsten carbide (WC), and NITINOL 60.