Saddlepoint approximation based structural reliability analysis with non-normal random variables

Introduction and Background Most engineering structures contain multiple failure modes or components in which the nonperformance of any of the events can lead to system failure. In addition, the different failure modes may be correlated due to common random variables between the events. In many cases the events are assumed independent and system failure is evaluated using unions and intersections of the events. However, neglecting correlation between events may lead to grossly over predicting the probability of failure. An upper and lower bound can be determined for the probability of system failure using first- and second-order techniques. While there is some contradiction in the literature, a complete treatment of first-order bounds is made in this research. The first- order bounds are identified for events with both positive and negative correlation. In practice the second-order bounds typically use a first-order approximation to the limit-state for each event and the intersection of the two events is estimated by the correlation between the events. For nonlinear limit-states, the bounds may not actually include the true solution based on the error of the event probabilities of failure or the predicted intersection of the events. Several numerical examples are used to evaluate the accuracy and effectiveness of

In many cases a two-state approach for reliability analysis of systems composed of multiple components is not appropriate. This is so especially when redundancy is applied or when some components have multiple failure modes having non-identical impacts on system performance. For example, consider a redundant rectifier system consisting of two diodes connected in series (Fig. 5.1(a)) and without repair. The diodes might suffer two distinct failure modes: a short-circuit failure D2allowing current to pass in both directions and an open failure D3. Figure 5.1(b) shows the states and the corresponding transitions for a diode. It will be obvious that the system is only redundant to short-circuit failures and that the redundancy is not active for open failures. Even worse, the probability of an open system failure has almost been doubled. This means that the two different failure modes of the diodes have different impacts on system performance and, therefore, cannot be treated as one single failure mode. For this simple system a combinatorial solution is still possible as:R = P r{Da1Db1∪ Da1Db2∪ Da2Db1}, however, the logic expression for system success contains dependent events since the diode states D1and D2are mutually exclusive. In Chapter 7 it is shown how to deal with systems of multistate items more generally while Chapter 8 explains how to perform probability calculations on large logical expressions with dependent events.

본 연구에서는 다중 파괴모드를 고려한 사면의 신뢰도해석에 대하여 연구한다. 해석은 크게 두 부분으로 나뉜다. 첫 번째, Der Kiureghian과 Dakessian이 제안한 barrier method를 사용하여 시스템 신뢰도에 크게 영향을 미치는 중요 파괴모드를 연속적으로 탐색하여 찾아낸다. 둘째로, 찾아낸 중요 파괴모드들과 이에 해당하는 설계점들을 바탕으로 사면의 파괴확률을 계산한다. 다중 파괴모드를 갖는 사면의 신뢰도해석에서 파괴확률은 다중점 일차신뢰도법, Ditlevsen의 구간해법 및 몬테카를로 시뮬레이션 등을 이용하여 평가할 수 있다. 본 연구에서는 예제 해석을 통하여 이들 방법들의 비교연구를 수행하였다. 해석결과는 토사사면에 많은 수의 잠재적인 파괴면이 존재할 수 있지만, 사면의 시스템 파괴확률은 소수의 중요 파괴면에 의해 지배된다는 것을 보여준다. 따라서 토사사면의 시스템 신뢰도해석을 위해서 가장 중요한 단계는 모든 중요 파괴모드를 효율적인 방법에 의해서 탐색하여 결정하는 것이다. This work studies the reliability analysis of a slope that considers multiple failure modes. The analysis consists of two parts. First, significant failure modes that contribute most to system reliability are determined. The so-called barrier method proposed by Der Kiureghian and Dakessian to identify significant failure modes successively is employed. Second, the failure probability for the slope is estimated on the basis of the identified significant failure modes and corresponding design points. For reliability problems entailing multiple design points, failure probability can be estimated by the multi-point first-order reliability method (FORM), Ditlevsen's bounds method, and Monte Carlo simulation. In this paper, a comparative study between these methods has been made through example problems. Analysis results showed that while a soil slope may have a large number of potential slip surfaces, its system failure probability is usually governed by a few significant slip surfaces. Therefore, the most important step in the system reliability analysis for a soil slope is to identify all the significant failure modes in an efficient way.

Efficient system reliability analysis illustrated for a retaining wall and a soil slope

This study proposes a system reliability analysis of rack storage facilities subjected to forklift colliding events. The proposed system reliability analysis consists of two steps: the first step is to identify dominant failure modes that most contribute to the failure of the whole rack facilities, and the second step is to evaluate the system failure probability. In the first step, dominant failure modes are identified by using a simulation-based selective searching technique where the contribution of a failure mode to the system failure is roughly estimated based on the distance from the origin in the space of the random variables. In the second step, the multi-scale system reliability method is used to compute the system reliability where the first-order reliability method (FORM) is initially used to evaluate the component failure probability (failure probability of one member), and then the probabilities of the identified failure modes and their statistical dependence are evaluated, which is called as the lower-scale reliability analysis. Since the system failure probability is comprised of the probabilities of the failure modes, a higher-scale reliability analysis is performed again based on the results of the lower-scale analyses, and the system failure probability is finally evaluated. The illustrative example demonstrates the results of the system reliability analysis of the rack storage facilities subjected to forklift impact loadings. The numerical efficiency and accuracy of the approach are compared with the Monte Carlo simulations. The results show that the proposed two-step approach is able to provide accurate reliability assessment as well as significant saving of computational time. The results of the identified failure modes additionally let us know the most-critical members and their failure sequence under the complicated configuration of the member connections.

A simplified method for reliability- and integrity-based design of engineering systems and its application to offshore mooring systems

Reliability based design optimization is a methodology for nding optimized designs that are characterized with a low probability of failure. Primarily, reliability based design optimization consists of optimizing a merit function while satisfying reliability constraints. The reliability constraints are constraints on the probability of failure corresponding to each of the failure modes of the system or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. During the last few years, a variety of dierent formulations have been developed for reliability based design optimization. Traditionally, these have been formulated as a doubleloop (nested) optimization problems. The upper level optimization loop generally involves optimizing a merit function subject to reliability constraints and the lower level optimization loop(s) compute the probabilities of failure corresponding to the failure mode(s) that govern the system failure. This formulation is, by nature, computationally intensive. Researchers have provided sequential strategies to address this issue, where the deterministic optimization and reliability analysis are decoupled, and the process is performed iteratively until convergence is achieved. These methods, though attractive in terms of obtaining a workable reliable design at considerably reduced computational costs, often lead to premature convergence and therefore, yield spurious optimal designs. In this paper, a novel unilevel formulation for reliability based design optimization is developed. In the proposed formulation, the lower level optimization (evaluation of reliability constraints in the double-loop formulation) is replaced by its corresponding rst order Karush-Kuhn-Tucker (KKT) necessary optimality conditions at the upper level optimization. Such a replacement is equivalent to solving the original nested optimization if the constraint qualication conditions are satised. It is shown through the use of test problems that the proposed formulation is numerically robust (stable) and computationally ecient compared to the existing approaches for reliability based design optimization.

Due to the growing concern on the issues of global warming, climate change, and energy shortage, more rigorous requirements are imposed on the energy conversion process of combustion. Due to the relative safety and convenience in transportation and storage, liquid fuels are commonly used in many practical combustion systems such as diesel engines, gas turbines and industrial furnaces. In these combustion devices, turbulent spray flows are involved. The character of spray dispersion, evaporation, mixing and combustion of fuel droplets strongly determines the performance of these systems with respect to the combustion efficiency, stability, and emissions. Therefore, an improved understanding of turbulent spray combustion and the development of predictive models are needed for a better design of more efficient combustion systems. The present work focuses on the development of a computational methodology based on the transported joint probability density function (PDF) method for the modeling and simulation of two-phase turbulent spray flows without and with chemical reactions. In the non-reacting situation, the dependent variables of the joint PDF include the gas velocity and the mixture fraction. For the simulation of reacting spray flows, a three-variate joint PDF transport equation is derived and modeled in order to account for the pre-vaporization effects and partially premixed regime in turbulent spray flames. The dependent variables include the gas phase mixture fraction, the reaction progress variable and gas enthalpy. Detailed combustion chemistry is considered through an extended spray flamelet model by including a reaction progress variable in addition to the classical formulation. The dilute spray is simulated using a Lagrangian discrete parcel method for the description of droplet motion, heating and evaporation. The infinite conductivity model with consideration of non-equilibrium effects based on the Langmuir-Knudsen law is considered. The spray evolution and flame structures in the frame of the polydisperse reacting spray flows are investigated. Numerical results are compared with experimental data provided by Prof. Masri at the University of Sydney, Australia, and the test cases include three different turbulent non-reacting acetone spray flows in air and turbulent spray flames with the liquid fuel ethanol. For the acetone spray flows, computational results generally show good agreement with experimental data in terms of the droplet size and mean velocity distribution, as well as the liquid volume flux. The results show that the inflow liquid mass loading hardly affects the droplet diameter distribution, whereas the inlet turbulence level has a pronounced effect. The tendency of droplet accumulation near the jet centerline is found with a somewhat overprediction of liquid volume flux at downstream locations. A more sophisticated turbulence model is expected to eliminate this discrepancy. Moreover, the local joint PDF of the gas velocity and the mixture fraction is analyzed. A linear correlation of the gas velocity and the mixture fraction exists close to the nozzle exit outside the main spray jet, and no regions are found where statistical independence prevails. For the simulation of spray flames, computations with the newly developed spray flamelet/progress variable approach and with the previous standard spray flamelet formulation are carried out and compared with the experiments by Prof. Masri at the University of Sydney. A good agreement between the computations with the new formulation and the experiments for gas temperature and droplet size and velocity is achieved. The major spray and combustion properties are correctly captured using this new formulation, which is compared with an unphysically attached flame near the nozzle exit predicted by the previous model. A partially premixed combustion prevails in this piloted turbulent spray flame. Due to the prevaporization of the ethanol droplets near the nozzle exit, a lean premixed gas mixture is found at the inner side of spray jet. Moving downstream, the lean-sided diffusion flame is promoted towards the inner fuel-rich side by heating up the inner premixed core that is controlled by the droplet evaporation. Additionally, it is observed that in the far-field region, the diffusion flame becomes the dominant combustion mode. In summary, an efficient computational model based on the transported joint PDF method is developed to two-phase turbulent spray flows. The combined transported joint PDF and a newly proposed spray flamelet/progress variable approach shows an improved performance in the prediction of complex turbulent spray flames, and new insights on the local flame structure influenced by evaporating sprays are obtained.

A new framework for system reliability-based design from quantile value perspective

A subset multicanonical Monte Carlo method for simulating rare failure events

An improved high order moment-based saddlepoint approximation method for reliability analysis

System reliability sensitivity analysis becomes difficult due to involving the issues of the correlation between failure modes whether using analytic method or numerical simulation methods. A fast conditional reduction method based on conditional probability theory is proposed to solve the sensitivity analysis based on the approximate analytic method. The relevant concepts are introduced to characterize the correlation between failure modes by the reliability index and correlation coefficient, and conditional normal fractile the for the multi-dimensional conditional failure analysis is proposed based on the two-dimensional normal distribution function. Thus the calculation of system failure probability can be represented as a summation of conditional probability terms, which is convenient to be computed by iterative solving sequentially. Further the system sensitivity solution is transformed into the derivation process of the failure probability correlation coefficient of each failure mode. Numerical examples results show that it is feasible to apply the idea of failure mode relevancy to failure probability sensitivity analysis, and it can avoid multi-dimension integral calculation and reduce complexity and difficulty. Compared with the product of conditional marginal method, a wider value range of correlation coefficient for reliability analysis is confirmed and an acceptable accuracy can be obtained with less computational cost.

The reliability analysis methods based on the Kriging model have been explored to update the design of experiments (DoE), but the number of calls to the performance function is high. In this paper, a new active learning method combining the weight information entropy function (WH) and the adaptive candidate sample pool is proposed to improve the reliability analysis efficiency. Based on the information entropy function (H), the learning function WH is constructed to consider Kriging variance and the joint probability density function (PDF). In this proposed method, the sample points can be assigned different weight by the learning function WH according to the important degrees of sample points. The point which not only nears the limit state function (LSF), but also has a high probability density function value and large Kriging variance is assigned more weight than others. To select the sample points with lower confidence level, the adaptive candidate sample pool is generated by Markov Chain Monte Carlo (MCMC) simulation. The Kriging model can be updated efficiently by the proposed method. Four numerical examples and an engineering example with implicit performance function are used to verify the efficiency and accuracy of the proposed method. The results show that the proposed method can significantly improve the computational efficiency of the reliability analysis without losing accuracy.

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.

New, exact expressions for the probability density and distribution functions of a bivariate Hoyt (Nakagami-q) process with arbitrary correlation pattern in a nonstationary environment are derived, a solution to a longstanding unsolved problem. More specifically, the following are obtained: joint probability density function, joint cumulative distribution function, power and envelope correlation coefficients, and some statistics related to the signal-to-noise ratio at the output of the selection combiner, namely, outage probability and probability density function. The exact expressions are given in infinite series form, but are mathematically tractable, easy to evaluate, and flexible enough to accommodate a myriad of correlation scenarios, useful in the analysis of a more general fading environment. The power correlation coefficient is found in an exact, simple, closed-form formula, whereas the envelope one, also exact, is more mathematically elaborate. Approximate, simple closed-form expressions for the joint distribution are also obtained that yield (i) excellent results for small to medium correlation coefficients and (ii) reasonable ones for high correlation coefficient.