Failure Probability Function Research Articles

Fast Simulation of the Functional Failure of an s – t-Network with Repair

An s – t-network with highly reliable edges with repair and variable external load is considered. A fast simulation method is proposed to evaluate the probability of functional failure when the real capacity of the network is less than the required capacity. It is proved that under some conditions the estimate has a bounded relative error as edges reliability increases. The numerical example illustrates the efficiency of the method.

A stochastic analysis of glass crack initiation under thermal loading

Window glass breakage plays a key role in compartment fire development as it opens a new vent for fresh air that accelerates the fire. To explore the correlation between temperature change and crack initiation, a series of experiments were preformed for float and Low-E glasses under uniform radiation conditions in a well designed facility. At the time of crack, the critical temperature differences of float glass and Low-E glass are 95 °C and 108 °C, respectively, with a same central temperature of 152 °C. The critical breaking stress occurs at 51 MPa and 70 MPa for float and Low-E glass, respectively. Detailed Weibull distribution functions on temperature and stress are obtained by fitting the experimental data. Failure probability function, survival probability function, probability density function are obtained and used to predict the glass failure with the survival probability at 0.9 and failure probability at 0.1. These results may be used to predict the crack probability of a glass bearing certain temperature and stress distributions.

Study on Power Failure Probability Optimization Function of Power Grid

Improving the reliability of power supply and reducing the power failure cases are eager required by demand-side, it is also important for power company. By studying on the track of power failure, it can provide some suggestions for improving power supply. Therefore, the function of power failure probability is built, combined probability theory and matlab software. Taking advantage of relative statistics in x region, it makes a deep research on the power failure events and creates the model for power failure. Based on the study, it is helpful for calculating power failure loss of demand-side. Meanwhile, it points out the future work target for power company.

Evaluating the Functional Failure Probability of System with the Varying Efficiency by the Fast Simulation Method

Evaluating the Functional Failure Probability of System with the Varying Efficiency by the Fast Simulation Method

Fracture Analysis and Distribution of Surface Cracks in Multicrystalline Silicon Wafers

Solar silicon wafers are mainly produced through multiwire sawing. The sawing process induces micro cracks on the wafer surface, which are responsible for brittle fracture. Hence, it is important to scrutinize the crack geometries most commonly generated in silicon wafer sawing or handling process and link the surface crack to the fracture of wafers. The fracture of a large number of multicrystalline silicon wafers has been investigated by means of 4-point bending and twisting tests and a failure probability function is presented. By neglecting the material property variation and assuming that one surface crack is dominating the wafer breakage, 3D finite element models with various crack sizes (depth, length, and orientation) have been analyzed to identify the distribution of surface crack geometries by fitting the failure probability from the experiments. With respect to the 63% probability, the existing surface cracks in the wafers studied appear to have depth and length ratios less than 0.042 and 0.19, respectively. Furthermore, it has been shown that the surface cracks with depth in the range from 10 to 20 μm, length up to 10 mm and angles in the range of 30 deg–60 deg, can be considered as the most common crack geometries in wafers we tested. Finally, it has been found that the mechanical strength of the wafers tested parallel to the sawing direction is approximately 15 MPa smaller than those tested perpendicular to the sawing direction.

Reliability assessment of the passive heat removal system of the VVER-1000 reactor at Kudankulam NPP

Abstract For analyzing the failure probability of the system, a model of the system was developed in best estimate T–H code RELAP5/Mod3.3. The parameters affecting the system performance were identified, their range and distribution was determined. Feasibility of various methods for calculating functional reliability was analyzed. The response surface method was selected. A response surface was constructed for the entire range of parameters for the model developed using Monte Carlo method. A large number of samples of the input parameters was generated, taking into account their individual distributions. They were classified as success/failure based on their position with respect to the failure surface. The ratio of the failed samples to the successful samples gives us the failure probability. An attempt was also made to integrate the functional failure probability with the component failure probability of the system to get an overall failure probability of the system.

The Stochastic and Integrative Prediction Methodology and Modeling for Reliability of Pedestrian Crossing on Roads

A very large number of pedestrians die every year across the globe in developed and developing countries. A pedestrian is under constant influence by various factors including the approaching vehicles equipped with Advanced Driver Assist Systems (ADAS), road infrastructures, environment and pedestrian's state, abilities, and conduct. The authors’ view in the analysis of road accidents has been to identify and address the root causes of pedestrian-to-vehicle accidents. It is a simplistic approach to blame human error as the cause of accidents whether on part of drivers or pedestrians and not to address the causes of human errors logically, systematically, and dynamically. The interfaces and dependencies among identified causes of pedestrian-to-vehicle accidents as unwanted events have been illustrated using fault tree in this research. Furthermore, the time-dependent reliability and failure probability functions of each of the contributing causes of this unwanted event have been modeled distinctively. Subsequently, the failure probability and reliability functions of pedestrian crossing on roads are determined. The results of this research will assist researchers and practitioners to logically and stochastically predict and analyze risk of the pedestrians involved accidents and provide measures in elimination or reduction of this risk.

Local estimation of failure probability function by weighted approach

In the reliability-based design of engineering systems, it is often required to evaluate the failure probability for different values of distribution parameters involved in the specification of design configuration. The failure probability as a function of the distribution parameters is referred as the ‘failure probability function (FPF)’ in this work. From first principles, this problem requires repeated reliability analyses to estimate the failure probability for different distribution parameter values, which is a computationally expensive task. A “weighted approach” is proposed in this work to locally evaluate the FPF efficiently by means of a single simulation. The basic idea is to rewrite the failure probability estimate for a given set of random samples in simulation as a function of the distribution parameters. It is shown that the FPF can be written as a weighted sum of sample values. The latter must be evaluated by system analysis (the most time-consuming task) but they do not depend on the distribution. Direct Monte Carlo simulation, importance sampling and Subset Simulation are incorporated under the proposed approach. Examples are given to illustrate their application.

Efficient estimation of the functional reliability of a passive system by means of an improved Line Sampling method

The Monte Carlo simulation based method requires considerable computational efforts for the estimation of functional reliability analysis. Efficient sampling techniques can be adopted for performing robust estimations with limited number of samples and associated with computational time. An innovative reliability analysis method known as improved Line Sampling, which incorporates the quadratic response surface taken as a fast-running surrogate in replacement of a long-running thermal hydraulic system code to improve the computational efficiency, is presented for the functional reliability analysis. In the present method, the important direction pointing towards failure region is obtained from the simulated samples with Markov Chain, instead of the original design point in the standard normal space. Simultaneously, the simulated samples can be used to evaluate the functional failure probability with the response surface. This application is illustrated for the functional reliability estimation of a passive residual heat removal system due to the epistemic uncertainties. The numerical results demonstrate the high level of computational efficiency and excellent computational accuracy by comparison to direct Monte Carlo simulation and original Line Sampling based on design point simulation. The sensitivity, defined as the partial derivative of the failure probability with respect to the distribution parameter can help to identify the relative contribution of the individual parameter to output response and guide the optimization design.

MEMBERSHIP FUNCTION OF FAILURE PROBABILITY USING MULTICUT-HIGH DIMENSIONAL MODEL REPRESENTATION

The structural reliability analysis in the presence of mixed uncertain variables demands more computation as the entire configuration fuzzy variables needs to be explored. Moreover the existence of multiple design points deviate the accuracy of results as the optimization algorithms may converge to a local design point by neglecting the main contribution from the global design point. Therefore, in this paper a novel uncertainty analysis method for estimating the membership function of failure probability of structural systems involving multiple design points in the presence of mixed uncertain variables is presented. The proposed method involves Multicut-High Dimensional Model Representation technique for the limit state function approximation, transformation technique to obtain the contribution of the fuzzy variables to the convolution integral and fast Fourier transform for solving the convolution integral. In the proposed method, efforts are required in evaluating conditional responses at a selected input determined by sample points, as compared to full scale simulation methods. Therefore, the proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The methodology developed is applicable for structural reliability analysis involving any number of fuzzy and random variables. The accuracy and efficiency of the proposed method is demonstrated through three examples.

SU-E-T-249: Theoretical Analysis of the Effects Uncertainties Have On Treatment Outcomes.

Mathematical models are used in Industrial & Systems Engineering to analyze complex integrated operational systems. Adapting this approach to radiation therapy can help quantify the precision and accuracy necessary to achieve optimal outcome of radiation treatment. The purpose of this work is to develop such a model using clinical data and assess the effect uncertainties have on treatment outcomes. The Taguchi Loss Function (TLF) is adapted to radiation therapy using conventional radiobiological models for tumor control probabilities (TCP) and normal tissues complication probabilities (NTCP) based on the equivalent uniform dose. The TCP and NTCP curves are combined to create a failure probability function for a given treatment plan. The composite effects of all uncertainties involved in treating a patient are modeled by a normal distribution. The standard deviation and mean of the normal distribution represent the precision and accuracy of a treatment. The failure probability function is convolved with the normal distribution to arrive at an expected failure probability. Precision was varied from 0.5% to 25% while accuracy ranged from ±5% to investigate uncertainties effects on complication-free local tumor control. 3D 4-field box plans where compared to IMRT plans for 18 prostate patients using this method. The average expected failure probability at the prescription dose for the 3D 4-field box plans was 30.02% and 18.13% for the IMRT plans at zero uncertainty. At 25% uncertainty the expected failure probabilities were 76.85% and 64.36%, respectively. On average the IMRT plans failure probability was 14.84% less than the 3D 4-field box plans for all uncertainty levels. This study demonstrates that uncertainty in radiotherapy procedures has a quantifiable effect on treatment outcome. To further improve complication-free local tumor control we must both improve treatment technologies and improve quality to minimize the uncertainties in radiation therapy.

Moment-independent importance measure of basic variable and its state dependent parameter solution

To analyze the effect of basic variable on output of the structure or system in reliability engineering, two moment-independent importance measures of the basic variable are proposed respectively on the failure probability and distribution function of the output. The importance measures proposed not only inherit the advantages of the traditional moment-independent importance measures, but also reflect the intrinsic relationship of the moment-independent measures and the corresponding variance-based importance measures. For the problem that the computational effort of the moment-independent importance measure is usually too high, the computation of the proposed moment-independent importance measures is transformed into that of the variance-based importance measures on their intrinsic relationship. And then combining the high efficient state dependent parameter (SDP) method for the calculation of the conditional moments of the model output, a SDP solution is established to solve two moment-independent importance measures. Several examples are used to demonstrate that the proposed importance measures can effectively describe the effect of the basic variable on the reliability of the structure system, and the established solution can obtain the two importance measures simultaneously with only a single set of model runs, which allows for a strong reduction of the computational cost.

Development of Injury Probability Functions for the Flexible Pedestrian Legform Impactor

The goal of this study was to develop injury probability functions for the leg bending moment and MCL (Medial Collateral Ligament) elongation of the Flexible Pedestrian Legform Impactor (Flex-PLI) based on human response data available from the literature. Data for the leg bending moment at fracture in dynamic 3-point bending were geometrically scaled to an average male using the standard lengths obtained from the anthropometric study, based on which the dimensions of the Flex-PLI were determined. Both male and female data were included since there was no statistically significant difference in bone material property. Since the data included both right censored and uncensored data, the Weibull Survival Model was used to develop a human leg fracture probability function. As for the MCL failure, since one of the two data sources does not provide tabulated data, two MCL failure probability functions as a function of the knee bending angle developed using the Weibull Survival Model were averaged over the knee bending angle. The functions developed for the human leg fracture and the MCL failure were converted to those for the Flex-PLI using the results of the previous study that investigated the correlation between human and Flex-PLI injury measures using human and Flex-PLI finite element (FE) models and simplified vehicle models. 10% increase in failure tolerance due to muscle tone estimated from the literature was taken into account when converting the function for the MCL. Since the conversion of the MCL failure probability function was made using the correlation function between different human and Flex-PLI injury measures, the correlation between the different measures both for human was investigated for validating the conversion. The injury thresholds proposed for the Flex-PLI were evaluated in terms of equivalence to the current injury thresholds for the TRL legform specified in the global technical regulation (gtr) on pedestrian safety by developing an injury probability function for the tibia fracture measure used for the TRL legform. The results showed that the protection level provided by the currently proposed injury thresholds for the Flex-PLI is similar to that for the TRL legform in terms of injury probability. Language: en

Reliability Measure Based on Failure Probability Function and Its Solution by Conditional Probability Simulation Method

The solution of the failure probability as a function of the distribution parameters is the key problem in reliability-based optimization.A novel method is proposed to obtain the failure probability function and a new reliability measure,which is the statistics characteristic value of the failure probability function.The key idea of the proposed method is using the conditional probability Markov chain simulation and the third order maximum entropy method to obtain the failure probability function.The conditional probability Markov chain simulation firstly transforms the failure probability into the product of a feature ratio factor and the probability of an introduced linear failure region,and then Markov chain algorithm is adopted to calculate the ratio factor by directly simulating the samples of the failure regions,the probability of the introduced linear failure region can be calculated easily.The third order maximum entropy method is implemented to estimate the conditional density function based on failure samples and obtain the failure probability function finally.The accuracy,efficiency and applicability of the proposed method are demonstrated by several examples.The results show that the proposed method can efficiently estimate the failure probability with high accuracy.The proposed method should be valuable for reliability-based optimization and reliability sensitivity analysis.

Reliability in Layered Networks With Random Link Failures

We consider network reliability in layered networks where the lower layer experiences random link failures. In layered networks, each failure at the lower layer may lead to multiple failures at the upper layer. We generalize the classical polynomial expression for network reliability to the multilayer setting. Using random sampling techniques, we develop polynomial-time approximation algorithms for the failure polynomial. Our approach gives an approximate expression for reliability as a function of the link failure probability, eliminating the need to resample for different values of the failure probability. Furthermore, it gives insight on how the routings of the logical topology on the physical topology impact network reliability. We show that maximizing the min cut of the (layered) network maximizes reliability in the low-failure-probability regime. Based on this observation, we develop algorithms for routing the logical topology to maximize reliability.

Reliability Analysis of Zero‐Failure Data with Poor Information

For costly and dangerous experiments, growing attention has been paid to the problem of the reliability analysis of zero‐failure data, with many new findings in world countries, especially in China. The existing reliability theory relies on the known lifetime distribution, such as the Weibull distribution and the gamma distribution. Thus, it is ineffective if the lifetime probability distribution is unknown. For this end, this article proposes the grey bootstrap method in the information poor theory for the reliability analysis of zero‐failure data under the condition of a known or unknown probability distribution of lifetime. The grey bootstrap method is able to generate many simulated zero‐failure data with the help of few zero‐failure data and to estimate the lifetime probability distribution by means of an empirical failure probability function defined in this article. The experimental investigation presents that the grey bootstrap method is effective in the reliability analysis only with the few zero‐failure data and without any prior information of the lifetime probability distribution. Copyright © 2011 John Wiley & Sons, Ltd.

Importance Measure Analysis in the Presence of Epistemic and Aleatory Uncertainties

In the presence of both epistemic and aleatory uncertainties, two importance measures are developed to quantify the effect of input variable on output response. One importance measure is defined as the effect of input variable on the membership function of structural failure probability, and the other is defined as the effect of the input variable on the membership function of the variance of structural performance function. Monte Carlo method and a novel point estimation method are investigated, to evaluate the two importance measures. A numerical example and an engineering example of wing model structure illustrate the directive significance of the two proposed importance measures in structure design, and the examples also demonstrate that the proposed point estimation is more efficient than the Monte Carlo simulation with acceptable precision.

Monte Carlo simulation-based sensitivity analysis of the model of a thermal–hydraulic passive system

Thermal–Hydraulic (T–H) passive safety systems are potentially more reliable than active systems, and for this reason are expected to improve the safety of nuclear power plants.However, uncertainties are present in the operation and modeling of a T–H passive system and the system may find itself unable to accomplish its function. For the analysis of the system functional failures, a mechanistic code is used and the probability of failure is estimated based on a Monte Carlo (MC) sample of code runs which propagate the uncertainties in the model and numerical values of its parameters/variables.Within this framework, sensitivity analysis aims at determining the contribution of the individual uncertain parameters (i.e., the inputs to the mechanistic code) to (i) the uncertainty in the outputs of the T–H model code and (ii) the probability of functional failure of the passive system. The analysis requires multiple (e.g., many hundreds or thousands) evaluations of the code for different combinations of system inputs: this makes the associated computational effort prohibitive in those practical cases in which the computer code requires several hours to run a single simulation.To tackle the computational issue, in this work the use of the Subset Simulation (SS) and Line Sampling (LS) methods is investigated. The methods are tested on two case studies: the first one is based on the well-known Ishigami function [1]; the second one involves the natural convection cooling in a Gas-cooled Fast Reactor (GFR) after a Loss of Coolant Accident (LOCA) [2].

On the effect of random defects on the fatigue notch factor at different stress ratios

The effect of randomly distributed defects of different sizes on the fatigue performance of a notched member has been studied. Both the effect of the volume density of defects and the size distribution of the defects have been studied. The fatigue life and the fatigue limit distributions of a smooth or notched specimen have been estimated. Using this probabilistic approach, the fatigue notch factor as function of failure probability can be estimated.It is shown that the approach with randomly distributed defects has better predictive capabilities than classical methods. The weakest link approach can be seen as an alternative to simulation of the randomly distributed defects. The two methods have similar predictive capabilities. The stress based methods generally do not take the load ratio into account when estimating the fatigue notch factor. It is also shown that the fatigue limit estimated with defect analysis is lower than the experimental one.

Convergence Rates of Distributed Average Consensus With Stochastic Link Failures

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We consider a distributed average consensus algorithm over a network in which communication links fail with independent probability. In such stochastic networks, convergence is defined in terms of the variance of deviation from average. We first show how the problem can be recast as a linear system with multiplicative random inputs which model link failures. We then use our formulation to derive recursion equations for the second order statistics of the deviation from average in networks with and without additive noise. We give expressions for the convergence behavior in the asymptotic limits of small failure probability and large networks. We also present simulation-free methods for computing the second order statistics in each network model and use these methods to study the behavior of various network examples as a function of link failure probability. </para>