Three-parameter Weibull Distribution Research Articles

A percolation model of unsaturated hydraulic conductivity using three-parameter Weibull distribution

Accurate estimation of unsaturated hydraulic conductivity, K(Sw), is crucial for modeling water and solute transport in variably-saturated soils. In this study, a new model for the estimation of K(Sw) based on percolation theory and critical path analysis is presented, in which the pore-throat size distribution is described by the three-parameter Weibull probability density function. To evaluate the performance of the proposed K(Sw) model, two datasets including more than three hundred samples are analyzed. The first dataset includes 240 pore-network simulations, while the second dataset consists of 101 soil samples from the UNSODA database. For the pore-network simulations, for which both pore-throat size distributions and capillary pressure curves are available, we estimate K(Sw) from the capillary pressure curve with RMSLE = 0.54 slightly better than the estimations obtained from the pore-throat size distribution with RMSLE = 0.6. For the soil samples, since only the capillary pressure curve is measured, we estimate K(Sw) from the capillary pressure curve and evaluate three previously proposed approaches for determining the exponent, α, in Poiseuille's law. We find that RMSLE = 0.98 for α = 3, RMSLE = 0.87 for α = 2(4 − D) − (3 − D)/(2D − 3), and RMSLE = 1.06 for α = 4 − 0.74D, in which D is the fractal dimension characterizing the pore space. Comparing the results with previous studies in which the power-law probability density function was used to characterize capillary pressure curve and pore sizes shows that the proposed three-parameter Weibull distribution, in conjunction with the CPA analysis, may more accurately estimate K(Sw) for a wide variety of soils ranging from very fine to very coarse textures.

A Computational Multiscale Modeling Method for Nanosilver-Sintered Joints with Stochastically Distributed Voids

A high power density is required in wide band gap power semiconductor packaging, which has led to the popularity of sintered nanosilver as an interconnecting material. However, affected by stochastically distributed voids in its microstructure, this material in practice exhibits instability leading to reduced reliability. In this paper, a computational multiscale modeling method is proposed to simulate the influence of micro-voids on macro-properties, providing an efficient tool to analyze the aforementioned problem. At the micro-scale, the three-parameter Weibull distribution of the equivalent Young’s modulus and the normal distribution of the equivalent Poisson’s ratio are captured by Monte Carlo-based finite element simulation on the reconstructed stochastic representative elements, where the density and distribution morphology of micro-voids are taken into consideration. At the macro-scale, the effect of the microscopic voids is transferred through a random sampling process to construct the multiscale model. The effectiveness and validity of the proposed method are verified through experimental case studies involving the modeling of nanosilver-sintered joints sintered at temperatures of 275°C and 300°C. In addition, the effects of the sintering temperature on the dispersion of the micro-voids, the distribution fluctuation of the constitutive parameters, and the mechanical properties are also discussed based on numerical and experimental results.Graphical

Development of Probabilistic Models Optimized for Korean Marine Environment Varying from Sea to Sea Based on the Three-parameter Weibull Distribution

In this study, probabilistic models for the wave- and lifting forces were derived directly from long-term in-situ wave data embedding the Korean marine environment characteristics varying from sea to sea based on the Three-Parameter Weibull distribution. Korean marine environment characteristics varying from sea to sea carved out their presence on the probability coefficients of probabilistic models for wave- and lifting forces. Energetic wave conditions along the southern coast of Korea distinguish themselves from the others with a relatively large scale coefficient, small location coefficient, and shape coefficient around 1.3. On the other hand, mild marine environment along the western coast has a small variability, leading to small scale-coefficient, large location coefficient and shape coefficient around 2.0. In the sea off Mokpo, near the boundary between the South- and West Seas, marine environment was characterized by small scale-coefficient, large location coefficient, and shape coefficient around 1.2, implying that marine environments characteristics of the South-and West Sea coexist in the sea off Mokpo.

Reliability model for key components of urban rail transit train based on improved hunter-prey optimization

The reliability of key components of urban rail transit (URT) plays an important role in the maintenance plans of URT. It is necessary to establish the reliability model of URT trains. In the current research, the reliability model has a limited scope of application and fails to accurately depict the reliability of key components in URT trains. To solve the above problem, a multi-peak type mixture Weibull distribution model is established using several three-parameter Weibull distributions based on fault modes of components sourced from historical lifetime data. Due to the complexity of this model, parameter estimation is challenging. For this purpose, an improved hunter-prey optimization (IHPO) was proposed to improve parameter estimation accuracy. Firstly, an improved Hénon chaos map was introduced to improve the distribution of the initial population. Secondly, the Lévy flight was introduced to increase the probability of the individual spreading to the whole range at the late stage. Lastly, a nonlinear balance factor was proposed to enhance the algorithm’s global search capability. The simulation experiment was carried out with examples of the balanced pressing wheel and the wheelset. The IHPO algorithm-based parameter estimation method shows the highest R-square with values of 0.996 and 0.999, respectively, and the lowest root mean square error with values of 0.019 and 0.008, respectively. The simulation results demonstrate that the stability and optimization of the HPO are improved, and the multi-peak mixture Weibull distribution model based on the IHPO can accurately depict URT trains’ reliability.

Effects of steel fibers on the flexural fatigue performance of recycled aggregate concrete

To explore the flexural fatigue performance of steel fibers reinforced recycled aggregate concrete (SF-RAC), the flexural fatigue test and residual strength test with different stress levels were conducted. Besides, the fatigue cracks propagation under cyclic loading were observed and analyzed by digital image correlation (DIC) technology. The results show that the flexural fatigue life of SF-RAC increases when the content of steel fibers increases, and it is proved to be well in accordance with the three-parameter Weibull distribution. The flexural fatigue life equation with different stress level and reliability (S-N-P equation) is obtained based on it, which shows that when the fiber volume content is 1.5%, the fatigue strength of SF-RAC is about 37.2% higher than that of RAC without fibers. Then, a fatigue strain-based damage evolution model is established, and the derived residual strength model can well predict the damage degree and residual life of SF-RAC. Moreover, the steel fibers can improve the critical crack propagation length and loading cycles of RAC. The crack propagation rate can be decreased by 97.63% when the fiber volume content is 1.5%, and it is increased with the development of stress level. Finally, the influence mechanism of steel fibers on flexural fatigue performance of RAC were explained through microstructure tests.

Reliability, maintainability and availability analysis of solar power plant in Pantai Baru using voltage measurement data

The hybrid power plant in Pantai Baru, Yogyakarta stands as a noteworthy exemplar within the realm of existing solar power installations throughout Indonesia. Analysis of reliability, maintenance, and availability is needed to minimize problems at solar power plant’s electrical energy production process, so that renewable energy security and sustainability can be maintained. This study aims to determine the reliability, maintainability, and availability of solar panels with a capacity of 48 Volts at the hybrid power plant, Pantai Baru using the control chart method, Weibull distribution. Calculation of reliability, maintainability, and availability of solar panels is carried out using solar panel output voltage measurement data collected over the last 4 years. Time to failure, TTF and time to repair, TTR were obtained from out-of-scope data on the control chart and processed using software, namely Reliasoft Weibull ++ Version 6. The simulation results and calculations show that the distribution obtained from TTF and TTR is a three-parameter Weibull distribution, with a system reliability value of 0.3, maintainability value is 1 and availability value is 0.997 with a solar panel operating time period of 5 days. The reliability of each solar panel unit is 0.122 or 12.2%.

Optimal Scheduling of Micro-Energy Grid Based on Pareto Frontier Under Uncertainty and Pollutant Emissions

With the continuous improvement of micro-energy grid, in order to reduce carbon emissions and environmental pollution, a multi-objective optimization operation strategy considering the distributed energy generation uncertainty, environmental factors, and self-supply rate is proposed in this paper. First, to simulate the uncertainty of distributed energy, we used the three-parameter Weibull distribution to simulate the wind power. Then, for the load side, an innovative electric and thermal comprehensive demand response model is proposed. Different from the traditional micro energy network that only considers the heat energy balance, this paper also considers the thermal inertia of buildings to more accurately describe the characteristic of the heat load. Meanwhile, on the basis of economic factors, the abandon rate of distributed energy, environmental coefficient and self-sufficiency rate are also regarded as three objective functions. Finally, in order to obtaining a set of solutions at the Pareto front, the entropy-weighted TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method is introduced to obtain the global optimal solution. Simulation study of multiple scenarios shows the effectiveness of the proposed method. Compared with the method that considers only economic costs, the pollutant emissions are reduced by 50kg and self-power supply rate is increased by 20%.

Reliability, Maintainability and Availability Analysis of Solar Power Plant in Pantai Baru using Voltage Measurement Data

The hybrid power plant in Pantai Baru, Yogyakarta stands as a noteworthy exemplar within the realm of existing solar power installations throughout Indonesia. Analysis of reliability, maintenance, and availability is needed to minimize problems at solar power plant’s electrical energy production process, so that renewable energy security and sustainability can be maintained. This study aims to determine the reliability, maintainability, and availability of solar panels with a capacity of 48 Volts at the hybrid power plant, Pantai Baru using the control chart method, Weibull distribution. Calculation of reliability, maintainability, and availability of solar panels is carried out using solar panel output voltage measurement data collected over the last 4 years. Time to failure, TTF and time to repair, TTR were obtained from out-of-scope data on the control chart and processed using software, namely Reliasoft Weibull ++ Version 6. The simulation results and calculations show that the distribution obtained from TTF and TTR is a three-parameter Weibull distribution, with a system reliability value of 0.3, maintainability value is 1 and availability value is 0.997 in day 5th. The reliability of each solar panel unit is 0.122 or 12.2%.

Quantification of micro-damage evolution process in ceramics through extensive analysis of multi-source heterogeneous data

Due to the randomness of hard and brittle material damage evolution, assessing damage and interpreting mechanisms is challenging. This study focuses on analyzing material damage mechanisms by employing extensive multi-source heterogeneous data to extract quantitative information. A machine vision-based method for detecting and assessing defects in ceramics was proposed, achieving 98 % accuracy. Leveraging this method in scratching experiments, the progression of damage and micro-failure mechanisms in Si3N4 was investigated through the analysis of defect area distribution, force and acoustical signals. Results revealed that the distribution of defect areas conforms to a three-parameter Weibull distribution. At scratch depths of 0.01, 0.02 mm, approximately 90 % of damage accumulated within 1e2–1e4 μm2; conversely, 0.03–0.06 mm led to a decrease to 30 – 40 %. Extensive crack propagation occurs within 0.02–0.03 mm, resulting in the intersecting formation of spalling blocks. Damage evolution occurs in three stages: shallow and small cracks with scattered spalling blocks in the first stage, deep and long cracks with continuous and irregularly shaped spalling blocks in the second stage, and nearly no cracks with continuous and regularly shaped blocks in the third stage. This study not only proposes an efficient method to detect and evaluate ceramic defects, but also deeply understands the evolution process of material damage mechanism. Considering defect distribution and statistical trends, this research not only provide guidance for the optimization of machining parameters, especially in the scene of specific precision and high removal rate. At the same time, the damage evolution and failure mechanism of Si3N4 provide useful experimental basis and data support for deepening the understanding of material behavior during ultra-precision grinding.

Effect of hygrothermal aging on compression behavior of CFRP material with different layups

This study investigates the mechanical degradation mechanism and statistical analysis of residual compressive strength of three lay-up laminates under more than one-year seawater environment. Fick and Langmuir model were used to describe the moisture absorption process with a maximum moisture equilibrium content of 3.78% and various characterization methods were used to reveal the microstructural evolution of CFRP during aging. Experimental results demonstrate that the multiple fiber directions in the MD specimens provide additional moisture absorption channels, leading to a stronger wicking effect and a higher equilibrium moisture content. The layups can significantly affect the compression failure mode after long-term aging. The compressive strength of the cross-ply and multidirectional laminates decreased by 50% while that of unidirectional laminates decreased by 28%, and surface cracking is easily observed in the aged specimens. An empirical prediction model based on the Langmuir model and experimental data was proposed and verified by the data from the literature, with at least a 10% improvement in prediction accuracy. Meanwhile the three-parameter Weibull distribution was used to describe the residual compressive strength after hygrothermal aging and tested by Kolmogorov-Smirnov test. It is expected to provide insight for the application of CFRP in seawater environment.

Utilizing jackknife and bootstrap to understand tensile stress to failure of an epoxy resin

A study was conducted on the tensile stress of an epoxy resin (Resoltech® 1050/1056). This was done by gathering a sample of 39 tensile strength data under consistent levels of stress. The tensile stress resistance is often characterized using a three-parameter Weibull distribution and the reliability of this characterization, given by confidence intervals (CIs). This approach commonly utilizes data-resampling techniques to estimate the CI of its parameters. CIs are constructed from six existing point-estimation methods. Herein, the jackknife was carried out to calculate the CIs using 39 subsamples and bootstrap methods using 100 or 200 subsamples. To date, there have been no studies exploring the effectiveness of subsampling methods for constructing CIs related to tensile strength. In this study, jackknifed and bootstrapped samples are used to implement the percentile method and three variations of the bias correction methods. We then performed simulations to evaluate the reliability of these methods using a Weibull random number generator. Our results showed that while the bias-corrected approach generated the most stable CIs from replicate samples, its accuracy was contingent on the point-estimation method employed. We also found that the different methods for calculating CIs resulted in significantly varying widths of the CIs.

Experimental Study on Flexural Fatigue Resistance of Recycled Fine Aggregate Concrete Incorporating Calcium Sulfate Whiskers

In order to study the flexural fatigue resistance of calcium sulfate whisker-modified recycled fine aggregate concrete (RFAC), flexural fatigue cyclic loading tests at different stress levels (0.6, 0.7, and 0.9) considering a calcium sulfate whisker (CSW) admixture as the main influencing factor were designed. Furthermore, the fatigue life was analyzed, and fatigue equations were established using the three-parameter Weibull distribution function theory. In addition, the micro-morphology of CSW-modified recycled fine aggregate concrete was observed and analyzed through Scanning Electron Microscopy (SEM), and the strengthening and toughening mechanisms of CSW on recycled fine aggregate concrete were further explored. The test results demonstrate that the inclusion of recycled fine aggregate reduces the fatigue life of concrete, while the incorporation of CSW can effectively improve the fatigue life of the recycled fine aggregate concrete, where 1% of CSW modification can extend the fatigue life of recycled fine aggregate concrete by 56.5%. Furthermore, the fatigue life of concrete under cyclic loading decreases rapidly as the maximum stress level increases. Fatigue life equations were established with double logarithmic curves, and P-S-N curves considering different survival probabilities (p = 0.5, 0.95) were derived. Microscopic analyses demonstrate that the CSW has a “bridging” effect at micro-seams in the concrete matrix, delaying the generation and enlargement of micro-cracks in the concrete matrix, thus resulting in improved mechanical properties and flexural fatigue resistance of the recycled fine aggregate concrete.

Prediction of Remaining Fatigue Life of In-Service Bridge Cranes

At present, bridge crane accidents occur frequently, resulting in significant losses and casualties; to ensure the safe use of in-service bridge cranes, it is necessary to predict the residual fatigue life of in-service bridge cranes. Firstly, a static analysis of the most dangerous working conditions of in-service bridge cranes is carried out to find the fatigue failure point. Subsequently, a three-parameter Weibull distribution model is established for the characteristic parameters affecting the acquisition of the stress spectrum. Latin hypercubic sampling is applied to randomly sample the characteristic parameters to produce a random sample set of characteristic parameters for use in obtaining the stress–time history. The amplitude and mean values of the stress spectra are obtained by cycle counting using the rainflow counting method. Finally, Forman’s formula and Miner’s continuous damage accumulation theory were used to derive the remaining life prediction equations for constant and variable amplitude loads. Comparing the remaining life obtained from the simulation and test, the error is about 9.145%, which proves that the remaining life obtained from the simulation is more accurate. The results show that the combined method of simulation and testing is feasible and can predict the remaining life more accurately. In the past, the prediction of residual life was performed with either testing or simulation, which is long and costly. Simulation is low-cost and takes a short time, but the accuracy is not high. In this paper, the combination method of testing and simulation improves the efficiency of production and reduces the cost of use.

3D numerical investigation on delamination behavior of the epoxy impregnated REBCO pancake coil

Superconducting coils made of rare-earth-barium-copper-oxide (REBCO) coated conductor (CC) exhibit superior electromagnetic performance. Employing epoxy impregnation can improve the structural integrity and mechanical property of the superconducting coils. However, due to the extreme work environment and weak adhesion strength of REBCO CC, the delamination induced by radial thermal stress and electromagnetic force significantly affects the electromagnetic property and the reliability of the superconducting coil. This study proposes a three-dimensional thermal-electromagnetic mechanical delamination model that incorporates the cohesive zone model to investigate the delamination mechanisms in epoxy impregnated REBCO pancake coils during the cooling and coil operation processes. The simulation employs a three-parameter Weibull distribution to account for the inhomogeneity of transverse tensile strength in the CCs. The delamination behavior and mechanisms of the coils under different conditions are analyzed. The simulation results show that the model considering random adhesion strength proves to be more effective in representing the delamination behavior of the coil. And large tensile radial stresses caused by thermal stresses and electromagnetic forces lead to the delamination behavior of the coil during cooling and operation. The main reason for the tensile radial stress is the mismatch in the thermal contraction among components of the coils during cooling process. Furthermore, we investigate the influence of the thermal expansion coefficient (CTE) and thickness of the mandrel, the CTE and prestress of the overband and the initial localized damage. The results indicate that these factors significantly affect the tensile radial stress and the extent of delamination in the windings. And the extent and distribution of delamination is related to the stress release caused by delamination to a certain degree.

From Zhentong Gao Method to Generalized Zhentong Gao Method

It is found that the data of structural fatigue life are mostly in accordance with Weibull distribution rather than Gaussian distribution, but the biggest obstacle affecting the application of Weibull distribution is that the estimation of its three- parameters will encounter difficulties in mathematical derivation and complicated calculation. Based on the estimation of correlation coefficient, the author has proposed Zhentong Gao method (Z.T. Gao method) which can avoid these difficulties. Further research found that the idea of Z.T. Gao method can be used to directly calculate the maximum value of the likelihood function of three-parameter Weibull distribution and estimate the three parameters simultaneously, which is called Generalized Zhentong Gao method. At the same time, it is also found that Nonlinear Programming method and Boltzmann Annealing method in machine learning can estimate the three parameters of Weibull distribution by directly calculating the maximum of likelihood function, so they can be regarded as Generalized Z. T. Gao methods in a sense. The results obtained by these methods are compared and some enlightening conclusions are obtained.

An improved crack probability model for silicon oxide layers using three-parameter Weibull analysis

For semiconductor dielectric layer crack detection in real-time we developed an efficient and very accurate method, which is based on acoustic emission (AE) testing and simulates the wafer probing process. The crack probability is determined using clustered and filtered AE data, that is collected by an advanced sensor-indenter system. The derived distribution function is based on two-parameter Weibull analysis, which is commonly applied for stress-induced failures of brittle materials like silicon oxide (SiO2). During indenter experiments, we observed, that the probability model for a higher number of data points is becoming inaccurate, especially at low fracture forces. Therefore, we introduce a more accurate statistical model for probing-induced SiO2 layer cracks, which is, according to a three-parameter Weibull distribution, considering a location parameter. Experimental results have shown, that the improved distribution function fits much better for predicting low-ppm failure rates of back-end-of-line (BEOL) stacks of CMOS technologies. This enables more reliable pad stack design rules and increases the reliability of complex chip designs regarding mechanical stress during test and assembly processes.

Sampling Design for Car Survey Using Stratified Random Sampling

Perumda Pasar Juara wants to know the number of cars that enter the Kosambi Market parking area for one month. However, limited access to available data means that Perumda has to count vehicles manually. Therefore, an appropriate sampling design is needed to estimate the number of cars. The data used is the result of observations based on preliminary sampling on October 14-20, 2022 at 08:00-17:00 WIB. The variables of the data used are car arrival time, day dummy, and hour dummy. The method used is dummy regression analysis and stratified sampling design. The regression analysis results show that there are four strata where each stratum has a three-parameter Weibull distribution. Based on the results, the minimum sample size required with a 5% error rate is 116 hours per day and is allocated according to the strata

Statistical analysis for predicting low-occurrence strong wind speeds at the pedestrian level in an actual urban case

This study validated the applicability of two statistical methods (3W and 2W methods) for estimating low-occurrence strong wind speeds (LOSWSs) of an actual urban case (Case-TPU) based on the three-parameter and two-parameter Weibull distributions, respectively. The 3W method necessitates the mean, standard deviation, and skewness of wind speed, while the 2W method only requires the mean and standard deviation. The large-eddy simulation (LES) results of Case-TPU were validated using the wind tunnel experimental results and applied to the statistical analysis of the 3W and 2W methods. An analysis was conducted to assess the suitability of the 3W and 2W for fitting wind speed data obtained from the LES. The results showed that even though the 3W had a better fit than the 2W owing to the location parameter of 3W enhancing its adaptability, both methods exhibited a satisfactory fit to the wind speed data at the majority of points. Furthermore, the 3W and 2W methods showed high estimation accuracies of LOSWSs. For LOSWSs with exceedance probabilities of q = 10%, 1%, and 0.1%, the 3W method produced relative errors within 10%, while the 2W method yielded relative errors within 20%. The high estimation accuracy of LOSWSs for Case-TPU proved the robustness of the 3W and 2W methods for the urban area case with complicated building layout.

Bayesian method for estimating Weibull parameters for wind resource assessment in a tropical region: a comparison between two-parameter and three-parameter Weibull distributions

Abstract. The two-parameter Weibull distribution has garnered much attention in the assessment of wind energy potential. The estimation of the shape and scale parameters of the distribution has brought forth a successful tool for the wind energy industry. However, it may be inappropriate to use the two-parameter Weibull distribution to assess energy at every location, especially at sites where low wind speeds are frequent, such as in tropical regions. In this work, a robust technique for wind resource assessment using a Bayesian approach for estimating Weibull parameters is first proposed. Secondly, the wind resource assessment techniques using a two-parameter Weibull distribution and a three-parameter Weibull distribution, which is a generalized form of two-parameter Weibull distribution, are compared. Simulation studies confirm that the Bayesian approach seems a more robust technique for accurate estimation of Weibull parameters. The research is conducted using data from seven sites in the tropical region from 1∘ N of the Equator to 21∘ S of the Equator. Results reveal that a three-parameter Weibull distribution with a non-zero shift parameter is a better fit for the wind data with a higher percentage of low wind speeds (0–1 m s−1) and low skewness. However, wind data with a smaller percentage of low wind speeds and high skewness showed better results with a two-parameter distribution that is a special case of the three-parameter Weibull distribution with a zero shift parameter. The proposed distribution can be incorporated into commercial software like WAsP to improve the accuracy of wind resource assessments. The results also demonstrate that the proposed Bayesian approach and application of a three-parameter Weibull distribution are extremely useful for accurate estimation of wind power density.

On interval estimation methods for the location parameter of the Weibull distribution: An application to alloy material fatigue failure data

The Weibull distribution is the most applied model in reliability field for lifetime analysis. The Weibull location parameter, characterizing the minimum possible life, plays a significant role in engineering applications. In this paper, we consider the interval estimation on the location parameter when the product’s lifetime follows the three-parameter Weibull distribution with a known shape parameter. A novel approach based on the relationship between the minimum order statistics, the location parameter, and the sample size is developed to construct confidence intervals for the Weibull location parameter. Thereafter, we compare it with other two interval estimation approaches by the performances of the coverage probability and the average length via simulations and a real application. The results show that the proposed method outperforms the pivot quantity (PQ) method and the bias-corrected and accelerated (Bca) bootstrap method in small and medium samples in terms of coverage probability.