Search Results

A simulation-based, system reliability-based design optimization (RBDO) method is presented that can handle problems with multiple failure regions and correlated random variables. Copulas are used to represent the correlation. The method uses a Probabilistic Re-Analysis (PRRA) approach in conjunction with a trust-region optimization approach and local metamodels covering each trust region. PRRA calculates very efficiently the system reliability of a design by performing a single Monte Carlo (MC) simulation per trust region. Although PRRA is based on MC simulation, it calculates “smooth” sensitivity derivatives, allowing therefore, the use of a gradient-based optimizer. The PRRA method is based on importance sampling. It provides accurate results, if the support of the sampling PDF contains the support of the joint PDF of the input random variables. The sequential, trust-region optimization approach satisfies this requirement.

There is randomness in both the applied loads and the strength of systems. Therefore, to account for the uncertainty, the safety of the system must be quantified using its reliability. Monte Carlo Simulation (MCS) is widely used for probabilistic analysis because of its robustness. However, the high computational cost limits the accuracy of MCS. Smarslok et al. [2010] developed an improved sampling technique for reliability assessment called Separable Monte Carlo (SMC) that can significantly increase the accuracy of estimation without increasing the cost of sampling. However, this method was applied to time-invariant problems involving two random variables. This paper extends SMC to problems with multiple random variables and develops a novel method for estimation of the standard deviation of the probability of failure of a structure. The method is demonstrated and validated on reliability assessment of an offshore wind turbine under turbulent wind loads.

This article presents an approach for comparing alternative repairable systems and calculating the value of information obtained by testing a specified number of such systems. More specifically, an approach is presented to determine the value of information that comes from field testing a specified number of systems in order to appropriately estimate the reliability metric associated with each of the respective repairable systems. Here the reliability of a repairable system will be measured by its failure rate. In support of the decision-making effort, the failure rate is translated into an expected utility based on a utility curve that represents the risk tolerance of the decision-maker. The algorithm calculates the change of the expected value of the decision with the sample size. The change in the value of the decision represents the value of information obtained from testing.

It is challenging to perform probabilistic analysis and design of large-scale structures because probabilistic analysis requires repeated finite element analyses of large models and each analysis is expensive. This paper presents a methodology for probabilistic analysis and reliability based design optimization of large scale structures that consists of two re-analysis methods; one for estimating the deterministic vibratory response and another for estimating the probability of the response exceeding a certain level. The deterministic re-analysis method can analyze efficiently large-scale finite element models consisting of tens or hundreds of thousand degrees of freedom and large numbers of design variables that vary in a wide range. The probabilistic re-analysis method calculates very efficiently the system reliability for many probability distributions of the design variables by performing a single Monte Carlo simulation.

A Reliability-Based Design Optimization (RBDO) method for multiple failure regions is presented. The method uses a Probabilistic Re-Analysis (PRRA) approach in conjunction with an approximate global metamodel with local refinements. The latter serves as an indicator to determine the failure and safe regions. PRRA calculates very efficiently the system reliability of a design by performing a single Monte Carlo (MC) simulation. Although PRRA is based on MC simulation, it calculates “smooth” sensitivity derivatives, allowing therefore, the use of a gradient-based optimizer. An “accurate-on-demand” metamodel is used in the PRRA that allows us to handle problems with multiple disjoint failure regions and potentially multiple most-probable points (MPP). The multiple failure regions are identified by using a clustering technique. A maximin “space-filling” sampling technique is used to construct the metamodel. A vibration absorber example highlights the potential of the proposed method.

An approach for Probabilistic Reanalysis (PRA) of a system is presented. PRA calculates very efficiently the system reliability or the average value of an attribute of a design for many probability distributions of the input variables, by performing a single Monte Carlo simulation. In addition, PRA calculates the sensitivity derivatives of the reliability to the parameters of the probability distributions. The approach is useful for analysis problems where reliability bounds need to be calculated because the probability distribution of the input variables is uncertain or for design problems where the design variables are random. The accuracy and efficiency of PRA is demonstrated on vibration analysis of a car and on system reliability-based optimization (RBDO) of an internal combustion engine.

Reliability is an important engineering requirement for consistently delivering acceptable product performance through time. As time progresses, the product may fail due to time-dependent operating conditions and material properties, component degradation, etc. The reliability degradation with time may increase the lifecycle cost due to potential warranty costs, repairs and loss of market share. Reliability is the probability that the system will perform its intended function successfully for a specified time interval. In this work, we consider the first-passage reliability which accounts for the first time failure of non-repairable systems. Methods are available in the literature, which provide an upper bound to the true reliability which may overestimate the true value considerably. Monte-Carlo simulations are accurate but computationally expensive.

In design of real-life systems, such as the suspension of a car, an offshore platform or a wind turbine, there are significant uncertainties in the model of the inputs. For example, scarcity of data leads to inaccuracies in the power spectral density function of the waves and the probability distribution of the wind speed. Therefore, it is necessary to evaluate the performance and safety of a system for different probability distributions. This is computationally expensive or even impractical. This paper presents a methodology to assess efficiently the fatigue life of structures for different power spectra of the applied loads. We accomplish that by reweighting the incremental damage calculated in one simulation. We demonstrate the accuracy and efficiency of the proposed method on an example which involves a nonlinear quarter car under a random dynamic load. The fatigue life of the suspension spring under loads generated by a sampling spectrum is calculated.

Reliability assessment of dynamic systems with low failure probability can be very expensive. This paper presents and demonstrates a method that uses the Metropolis-Hastings algorithm to sample from an optimal probability density function (PDF) of the random variables. This function is the true PDF truncated over the failure region. For a system subjected to time varying excitation, Shinozuka's method is employed to generate time histories of the excitation. Random values of the frequencies and the phase angles of the excitation are drawn from the optimal PDF. It is shown that running the subset simulation by the proposed approach, which uses Shinozuka's method, is more efficient than the original subset simulation. The main reasons are that the approach involves only 10 to 20 random variables, and it takes advantage of the symmetry of the expression of the displacement as a function of the inputs. The paper demonstrates the method on two examples.

Estimation of the probability of failure of mechanical systems under random loads is computationally expensive, especially for very reliable systems with low probabilities of failure. Importance Sampling can be an efficient tool for static problems if a proper sampling distribution is selected. This paper presents a methodology to apply Importance Sampling to dynamic systems in which both the load and response are stochastic processes. The method is applicable to problems for which the input loads are stationary and Gaussian and are represented by power spectral density functions. Shinozuka's method is used to generate random time histories of excitation. The method is demonstrated on a linear quarter car model. This approach is more efficient than standard Monte Carlo simulation by several orders of magnitude.

Multi-attribute decision making and multi-objective optimization complement each other. Often, while making design decisions involving multiple attributes, a Pareto front is generated using a multi-objective optimizer. The end user then chooses the optimal design from the Pareto front based on his/her preferences. This seemingly simple methodology requires sufficient modification if uncertainty is present. We explore two kinds of uncertainties in this paper: uncertainty in the decision variables which we call inherent design problem (IDP) uncertainty and that in knowledge of the preferences of the decision maker which we refer to as preference assessment (PA) uncertainty. From a purely utility theory perspective a rational decision maker maximizes his or her expected multi attribute utility.

Importance Sampling is a popular method for reliability assessment. Although it is significantly more efficient than standard Monte Carlo simulation if a suitable sampling distribution is used, in many design problems it is too expensive. The authors have previously proposed a method to manage the computational cost in standard Monte Carlo simulation that views design as a choice among alternatives with uncertain reliabilities. Information from simulation has value only if it helps the designer make a better choice among the alternatives. This paper extends their method to Importance Sampling. First, the designer estimates the prior probability density functions of the reliabilities of the alternative designs and calculates the expected utility of the choice of the best design. Subsequently, the designer estimates the likelihood function of the probability of failure by performing an initial simulation with Importance Sampling.

This paper presents a methodology to evaluate and optimize discrete event systems, such as an assembly line or a call center. First, the methodology estimates the performance of a system for a single probability distribution of the inputs. Probabilistic Reanalysis (PRRA) uses this information to evaluate the effect of changes in the system configuration on its performance. PRRA is integrated with a program to optimize the system. The proposed methodology is dramatically more efficient than one requiring a new Monte Carlo simulation each time we change the system. We demonstrate the approach on a drilling center and an electronic parts factory.

Finite element analysis is a well-established methodology in structural dynamics. However, optimization and/or probabilistic studies can be prohibitively expensive because they require repeated FE analyses of large models. Various reanalysis methods have been proposed in order to calculate efficiently the dynamic response of a structure after a baseline design has been modified, without recalculating the new response. The parametric reduced-order modeling (PROM) and the combined approximation (CA) methods are two re-analysis methods, which can handle large model parameter changes in a relatively efficient manner. Although both methods are promising by themselves, they can not handle large FE models with large numbers of DOF (e.g. 100,000) with a large number of design parameters (e.g. 50), which are common in practice. In this paper, the advantages and disadvantages of the PROM and CA methods are first discussed in detail.

An efficient approach for series system reliability-based design optimization (RBDO) is presented. The key idea is to apportion optimally the system reliability among the failure modes by considering the target values of the failure probabilities of the modes as design variables. Critical failure modes that contribute the most to the overall system reliability are identified. This paper proposes a computationally efficient, system RBDO approach using a single-loop method where the searches for the optimum design and for the most probable failure points proceed simultaneously. Specifically, at each iteration the optimizer uses approximated most probable failure points from the previous iteration to search for the optimum. A second-order Ditlevsen upper bound is used for the joint failure probability of failure modes. Also, an easy to implement active strategy set is employed to improve algorithmic stability.

In reliability design, often, there is scarce data for constructing probabilistic models. Probabilistic models whose parameters vary in known intervals could be more suitable than Bayesian models because the former models do not require making assumptions that are not supported by the available evidence. If we use models whose parameters vary in intervals we need to calculate upper and lower bounds of the failure probability (or reliability) of a system in order to make design decisions. Monte Carlo simulation can be used for this purpose, but it is too expensive for all but very simple systems. This paper proposes an efficient Monte-Carlo simulation approach for estimation of upper and lower probabilities. This approach is based on two ideas: a) use an efficient approach for reliability reanalysis of a system, which is introduced in this paper, and b) approximate the probability distribution of the minimum and maximum failure probabilities using extreme value statistics.

Probabilistic studies can be prohibitively expensive because they require repeated finite element analyses of large models. Re-analysis methods have been proposed with the premise to estimate accurately the dynamic response of a structure after a baseline design has been modified, without recalculating the new response. Although these methods increase computational efficiency, they are still not efficient enough for probabilistic analysis of large-scale dynamic systems with low failure probabilities (less or equal to 10-3). This paper presents a methodology that uses deterministic and probabilistic re-analysis methods to generate sample points of the response. Subsequently, tail modeling is used to estimate the right tail of the response PDF and the probability of failure a highly reliable system. The methodology is demonstrated on probabilistic vibration analysis of a realistic vehicle FE model.

In reliability design, often, there is scarce data for constructing probabilistic models. It is particularly challenging to model uncertainty in variables when the type of their probability distribution is unknown. Moreover, it is expensive to estimate the upper and lower bounds of the reliability of a system involving such variables. A method for modeling uncertainty by using Polynomial Chaos Expansion is presented. The method requires specifying bounds for statistical summaries such as the first four moments and credible intervals. A constrained optimization problem, in which decision variables are the coefficients of the Polynomial Chaos Expansion approximation, is formulated and solved in order to estimate the minimum and maximum values of a system’s reliability. This problem is solved efficiently by employing a probabilistic re-analysis approach to approximate the system reliability as a function of the moments of the random variables.

A complete probabilistic model of uncertainty in probabilistic analysis and design problems is the joint probability distribution of the random variables. Often, it is impractical to estimate this joint probability distribution because the mechanism of the dependence of the variables is not completely understood. This paper proposes modeling dependence by using copulas and demonstrates their representational power. It also compares this representation with a Monte-Carlo simulation using dispersive sampling.

Methods for designing targeted tests for reliability validation of structures obtained from reliability-based design are presented. These methods optimize the test parameters to minimize the variance in the estimated reliability (or equivalently the failure probability) estimated from the tests. The tests are designed using information from analytical models used to design the structure. Both analytical tests, in which very detailed models are used as reference, or physical tests can be designed using the methods presented. The methods are demonstrated on examples and their robustness to errors in the analytical models used to design the tests is assessed.