Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

Fuzzy stochastic generalized reliability studies on embankment systems based on first-order approximation theorem

In order to address the complex uncertainties caused by interfacing between the fuzziness and randomness of the safety problem for embankment engineering projects, and to evaluate the safety of embankment engineering projects more scientifically and reasonably, this study presents the fuzzy logic modeling of the stochastic finite element method （SFEM） based on the harmonious finite element （HFE） technique using a first-order approximation theorem. Fuzzy mathematical models of safety repertories were introduced into the SFEM to analyze the stability of embankments and foundations in order to describe the fuzzy failure procedure for the random safety performance function. The fuzzy models were developed with membership functions with half depressed gamma distribution, half depressed normal distribution, and half depressed echelon distribution. The fuzzy stochastic mathematical algorithm was used to comprehensively study the local failure mechanism of the main embankment section near Jingnan in the Yangtze River in terms of numerical analysis for the probability integration of reliability on the random field affected by three fuzzy factors. The result shows that the middle region of the embankment is the principal zone of concentrated failure due to local fractures. There is also some local shear failure on the embankment crust. This study provides a referential method for solving complex multi-uncertainty problems in engineering safety analysis.

An Uncertainty Analysis and Reliability-Based Multidisciplinary Design Optimization Method Using Fourth-Moment Saddlepoint Approximation

In uncertainty analysis and reliability-based multidisciplinary design and optimization(RBMDO)of engineering structures,the saddlepoint approximation(SA)method can be utilized to enhance the accuracy and efficiency of reliability evaluation.However,the random variables involved in SA should be easy to handle.Additionally,the corresponding saddlepoint equation should not be complicated.Both of them limit the application of SA for engineering problems.The moment method can construct an approximate cumulative distribution function of the performance function based on the first few statistical moments.However,the traditional moment matching method is not very accurate generally.In order to take advantage of the SA method and the moment matching method to enhance the efficiency of design and optimization,a fourth-moment saddlepoint approximation(FMSA)method is introduced into RBMDO.In FMSA,the approximate cumulative generating functions are constructed based on the first four moments of the limit state function.The probability density function and cumulative distribution function are estimated based on this approximate cumulative generating function.Furthermore,the FMSA method is introduced and combined into RBMDO within the framework of sequence optimization and reliability assessment,which is based on the performance measure approach strategy.Two engineering examples are introduced to verify the effectiveness of proposed method.

Inverse reliability analysis and design for tunnel face stability considering soil spatial variability

The traditional deterministic analysis for tunnel face stability neglects the uncertainties of geotechnical parameters,while the simplified reliability analysis which models the potential uncertainties by means of random variables usually fails to account for soil spatial variability.To overcome these limitations,this study proposes an efficient framework for conducting reliability analysis and reliability-based design(RBD)of tunnel face stability in spatially variable soil strata.The three-dimensional(3D)rotational failure mechanism of the tunnel face is extended to account for the soil spatial variability,and a probabilistic framework is established by coupling the extended mechanism with the improved Hasofer-Lind-Rackwits-Fiessler recursive algorithm(iHLRF)as well as its inverse analysis formulation.The proposed framework allows for rapid and precise reliability analysis and RBD of tunnel face stability.To demonstrate the feasibility and efficacy of the proposed framework,an illustrative case of tunnelling in frictional soils is presented,where the soil's cohesion and friction angle are modelled as two anisotropic cross-correlated lognormal random fields.The results show that the proposed method can accurately estimate the failure probability(or reliability index)regarding the tunnel face stability and can efficiently determine the required supporting pressure for a target reliability index with soil spatial variability being taken into account.Furthermore,this study reveals the impact of various factors on the support pressure,including coefficient of variation,cross-correlation between cohesion and friction angle,as well as autocorrelation distance of spatially variable soil strata.The results also demonstrate the feasibility of using the forward and/or inverse first-order reliability method(FORM)in high-dimensional stochastic problems.It is hoped that this study may provide a practical and reliable framework for determining the stability of tunnels in complex soil strata.

A POSTERIORI ERROR ESTIMATE OF THE DSD METHOD FOR FIRST-ORDER HYPERBOLIC EQUATIONS

A posteriori error estimate of the discontinuous-streamline diffusion method for first-order hyperbolic equations was presented, which can be used to adjust space mesh reasonably. A numerical example is given to illustrate the accuracy and feasibility of this method.

Fast First-Order Methods for Minimizing Convex Composite Functions

Two new versions of accelerated first-order methods for minimizing convex composite functions are proposed. In this paper, we first present an accelerated first-order method which chooses the step size 1/ Lk to be 1/ L0 at the beginning of each iteration and preserves the computational simplicity of the fast iterative shrinkage-thresholding algorithm. The first proposed algorithm is a non-monotone algorithm. To avoid this behavior, we present another accelerated monotone first-order method. The proposed two accelerated first-order methods are proved to have a better convergence rate for minimizing convex composite functions. Numerical results demonstrate the efficiency of the proposed two accelerated first-order methods.

Comparative Studies between Picard’s and Taylor’s Methods of Numerical Solutions of First Ordinary Order Differential Equations Arising from Real-Life Problems

To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.

Distributed Collaborative Response Surface Method for Mechanical Dynamic Assembly Reliability Design

Because of the randomness of many impact factors influencing the dynamic assembly relationship of complex machinery, the reliability analysis of dynamic assembly relationship needs to be accomplished considering the randomness from a probabilistic perspective. To improve the accuracy and efficiency of dynamic assembly relationship reliability analysis, the mechanical dynamic assembly reliability（MDAR） theory and a distributed collaborative response surface method（DCRSM） are proposed. The mathematic model of DCRSM is established based on the quadratic response surface function, and verified by the assembly relationship reliability analysis of aeroengine high pressure turbine（HPT） blade-tip radial running clearance（BTRRC）. Through the comparison of the DCRSM, traditional response surface method（RSM） and Monte Carlo Method（MCM）, the results show that the DCRSM is not able to accomplish the computational task which is impossible for the other methods when the number of simulation is more than 100 000 times, but also the computational precision for the DCRSM is basically consistent with the MCM and improved by 0.40-4.63% to the RSM, furthermore, the computational efficiency of DCRSM is up to about 188 times of the MCM and 55 times of the RSM under 10000 times simulations. The DCRSM is demonstrated to be a feasible and effective approach for markedly improving the computational efficiency and accuracy of MDAR analysis. Thus, the proposed research provides the promising theory and method for the MDAR design and optimization, and opens a novel research direction of probabilistic analysis for developing the high-performance and high-reliability of aeroengine.

RELIABILITY ASSESSMENT OF ENTROPY METHOD FOR SYSTEM CONSISTED OF IDENTICAL EXPONENTIAL UNITS

The reliability assessment of unit-system near two levels is the mostimportant content in the reliability multi-level synthesis of complex systems. Introducing theinformation theory into system reliability assessment, using the addible characteristic ofinformation quantity and the principle of equivalence of information quantity, an entropy method ofdata information conversion is presented for the system consisted of identical exponential units.The basic conversion formulae of entropy method of unit test data are derived based on the principleof information quantity equivalence. The general models of entropy method synthesis assessment forsystem reliability approximate lower limits are established according to the fundamental principleof the unit reliability assessment. The applications of the entropy method are discussed by way ofpractical examples. Compared with the traditional methods, the entropy method is found to be validand practicable and the assessment results are very satisfactory.

Advanced method to estimate reliability-based sensitivity of mechanical components with strongly nonlinear performance function

Based on the random perturbation technique for reliability sensitivity design,some realistic reliability-based sensitivity issues are discussed,some of which have a structure of high nonlinear performance functions.Combining the related theories of the moment method of the reliability analysis,the matrix differential,and the Kronecker algebra,the reliability-based sensitivity method based on the perturbation method is modified if the first four moments of random variables are given.Meanwhile,a reliability-based sensitivity computation method is proposed.Some examples are used to show that using this method can effectively improve the accuracy of the reliability-based sensitivity computation and offer a reliable theoretic basis in engineering.

An Intelligent Method for Structural Reliability Analysis Based on Response Surface

As water depth increases, the structural safety and reliability of a system become more and more important and challenging. Therefore, the structural reliability method must be applied in ocean engineering design such as offshore platform design. If the performance function is known in structural reliability analysis, the first-order second-moment method is often used. If the performance function could not be definitely expressed, the response surface method is always used because it has a very clear train of thought and simple programming. However, the traditional response surface method fits the response surface of quadratic polynomials where the problem of accuracy could not be solved, because the true limit state surface can be fitted well only in the area near the checking point. In this paper, an intelligent computing method based on the whole response surface is proposed, which can be used for the situation where the performance function could not be definitely expressed in structural reliability analysis. In this method, a response surface of the fuzzy neural network for the whole area should be constructed first, and then the structural reliability can be calculated by the genetic algorithm. In the proposed method, all the sample points for the training network come from the whole area, so the true limit state surface in the whole area can be fitted. Through calculational examples and comparative analysis, it can be known that the proposed method is much better than the traditional response surface method of quadratic polynomials, because, the amount of calculation of finite element analysis is largely reduced, the accuracy of calculation is improved, and the true limit state surface can be fitted very well in the whole area. So, the method proposed in this paper is suitable for engineering application.

A Bayesian Updating Method for Non-Probabilistic Reliability Assessment of Structures with Performance Test Data

For structures that only the predicted bounds of uncertainties are available,this study proposes a Bayesianmethod to logically evaluate the nonprobabilistic reliability of structures based on multi-ellipsoid convex model and performance test data.According to the given interval ranges of uncertainties,we determine the initial characteristic parameters of a multi-ellipsoid convex set.Moreover,to update the plausibility of characteristic parameters,a Bayesian network for the information fusion of prior uncertainty knowledge and subsequent performance test data is constructed.Then,an updated multi-ellipsoid set with the maximum likelihood of the performance test data can be achieved.The credible non-probabilistic reliability index is calculated based on the Kriging-based surrogate model of the performance function.Several numerical examples are presented to validate the proposed Bayesian updating method.

Experimental and Finite Element Method Studies of J-Lead Solder Joint Reliability

A comprehensive experimental and numerical study of solder joints for plastic leaded chip carrier (PLCC) 84-Pin, 1.27 mm pitch was carried out. The reliability of solder joints was assessed through accelerated thermal cycling at the temperature range of - 55℃-125℃. The samples were taken out to observe the evolution in microstructure, such as grain coarsening, initiation and propagation of cracks. It was found that the Pb-rich phases segregated gradually and formed a continuous layer adjacent to the intermetallic compound (IMC) layer with increasing the number of thermal cycles, resulting in cracks near the solder/lead interface. The response of stress and strain was studied using nonlinear finite element method (FEM), and the results agreed well with the experimental data.

Reliability analysis for aeroengine turbine disc fatigue life with multiple random variables based on distributed collaborative response surface method

The fatigue life of aeroengine turbine disc presents great dispersion due to the randomness of the basic variables,such as applied load,working temperature,geometrical dimensions and material properties.In order to ameliorate reliability analysis efficiency without loss of reliability,the distributed collaborative response surface method(DCRSM) was proposed,and its basic theories were established in this work.Considering the failure dependency among the failure modes,the distributed response surface was constructed to establish the relationship between the failure mode and the relevant random variables.Then,the failure modes were considered as the random variables of system response to obtain the distributed collaborative response surface model based on structure failure criterion.Finally,the given turbine disc structure was employed to illustrate the feasibility and validity of the presented method.Through the comparison of DCRSM,Monte Carlo method(MCM) and the traditional response surface method(RSM),the results show that the computational precision for DCRSM is more consistent with MCM than RSM,while DCRSM needs far less computing time than MCM and RSM under the same simulation conditions.Thus,DCRSM is demonstrated to be a feasible and valid approach for improving the computational efficiency of reliability analysis for aeroengine turbine disc fatigue life with multiple random variables,and has great potential value for the complicated mechanical structure with multi-component and multi-failure mode.

Singularity of Some Software Reliability Models and Parameter Estimation Method

According to the principle, “The failure data is the basis of software reliability analysis”, we built a software reliability expert system (SRES) by adopting the artificial intelligence technology. By reasoning out the conclusion from the fitting results of failure data of a software project, the SRES can recommend users “the most suitable model” as a software reliability measurement model. We believe that the SRES can overcome the inconsistency in applications of software reliability models well. We report investigation results of singularity and parameter estimation methods of experimental models in SRES.

IMPROVED STRATIFIED SAMPLING MONTE CARLO METHOD TO ANALYZE RELIABILITY OF STRUCTURAL SYSTEMS

A stratified sampling Monte Carlo method to analyze the reliability of structural systems is presented. Introducing a small exploratory simulation, this method overcomes the difficulties for getting the systematic sampling probability of all the strata. Several useful and efficient stratification methods are given and the strategies of stratification and simulation are studied. A general conclusion has been presented corresponding to actual engineering structures. The strict theoretical proof has been given,and it is especially effective to solve probabilistic integration. Statistic error of evaluating failure probability is reduced obviously. Especially in highly non-linear and nonreonvex problems, it is more accurate than other methods. Compared with other variance reduction techniques, this method can obtain a more obvious variance reduction and an increased sampling efficiency. Moreover, without strict limiting condition, it is convenient to use. This method is especially suitable to solve the reliability problem of structural systems with multiple failure modes and highly non-linear safety margin equations.

Computer System Reliability Allocation Method and Supporting Tool

This paper presents a computer system reliability allocation method that is based on the theory of statistic and Markovian chain,which can be used to allocate reliability to subsystem, to hybrid system and software modules. A relevant supporting tool built by us is introduced.

A new look at the response surface method for reliability analysis using chaos theory

To overcome excessive computation errors and convergence failures encountered in an iterative calculation of the reliability index using the response surface method （RSM） for some nonlinear limit state functions, this study investigates an essential factor based on chaotic dynamics theory. The bifurcation diagrams of the reliability index are presented for some typical nonlinear limit state functions, and the computation results from the mapping functions due to the RSM iterations show the complicated dynamic phenomena such as the periodic oscillation, as well as bifurcation and chaos. From the numerical examples, it is concluded that the parameter of selection range fplays an important role in the convergence of the RSM iteration, and an improved RSM iterative algorithm is proposed with regard to the incorporation of the iterative sequential function of selection rangef The proposed method is shown to be efficient and to yield accurate results.

An Artificial Neural Network-Based Response Surface Method for Reliability Analyses of c-φ Slopes with Spatially Variable Soil

This paper presents an artificial neural network（ANN）-based response surface method that can be used to predict the failure probability of c-φslopes with spatially variable soil.In this method,the Latin hypercube sampling technique is adopted to generate input datasets for establishing an ANN model;the random finite element method is then utilized to calculate the corresponding output datasets considering the spatial variability of soil properties;and finally,an ANN model is trained to construct the response surface of failure probability and obtain an approximate function that incorporates the relevant variables.The results of the illustrated example indicate that the proposed method provides credible and accurate estimations of failure probability.As a result,the obtained approximate function can be used as an alternative to the specific analysis process in c-φslope reliability analyses.

Extremum response surface method of reliability analysis on two-link flexible robot manipulator

In order to present a new method for analyzing the reliability of a two-link flexible robot manipulator,Lagrange dynamics differential equations of the two-link flexible robot manipulator were established by using the integrated modal method and the multi-body system dynamics method.By using the Monte Carlo method,the random sample values of the dynamic parameters were obtained and Lagrange dynamics differential equations were solved for each random sample value which revealed their displacement,speed and acceleration.On this basis,dynamic stresses and deformations were obtained.By taking the maximum values of the stresses and the deformations as output responses and the random sample values of dynamic parameters as input quantities,extremum response surface functions were established.A number of random samples were then obtained by using the Monte Carlo method and then the reliability was analyzed by using the extremum response surface method.The results show that the extremum response surface method is an efficient and fast reliability analysis method with high-accuracy for the two-link flexible robot manipulator.