Reliability Analysis of Multiple Causes of Failure in Presence of Independent Competing Risks

The failures of complex systems always arise from different causes in reliability test. However, it is difficult to evaluate the failure effect caused by a specific cause in presence of other causes. Therefore, a generalize reliability analysis model, which takes into account of the multiple competing causes, is highly needed. This paper develops a statistical reliability analysis procedure to investigate the reliability characteristics of multiple failure causes under independent competing risks. We mainly consider the case when the lifetime data follow log‐location‐scale distributions and may also be right‐censored. Maximum likelihood (ML) estimators of unknown parameters are derived by applying the Newton–Raphson method. With the large‐sample assumption, the normal approximation of the ML estimators is used to construct the asymptotic confidence intervals in which the standard error of the variance‐covariance matrix is calculated by using the delta method. In particular, the Akaike information criterion is utilized to determine the appropriate fitted distribution for each cause of failure. An illustrative numerical experiment about the fuel cell engine (FCE) is presented to demonstrate the feasibility and effectiveness of the proposed model. The results can facilitate continued advancement in reliability prediction and reliability allocation for FCE, and also provide theoretical basis for the application of reliability concepts to many other complex systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Bayes estimation of residual life by fusing multisource information

Residual life estimation is essential for reliability engineering. Traditional methods may experience difficulties in estimating the residual life of products with high reliability, long life, and small sample. The Bayes model provides a feasible solution and can be a useful tool for fusing multisource information. In this study, a Bayes model is proposed to estimate the residual life of products by fusing expert knowledge, degradation data, and lifetime data. The linear Wiener process is used to model degradation data, whereas lifetime data are described via the inverse Gaussian distribution. Therefore, the joint maximum likelihood (ML) function can be obtained by combining lifetime and degradation data. Expert knowledge is used according to the maximum entropy method to determine the prior distributions of parameters, thereby making this work different from existing studies that use non-informative prior. The discussion and analysis of different types of expert knowledge also distinguish our research from others. Expert knowledge can be classified into three categories according to practical engineering. Methods for determining prior distribution by using the aforementioned three types of data are presented. The Markov chain Monte Carlo is applied to obtain samples of the parameters and to estimate the residual life of products due to the complexity of the joint ML function and the posterior distribution of parameters. Finally, a numerical example is presented. The effectiveness and practicability of the proposed method are validated by comparing it with residual life estimation that uses non-informative prior. Then, its accuracy and correctness are proven via simulation experiments.     

On a new extension of Weibull distribution: Properties,estimation, and applications to one and two causes of failures

In this paper, we try to supplement the distribution theory literature by introducing a new distribution, called the logarithmic transformed Weibull (LTW) distribution by logarithm transformation method. The proposed distribution exhibits constant, decreasing, increasing, unimodal and unimodal then bathtub-shaped hazard rates. Although our main focus is on the estimation from the frequentist point of view, in addition, we derive some useful structural and statistical properties of the proposed LTW distribution. The LTW parameters are estimated by eight frequentist methods of estimation. We perform extensive simulation experiments to show the performances of the proposed estimators. Applications of the model are presented by reanalyzing two real data sets, and comparisons are made with the fit attained by some other well-known distributions for illustrative purposes. As an illustration, one data set is analyzed for competing risks to demonstrate the flexibility of the proposed model.     

Eccentrically loaded sandwich panels

The paper deals with the design of eccentrically loaded sandwich panels in civil engineering structures. A theoretical analysis taking into account the non-linear stress–strain properties of the materials and the influence of shear deformations is proposed. A computer program based on the described algorithm is used to solve a numerical example. It is found that the Newton–Raphson procedure both in the solution of strains at individual sections and in the step by step procedure over the whole structure converges rapidly. Thus the method is suitable for the evaluation of the critical load due to strength failure or loss of stability.     

A new hybrid method for solving linear–non‐linear systems of equations

This paper presents a new method for solving any combination of linear–non‐linear equations. The method is based on the separation of linear equations in terms of some selected variables from the non‐linear ones. The linear group is solved by means of any method suitable for the linear system. This operation needs no iteration. The non‐linear group, however, is solved by an iteration technique based on a new formula using the Taylor series expansion. The method has been described and demonstrated in several examples of analytical systems with very good results. The new method needs the initial approximations for non‐linear variables only. This requires far less computation than the Newton–Raphson method. The method also has a very good convergence rate. The proposed method is most beneficial for engineering systems that very often involve a large number of linear equations with limited number of non‐linear equations. Copyright © 2005 John Wiley & Sons, Ltd.     

EM algorithm for estimating the Burr XII parameters with multiple censored data

The two‐parameter Burr XII distribution has been widely used in various practical applications such as business, chemical engineering, quality control, medical research and reliability engineering. In this paper, we present maximum likelihood estimation (MLE) via the expectation–maximization (EM) algorithm to estimate the Burr XII parameters with multiple censored data. We also provide a method that can be used to construct the confidence intervals of the parameters, a method that computes the asymptotic variance and the covariance of the MLE from the complete and missing information matrices. A simulation study is conducted to compare the performance of the MLE via the EM algorithm and the Netwon–Raphson (NR) algorithm. The simulation results show that the EM algorithm outperforms the NR algorithm in most cases in terms of bias and errors in the root mean square. A numerical example is also used to demonstrate the performance of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

An additive Chen-Weibull distribution and its applications in reliability modeling

In this article, we introduce a new lifetime distribution with increasing and bathtub-shaped failure rates. Some statistical properties of the proposed distribution are studied. We use the method of maximum likelihood for estimating the model parameters and reliability characteristics and discuss the interval estimates using asymptotic confidence intervals and bootstrap confidence intervals on one hand, and we provide Bayes estimators and highest posterior density intervals for the parameters via Hamiltonian Monte Carlo simulation method on the other hand. We demonstrate the superiority of the proposed distribution by fitting two reliability data sets well-known from references.     

Online parameter estimation and run-to-run process adjustment using categorical observations

Categorical observations are frequently observed in run-to-run processes where obtaining accurate measurements of quality characteristics is difficult. In such circumstances, the use of categorical observations to estimate a process model and generate an adjustment recipe becomes inevitable. However, most conventional run-to-run controllers cannot be applied if no continuous observations are available; some parameter estimation methods that can handle categorical data only use historical dataset in an offline manner. In practice, it is common to see observations collected following a time sequence in a run-to-run process. Taking the lapping process in semiconductor manufacturing as an example, this paper develops an online approach for parameters estimation and run-to-run process adjustment using categorical observations. The proposed method optimises a penalised Maximum Likelihood (ML) function and updates parameters step by step when new categorical observations become available. A control strategy is also derived to generate receipts for process update between runs. The computational results of performance evaluation show that the proposed method is capable of estimating unknown parameters and control output quality online when initial bias exists.     

A modified Weibull extension with bathtub-shaped failure rate function

Models with bathtub-shaped failure rate function are useful in reliability analysis, and particularly in reliability related decision making and cost analysis. The traditional Weibull distribution is, however, unable to model the complete lifetime of systems with a bathtub-shaped failure rate function. In this paper, a new model, which is useful for modeling this type of failure rate function, is presented. The model can also be seen as a generalization of the Weibull distribution. Parameter estimation methods are studied for this new distribution. Examples and results of comparison are shown to illustrate the applicability of this new model.     

Reliability analysis of the dynamic system for the Chen model through sequential order statistics

Recently, the two-parameter Chen distribution has widely been used for reliability studies in various engineering fields. In this article, we have developed various statistical inferences on the composite dynamic system, assuming Chen distribution as a baseline model. In this dynamic system, failure of a component induces a higher load on the surviving components and thus increases component hazard rate through a power-trend process. The classical and Bayesian point estimates of the unknown parameters of the composite system are obtained by the method of maximum likelihood and Markov chain Monte Carlo techniques, respectively. In the Bayesian framework, we have used gamma priors to obtain Bayes estimates of unknown parameters under the squared error and generalized entropy loss functions. The interval estimates of the baseline reliability function are obtained by using the Fisher information matrix and Bayesian method. A parametric hypothesis test is presented to test whether the failed components change the hazard rate function. A compact simulation study is carried out to examine the behavior of the proposed estimation methods. Finally, one real data analysis is performed for illustrative purposes.     

An improved modified Weibull distribution applied to predict the reliability evolution of an aircraft lock mechanism

A new improved modified Weibull distribution (IMW), which consists of two tandem failure models is proposed in this paper. The order statistics, moment estimate and maximum likelihood estimate of the new distribution are investigated. The distribution is flexible for modeling the three phases of the modified bathtub-shaped hazard function. The modified distribution allows for enhanced statistical simulation. The new distribution is analyzed and applied to two sets of known lifetime data to demonstrate the advantages of the model. To verify the practical engineering applications of the model, reliability evolution test data for an aircraft door lock mechanism are analyzed and fitted. Based on the obtained trends of the lock hook angle and lock hook distance, we predict the evolutionary data for numerous mechanism samples, thus reducing the test cost. By considering the subdistribution and removing one parameter, the fitting effect is improved over that of the traditional distribution model.     

A new model with bathtub-shaped failure rate using an additive Burr XII distribution

It is a common situation that the failure rate function has a bathtub shape for many mechanical and electronic components. A simple model based on adding two Burr XII distributions is presented for modeling this type data. The graphical estimation on probability paper is illustrated, and examples of its usage are presented. The Akaike Information Criterion was used for judging the adequacy of the models presented for the numerical examples. It can be seen that the proposed model is a very competitive model for describing the bathtub-shaped failure rate lifetime data.     

Assessment of methods for computing the closest point projection,penetration, and gap functions in contact searching problems

In computational contact mechanics problems, local searching requires calculation of the closest point projection of a contactor point onto a given target segment. It is generally supposed that the contact boundary is locally described by a convex region. However, because this assumption is not valid for a general curved segment of a three‐dimensional quadratic serendipity element, an iterative numerical procedure may not converge to the nearest local minimum. To this end, several unconstrained optimization methods are tested: the Newton–Raphson method, the least square projection, the sphere and torus approximation method, the steepest descent method, the Broyden method, the Broyden–Fletcher–Goldfarb–Shanno method, and the simplex method. The effectiveness and robustness of these methods are tested by means of a proposed benchmark problem. It is concluded that the Newton–Raphson method in conjunction with the simplex method significantly increases the robustness of the local contact search procedure of pure penalty contact methods, whereas the torus approximation method can be recommended for contact searching algorithms, which employ the Lagrange method or the augmented Lagrangian method. Copyright © 2015 John Wiley & Sons, Ltd.     

E-Bayesian estimation of reliability characteristics of two-parameter bathtub-shaped lifetime distribution with application

This article presents the expected Bayesian (E-Bayesian) estimation of the scale parameter, reliability and failure rate functions of two-parameter bathtub-shaped lifetime distribution under type-II censoring data with. Squared error loss function and gamma distribution as a conjugate prior distribution for the unknown parameter are used to obtain the E-Bayesian estimators. Also, three different prior distributions for the hyperparameters for the E-Bayesian estimators are considered. Some properties of the E-Bayesian estimators are studied. Using minimum mean square error criteria, a simulation study is conducted to compare the performance of the E-Bayesian estimators and the corresponding Bayes and maximum likelihood estimators. A real data set is analysed to show the applicability of the different proposed estimators. The numerical results show that the E-Bayesian estimators perform better than the classical and Bayesian estimators.     

Trust‐region based return mapping algorithm for implicit integration of elastic‐plastic constitutive models

A new method for the solution of the non‐linear equations forming the core of constitutive model integration is proposed. Specifically, the trust‐region method that has been developed in the numerical optimization community is successfully modified for use in implicit integration of elastic‐plastic models. Although attention here is restricted to these rate‐independent formulations, the proposed approach holds substantial promise for adoption with models incorporating complex physics, multiple inelastic mechanisms, and/or multiphysics. As a first step, the non‐quadratic Hosford yield surface is used as a representative case to investigate computationally challenging constitutive models. The theory and implementation are presented, discussed, and compared with other common integration schemes. Multiple boundary value problems are studied and used to verify the proposed algorithm and demonstrate the capabilities of this approach over more common methodologies. Robustness and speed are then investigated and compared with existing algorithms. Through these efforts, it is shown that the utilization of a trust‐region approach leads to superior performance versus a traditional closest‐point projection Newton–Raphson method and comparable speed and robustness to a line search augmented scheme. Copyright © 2017 John Wiley & Sons, Ltd.     

A radial point interpolation meshless method extended with an elastic rate-independent continuum damage model for concrete materials

An advanced discretization meshless technique, the radial point interpolation method (RPIM), is applied to analyze concrete structures using an elastic continuum damage constitutive model. Here, the theoretical basis of the material model and the computational procedure are fully presented. The plane stress meshless formulation is extended to a rate-independent damage criterion, where both compressive and tensile damage evolutions are established based on a Helmholtz free energy function. Within the return-mapping damage algorithm, the required variable fields, such as the damage variables and the displacement field, are obtained. This study uses the Newton–Raphson nonlinear solution algorithm to achieve the nonlinear damage solution. The verification, where the performance is assessed, of the proposed model is demonstrated by relevant numerical examples available in the literature.     

0-1型二次规划的光滑函数法

本文针对工程设计、经济分析及计算机辅助设计等领域出现的0-1型二次规划问题,提出了Newton型的光滑迭代算法.首先利用NCP函数将0-1规划转化为不可微优化问题,然后通过构造不可微问题的光滑一致逼近,将组合优化问题转化成了可微的无约束优化问题,克服了已有算法收敛速度慢且计算结构复杂的缺点.文中给出了算法的迭代格式,证...     

On the equivalence of dynamic relaxation and the Newton‐Raphson method

Dynamic relaxation is an iterative method to solve nonlinear systems of equations, which is frequently used for form finding and analysis of structures that undergo large displacements. It is based on the solution of a fictitious dynamic problem where the vibrations of the structure are traced by means of a time integration scheme until a static equilibrium is reached. Fictitious values are used for the mass and damping parameters. Heuristic rules exist to determine these values in such a way that the time integration procedure converges rapidly without becoming unstable. Central to these heuristic rules is the assumption that the highest convergence rate is achieved when the ratio of the highest and lowest eigenfrequency of the structure is minimal. This short communication shows that all eigenfrequencies become identical when a fictitious mass matrix proportional to the stiffness matrix is used. If, in addition, specific values are used for the fictitious damping parameters and the time integration step, the dynamic relaxation method becomes completely equivalent to the Newton‐Raphson method. The Newton‐Raphson method can therefore be regarded as a specific form of dynamic relaxation. This insight may help to interpret and improve nonlinear solvers based on dynamic relaxation and/or the Newton‐Raphson method.     

Parameter estimation of the intensity process of self-excitingpoint processes using the EM algorithm

This paper presents a method of estimating the parameters of intensity processes in the self-exciting point process (SEPP) with the expectation-maximization (EM) algorithm. In the present paper, the case is considered where the intensity process of SEPPs is dependent only on the latest occurrence, i.e., one-memory SEPPs, as well as where the impulse response function characterizing the intensity process is parameterized as a single exponential function having a constant coefficient that takes a positive or negative value, i.e., making it possible to model a self “-exciting” or “-inhibiting” point process. Then, an explicit formula is derived for estimating the parameters specifying the intensity process on the basis of the EM algorithm, which in this instance gives the maximum likelihood (ML) estimates without solving nonlinear optimization problems. In practical computations, the parameters of interest can be estimated from the histogram of time intervals between point events. Monte Carlo simulations illustrate the validity of the derived estimation formulas and procedures