Probability Distribution Of Height Research Articles

A numerical method for generation of shot peened surfaces

Shot peening is a widely utilized surface modification technique, and the surface morphology of the treated materials plays a crucial role in determining their friction and fatigue properties. Surface morphology generation is a pivotal process in simulating shot-peened surface friction and other performance characteristics. However, there are limited available methods for shot-peened surfaces until now. In this paper, we present a method that directly specifies the height probability distribution and power spectrum to generate shot-peened surfaces. Compared to methods that specify height parameters, this method has more flexibility. This approach replaces the traditional Johnson transformation method with an optimization algorithm to generate height data that follows a high probability distribution, while the power spectrum is derived from the Fourier transform of an exponential autocorrelation function. The results show that this approach can generate shot-peened surfaces using any conventional height parameters, with a maximum difference in height parameters between the reconstructed surface and the original surface of no more than 18.63%. Compared to the Johnson transformation method, this method significantly reduces the errors in skewness and kurtosis of the generated surface. It offers rapid surface generation for shot-peened surfaces, leading to significant time and cost savings in experiments.

Restoring the Fluctuation-Dissipation Theorem in Kardar-Parisi-Zhang Universality Class through a New Emergent Fractal Dimension.

The Kardar-Parisi-Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth exponents; and the apparent absence of a fluctuation-dissipation theorem (FDT) for spatial dimension d>1. Notably, these questions were answered exactly only for 1+1 dimensions. In this work, we propose a new FDT valid for the KPZ problem in d+1 dimensions. This is achieved by rearranging terms and identifying a new correlated noise which we argue to be characterized by a fractal dimension dn. We present relations between the KPZ exponents and two emergent fractal dimensions, namely df, of the rough interface, and dn. Also, we simulate KPZ growth to obtain values for transient versions of the roughness exponent α, the surface fractal dimension df and, through our relations, the noise fractal dimension dn. Our results indicate that KPZ may have at least two fractal dimensions and that, within this proposal, an FDT is restored. Finally, we provide new insights into the old question about the upper critical dimension of the KPZ universality class.

Effect of lake waves on manoeuvring performance of a model ship

ABSTRACT In this study, we investigate how natural environmental conditions affect a training vessel using a ‘mathematical model of wind and wave’ specific to Lake Silm near Kamionka, Poland. Wind and wave measurements on the lake were conducted. We present the vessel’s trajectory under varying conditions: navigating in restricted waters simulating port conditions with speed readings, and during propeller operation. The control system of the nonlinear autonomous object employed two controllers: a linear matrix inequality controller and an overriding trajectory controller. We assess the impact of nonlinearities in the measured waves. It is shown that the maximum entropy probability distribution conforms more closely to the empirical frequency histogram than the Gaussian distribution. The wave height probability distributions diverge from Rayleigh distributions, indicating slight nonlinear effects. These results enhance the mathematical model of Silm Lake’s wave dynamics.

Joint Probability Distribution of Significant Wave Height and Peak Wave Period Using Gaussian Copula Method

Joint Probability Distribution of Significant Wave Height and Peak Wave Period Using Gaussian Copula Method

A Novel Method of Designing Random Rough Surface Considering Its Contact Force-Deformation Characteristic Requirement

Abstract The height probability distribution (HPD) of random rough surface topography has significant effect on its contact behaviors. In this paper, an optimization model to calculate the optimal HPD of random rough surface topography to make its contact force-deformation characteristic satisfy the given target force-deformation characteristic was established. In the solution to calculate the optimal HPD, using Bezier interpolation curve to represent the curve of HPD function and using the positions of the control points of Bezier interpolation curve as optimization variables were proposed. The solution was validated by numerical simulations implemented using MATLAB. The effect of the number of the control points on the minimum objective function value was investigated. Through analyzing the obtained result, we found the most appropriate number of the control points is 35. Comparison with the method using height parameters as optimization variables in literature was implemented. It was found that the proposed method is applicable to broader types of contact force-deformation characteristic requirements than the method using height parameters as optimization variables.

The dependence of wave height probability distributions on physical parameters from measurements near Sakhalin Island

The data of long-term surface waves measurements with bottom sensors near Sakhalin Island were used to build instrumental probability distributions for exceedance of wave heights. Waves with heights exceeding the significant wave height by more than three times were recorded. Specific features of the observations conducted during the periods of open and ice-covered sea surfaces are discussed. The subsets of statistically homogeneous data are arranged through selection considering the natural physical dimensionless parameters of the task, that control the effects of finite depth and nonlinearity (namely, the wave steepness, the wave amplitude to water depth ratio, the Ursell parameter). The manifestation of these and composite parameters in theoretical probability distributions of wave heights is discussed. The effects of nonlinearity and the measurement point depth on probability distributions are estimated with a focus on abnormally high waves. In particular, it is shown that for the place of registration an increase in the wave amplitude to water depth ratio parameter leads to a decrease in the probability of abnormally high waves. This behavior is consistent with the Glukhovsky theoretical distribution. The waves characterized by relatively large dimensionless depth parameter exhibit a higher probability of substantial exceedance of the significant height and are better described by the Rayleigh distribution.

Validation of short-term prediction of added resistance in head seas considering wave steepness nonlinearity with probability density function method

The added resistance in waves, which is an important component in the estimation of actual sea performance of ships, has been evaluated as the average value by short-term prediction methods. In this paper, a novel probabilistic method to account for non-linearities of added resistance with respect to wave height in short-term sea conditions is proposed using the joint occurrence probability distribution of wave height and period and non-linear added resistance response in the frequency region. It is a method of averaging the combination of crest and trough that ships encounter in irregular waves, which is considered as one wave of the regular waves, and the resulting added resistance is combined stochastically. The non-linearity with respect to wave height is empirically expressed as a “correction coefficient” obtained from the comparison of CFD data and potential-based linear calculation results in regular waves. The results of the proposed method have a reasonable agreement in the acceptable range of accuracy with results of RANSE based CFD numerical simulations in long crested irregular waves and the experimental model tests. As a result, the proposed method is applicable for obtaining added resistance in irregular head waves having the non-linearity with respect to wave height with a scope of further improvement in the method.

Critical effects and scaling at meniscus osculation transitions.

We propose a simple scaling theory describing critical effects at rounded meniscus osculation transitions which occur when the Laplace radius of a condensed macroscopic drop of liquid coincides with the local radius of curvature R_{w} in a confining parabolic geometry. We argue that the exponent β_{osc} characterizing the scale of the interfacial height ℓ_{0}∝R_{w}^{β_{osc}} at osculation, for large R_{w}, falls into two regimes representing fluctuation-dominated and mean-field-like behavior, respectively. These two regimes are separated by an upper critical dimension, which is determined here explicitly and depends on the range of the intermolecular forces. In the fluctuation-dominated regime, representing the universality class of systems with short-range forces, the exponent is related to the value of the interfacial wandering exponent ζ by β_{osc}=3ζ/(4-ζ). In contrast, in the mean-field regime, which was not previously identified and which occurs for systems with longer-range forces (and higher dimensions), the exponent β_{osc} takes the same value as the exponent β_{s}^{co} for complete wetting, which is determined directly by the intermolecular forces. The prediction β_{osc}=3/7 in d=2 for systems with short-range forces (corresponding to ζ=1/2) is confirmed using an interfacial Hamiltonian model which determines the exact scaling form for the decay of the interfacial height probability distribution function. A numerical study in d=3, based on a microscopic model density-functional theory, determines that β_{osc}≈β_{s}^{co}≈0.326 close to the predicted value of 1/3 appropriate to the mean-field regime for dispersion forces.

Statistical technique to improve the wave potential estimation for the design of wave electric generators

• Wave measurements are used for the design and simulation of the electric generators. • The analysis is based on the joint probability distribution of height and period. • Fluctuations in wave power estimations, reduced; original core statistics, preserved. • The uncertainty in the generator’s electrical supply is reduced. • Year-long datasets recorded at a maritime buoy are used for realistic estimates. This study presents a statistical approach that allows estimating the sea wave power ( P max ) by using the joint probability distribution of the wave’s significant height ( H m ) and period ( T ), p H m , T . Unlike other works in the literature, the goal of the present paper is not to find the best locations for wave farming or to extrapolate the data in space or time, but to achieve a more suitable design of wave electric generators in terms of a reduced delivery uncertainty. When the raw values of H m and T are used, the estimations of P max end up spanning over several orders of magnitude owing to the nonlinear amplification of the fluctuations through the calculations. This typically results in large overestimations of P max that preclude their use for a modest design. Another associated issue is the low temporal resolution of the data available, which forbids using measurements of H m and T in numerical simulations directly. These issues have been overcome with the proposed methodology, whose core feature consists of generating a new series of random H m , T pairs, based on p H m , T , which allow estimating the wave power potential realistically while reducing the chances of over/underrating. In this new series, large fluctuations are suppressed, but the essential statistics of the measurements are preserved. Additionally, since the new series of H m , T pairs is time-independent, it can be fed directly into numerical simulations. To illustrate our proposal, we used one-year long datasets of hourly measurements of H m and T recorded at a maritime buoy, for the design and simulation of a linear electric generator with a single degree of freedom.

Probability distribution of wind speed and wave height in Nowshahr Port using the data acquired from wave scan buoy

The present study aimed to determine probability density function (Pdf) of wind speed and wave height in Nowshahr Port using the data obtained from wave scan buoy. Three Pdfs, i.e. Weibull, Rayleigh and Lognormal, were adopted for estimation of data frequency. The shape (k) and scale (c) parameters of the Weibull function were calculated by four methods, i.e. Graphical method (Gm), Moment method (Mm), Maximum likelihood method (Mlm) and Empirical method (Em) whereas the parameters of the other two functions were estimated only through Mlm. In order to analyze the efficiency of pdfs root mean square error (Rmse), determination coefficient (R2) and chi-square test (χ2) were considered. The results revealed that Mm, Em and Mlm presented almost similar ‘k’ and ‘c’ values. However, Gm yielded smaller values. Accuracy of Weibull Pdf was generally higher than those in Rayleigh one. However, it was seen in the present study that Rayleigh Pdf is better only when Weibull parameters in wind data were measured via Gm. Lognormal Pdf was the most efficient in both wind and wave data. Finally, cumulative distribution function was estimated based upon Weibull and Lognormal probability models and the former was again found to be stronger and more accurate.

Surface roughness characteristics and stochastic model of corroded steel foundation in submarine soil environment

With the rapid development of ocean engineering, many types of steel foundation have been developed to support above ground structures, which encounter great corrosion risk and serious safety hazards in whole service life. Though lots of studies have been devoted to corrosion loss of sectional area and deterioration of strength, little effect focuses on the surface roughness characteristics of corroded steel foundation in submarine soil, which have significant impact on the soil-structure interaction and even on the bearing capacity of overall structure. Electrolytic accelerated corrosion experiments are conducted to investigate the surface roughness characteristics of corroded steel in marine clay soil. The experiment results show that the roughness parameters, i.e. the arithmetic mean height, the maximum height and the standard deviation, have a S-shape growth with the degree of corrosion represented by mean corrosion depth. Moreover, the probability distribution of corrosion height on each corroded surface is statistically analyzed, which generally follows Gauss distribution. Based on the experimental results and roughness parameter change law, a stochastic model is constructed to reconstruct the surface topography of corroded steel with considering different degrees of corrosion, which is verified by comparing the predicated results with experiment data. The findings of surface roughness characteristics and developed stochastic model are helpful for studying the soil-steel foundation interaction characteristics and bearing capacity of ocean engineering structure with considering the corrosion effect.

Study on Wind Field Characteristics in a Coastal Plain Based on a New Three-Dimensional Joint Distribution Model

This paper studied the joint probability distribution of wind speed, wind direction, and wind height. The measured wind field data of a coastal plain in Zhongshan city, Guangdong Province, China, were taken as the research object. A three-dimensional joint distribution modeling method, based on the copula function and the AL (angular–linear) model, is proposed. Firstly, the wind speed is modeled by the common distribution model, and the Weibull distribution is selected. Secondly, the mvM (mixed von Mises distribution) was used to fit the wind direction probability density, and the joint distribution of wind speed and wind direction was established based on the AL model. Finally, a three-dimensional joint distribution model of wind speed, wind direction, and height was established by considering the effect of height through the copula function. The results showed that Weibull distribution can better describe the wind speed distribution in this region. The north–south wind prevailed in this region, and the probability of the main wind direction decreased with the increase in height. The joint distribution of wind speed and direction, based on the AL model, fitted well with the measured data, and the final three-dimensional distribution model had a good fitting effect.

Instabilities in quasiperiodic motion lead to intermittent large-intensity events in Zeeman laser.

We report intermittent large-intensity pulses that originate in Zeeman laser due to instabilities in quasiperiodic motion, one route follows torus-doubling to chaos and another goes via quasiperiodic intermittency in response to variation in system parameters. The quasiperiodic breakdown route to chaos via torus-doubling is well known; however, the laser model shows intermittent large-intensity pulses for parameter variation beyond the chaotic regime. During quasiperiodic intermittency, the temporal evolution of the laser shows intermittent chaotic bursting episodes intermediate to the quasiperiodic motion instead of periodic motion as usually seen during the Pomeau-Manneville intermittency. The intermittent bursting appears as occasional large-intensity events. In particular, this quasiperiodic intermittency has not been given much attention so far from the dynamical system perspective, in general. In both cases, the infrequent and recurrent large events show non-Gaussian probability distribution of event height extended beyond a significant threshold with a decaying probability confirming rare occurrence of large-intensity pulses.

Urban Morphological Parameters of the Main Cities in China and Their Application in the WRF Model

AbstractAs one of the effective ways to study urban climate and environmental problems, the performance of numerical model could be improved in urban areas by enhancing the description of urban morphology. One of the largest challenges faced by researchers is the lack of real urban canopy parameters (UCPs) to represent urban morphological characteristics. In this study, the urban morphological parameters were estimated and a high‐resolution urban morphological parameter data set for the main cities in China was developed. The results show that the distribution of urban morphological parameters is of regional characteristics and cities in China are mainly dominated by low‐rise buildings. Chinese cities can be broadly divided into two types based on the probability distribution of building height: single‐peak and double‐peak. The number of low‐rise buildings in single‐peak cities is much larger than that of double‐peak cities, which contributes to the difference between the two types of cities. The equivalent street width of northern cities in China is higher than that of southern cities, and the equivalent building width is almost the same. However, the default UCPs of the Weather Research and Forecasting (WRF) model distort the morphology of Chinese cities. The sensitivity of the WRF model to urban morphological parameters was also investigated in Nanjing. Overall, our results indicate that urban morphological parameters have a significant influence on the energy balance process in cities and improve the performance of the WRF model in simulating the air temperature and canopy wind speed.

Conditions of the Presence of Bimodal Amplitude Distribution of Two-Process Surfaces

Two-process surfaces are functionally important. They contain plateau and valley parts. They are created by superimpositions of two one-process textures of Gaussian probability height distributions. It is expected that the resulting two-process surface would have bimodal height probability distribution. However, typically two-process textures have unimodal ordinate distribution. The present authors developed limiting conditions of presence of bimodal ordinate distribution. These conditions depend on the material ratio at the plateau-to-valley transition (the Smq parameter), and on the ratio of heights of the plateau and valley surface parts (Spq/Svq). Generated stratified textures and measured two-process surfaces of cylinder liners were taken into consideration.

Feedback control of surface roughness in a one-dimensional Kardar-Parisi-Zhang growth process.

Control of generically scale-invariant systems, i.e., targeting specific cooperative features in nonlinear stochastic interacting systems with many degrees of freedom subject to strong fluctuations and correlations that are characterized by power laws, remains an important open problem. We study the control of surface roughness during a growth process described by the Kardar-Parisi-Zhang (KPZ) equation in (1+1) dimensions. We achieve the saturation of the mean surface roughness to a prescribed value using nonlinear feedback control. Numerical integration is performed by means of the pseudospectral method, and the results are used to investigate the coupling effects of controlled (linear) and uncontrolled (nonlinear) KPZ dynamics during the control process. While the intermediate time kinetics is governed by KPZ scaling, at later times a linear regime prevails, namely the relaxation toward the desired surface roughness. The temporal crossover region between these two distinct regimes displays intriguing scaling behavior that is characterized by nontrivial exponents and involves the number of controlled Fourier modes. Due to the control, the height probability distribution becomes negatively skewed, which affects the value of the saturation width.

Toppling and height probabilities in sandpiles

We study Abelian sandpiles numerically, using exact sampling. Our method uses a combination of Wilson’s algorithm to generate uniformly distributed spanning trees, and Majumdar and Dhar’s bijection with sandpiles. We study the probability of topplings of individual vertices in avalanches initiated at the centre of large cubic lattices in dimensions and 5. Based on these, we estimate the values of the toppling probability exponent in the infinite volume limit in dimensions , and find good agreement with theoretical results on the mean-field value of the exponent in . We also study the distribution of the number of waves in 2D avalanches. Our simulation method, combined with a variance reduction idea, lends itself well to studying some problems even in very high dimensions. We illustrate this with an estimation of the single site height probability distribution in d = 32, and compare this to the asymptotic behaviour as .

Contact Mechanics for Solids with Randomly Rough Surfaces and Plasticity

We present experimental results for the elastic and plastic deformation of sandblasted polymer balls resulting from contacts with flat smooth steel and silica glass surfaces. Nearly symmetric, Gaussian-like height probability distributions were observed experimentally before and remarkably, also after the polymer balls were deformed plastically. For all the polymers studied we find that the surface roughness power spectra for large wavenumbers (short length scales) are nearly unchanged after squeezing the polymer balls against flat surfaces. We attribute this to non-uniform plastic flow processes at the micrometer length scale. The experimental data are analyzed using the Persson contact mechanics theory with plasticity and with finite-element method (FEM) calculations.

A parameterization of DNV-GL storm profile for the calculation of design wave of marine structures

This paper proposes a parameterization of the Det Norske Veritas and Germanischer Lloyd (DNV GL) storm profile for the development of analytical solution for the calculation of the associated return period. This solution allows the possibility of calculating the design wave for marine structures by means of a very simple approach. The basic concept is to substitute the sequence of actual storms at a given site with a sequence of trapezoidal storms (TS) defined on the basis of the DNV GL storm temporal evolution and derive analytically the return period referring to the TSs sequence. The DNV GL storm is represented by a trapezium which may be totally defined by means of three parameters: the trapezium height related to the storm intensity, and its minor and major bases, which represent two durations Dp, D* pertaining to the maximum significant wave height Hsmax and to 50% of Hsmax, respectively. It is assumed that the parameters Dp, D* are uncorrelated to the storm intensity and are the same for all the storms. With this assumption, the return period is obtained in a closed form as a function of the ratio n between Dp and D*, D* itself, and of the exceedance probability distribution of significant wave height. The model is validated by comparison with Equivalent Triangular Storm (ETS) model by processing buoys data at four locations representative of different wave climates: Haltenbanken in North Sea, Alghero in Mediterranean Sea, stations 41001 and 46006 in Atlantic and Pacific Oceans, respectively. Results reveal that although its significant simplicity, the TS model provides reliable estimate of return values which are in agreement with those achieved by ETS model. Further, the employment of TS model with the values of parameters n and D* corresponding to the DNV GL profile is suitable for any location.

Effects of surface shapes on properties of turbulent/non-turbulent interface in turbulent boundary layers

The effects of directional riblets surfaces on the turbulent/non-turbulent (T/NT) interface in turbulent boundary layers are experimentally investigated using two-dimensional time-resolved particle image velocimetry (PIV). The velocity field of streamwise—wall-normal plane for the smooth surface, converging and diverging riblets surfaces are measured. The interface is detected using the criterion of local kinetic energy. The statistical properties of interface height and conditional averaged velocity for different surfaces are analyzed. It is shown that, the converging and diverging riblets surfaces have little effect on the fractal dimension of the T/NT interface, but they cause the intermittency profile deviate from error function and the probability distribution of interface height deviate from Gaussian function. To be specific, the distribution of interface height for the converging riblets surface shows a positive skewness while it shows a negative skewness for the diverging riblets surface. Moreover, the conditional averaged streamwise velocity and spanwise vorticity across the interface are analyzed, and it is found that their self-similarities are preserved for different surfaces when normalized with respective friction velocity. The correlation analysis reveals that near-wall streamwise velocity fluctuation and interface height show a negative correlation.