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Technical Paper

A 3-D Joint Model for Automotive Structures

A simple, design-oriented model of joints in vehicles structures is developed. This model accounts for the flexibility, the offsets of rotation centers of joint branches from geometric center, and the coupling between rotations of a joint branch in different planes. A family of joint models with different levels of complexity is also defined. A probabilistic system identification is used to estimate the joint model parameters by using the measured displacements. Statistical tools which identify important parameters are also presented. The identification methodology is applied to the estimation of parameters of a B-pillar to rocker joint.
Technical Paper

A New Approach for System Reliability-Based Design Optimization

An efficient approach for Reliability-Based Design Optimization (RBDO) of series systems is presented. A modified formulation of the RBDO problem is employed in which the required reliabilities of the failure modes of a system are design variables. This allows for an optimal apportionment of the reliability of a system among its failure modes. A sequential optimization and reliability assessment method is used to efficiently determine the optimum design. Here, the constraints on the reliabilities of the failure modes of the RBDO problem are replaced with deterministic constraints. The method is demonstrated on an example problem that has been solved in a previous study that did not treat the required reliability levels of the failure modes as design variables. The new approach finds designs with lower mass than designs found in the previous study without reducing their system reliability.
Journal Article

A Re-Analysis Methodology for System RBDO Using a Trust Region Approach with Local Metamodels

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.
Technical Paper

An Efficient Re-Analysis Methodology for Vibration of Large-Scale Structures

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.
Journal Article

An RBDO Method for Multiple Failure Region Problems using Probabilistic Reanalysis and Approximate Metamodels

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.
Journal Article

Assessing the Value of Information for Multiple, Correlated Design Alternatives

Design optimization occurs through a series of decisions that are a standard part of the product development process. Decisions are made anywhere from concept selection to the design of the assembly and manufacturing processes. The effectiveness of these decisions is based on the information available to the decision maker. Decision analysis provides a structured approach for quantifying the value of information that may be provided to the decision maker. This paper presents a process for determining the value of information that can be gained by evaluating linearly correlated design alternatives. A unique approach to the application of Bayesian Inference is used to provide simulated estimates in the expected utility with increasing observations sizes. The results provide insight into the optimum observation size that maximizes the expected utility when assessing correlated decision alternatives.
Technical Paper

Assessment of Imprecise Reliability Using Efficient Probabilistic Reanalysis

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.
Journal Article

Bootstrapping and Separable Monte Carlo Simulation Methods Tailored for Efficient Assessment of Probability of Failure of Structural Systems

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.
Technical Paper

Combined Approximation for Efficient Reliability Analysis of Linear Dynamic Systems

The Combined Approximation (CA) method is an efficient reanalysis method that aims at reducing the cost of optimization problems. The CA uses results of a single exact analysis, and it is suitable for different types of structures and design variables. The second author utilized CA to calculate the frequency response function of a system at a frequency of interest by using the results at a frequency in the vicinity of that frequency. He showed that the CA yields accurate results for small frequency perturbations. This work demonstrates a methodology that utilizes CA to reduce the cost of Monte Carlo simulation (MCs) of linear systems under random dynamic loads. The main idea is to divide the power spectral density function (PSD) of the input load into several frequency bins before calculating the load realizations.
Journal Article

Efficient Probabilistic Reanalysis and Optimization of a Discrete Event System

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.
Journal Article

Efficient Random Vibration Analysis Using Markov Chain Monte Carlo Simulation

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.
Journal Article

Efficient Re-Analysis Methodology for Probabilistic Vibration of Large-Scale Structures

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.
Journal Article

Estimation of High-Cycle Fatigue Life by using Re-analysis

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.
Technical Paper

Evidence Theory Approach and Bayesian Approach for Modeling Uncertainty when Information is Imprecise

This paper investigates the potential of Evidence Theory (ET) and Bayesian Theory (BT) for decision under uncertainty, when the evidence about uncertainty is imprecise. The basic concepts of ET and BT are introduced and the ways these theories model uncertainties, propagate them through systems and assess the safety of these systems are presented. ET and BT approaches are demonstrated and compared on examples involving an algebraic function when the evidence about the input variables consists of intervals provided by experts. It is recommended that a decision maker compute both the Bayesian probability of events and their lower and upper probabilities using ET when evidence from experts is imprecise. A large gap between the lower and upper probability suggests that more information should be collected before making a decision. If this is not feasible, then Bayesian probabilities can help make a decision.
Technical Paper

Imprecise Reliability Assessment When the Type of the Probability Distribution of the Random Variables is Unknown

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.
Technical Paper

Inverse Modeling: Theory and Engineering Examples

Over the last two decades inverse problems have become increasingly popular due to their widespread applications. This popularity continuously demands designers to find alternative methods, to solve the inverse problems, which are efficient and accurate. It is important to use effective techniques that are both accurate and computationally efficient. This paper presents a method for solving inverse problems through Artificial Neural Network (ANN) theory. The paper also presents a method to apply Grey Wolf optimizer (GWO) algorithm to inverse problems. GWO is a recent optimization method producing superior results. Both methods are then compared to traditional methods such as Particle Swarm Optimization (PSO) and Markov Chain Monte Carlo (MCMC). Four typical engineering design problems are used to compare the four methods. The results show that the GWO outperforms other methods both in terms of efficiency and accuracy.
Technical Paper

Managing the Computational Cost in a Monte Carlo Simulation by Considering the Value of Information

Monte Carlo simulation is a popular tool for reliability assessment because of its robustness and ease of implementation. A major concern with this method is its computational cost; standard Monte Carlo simulation requires quadrupling the number of replications for halving the standard deviation of the estimated failure probability. Efforts to increase efficiency focus on intelligent sampling procedures and methods for efficient calculation of the performance function of a system. This paper proposes a new method to manage cost that views design as a decision among alternatives with uncertain reliabilities. Information from a simulation has value only if it enables the designer to make a better choice among the alternative options. Consequently, the value of information from the simulation is equal to the gain from using this information to improve the decision. A designer can determine the number of replications that are worth performing by using the method.
Journal Article

Managing the Computational Cost of Monte Carlo Simulation with Importance Sampling by Considering the Value of Information

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.
Technical Paper

Modeling Dependence and Assessing the Effect of Uncertainty in Dependence in Probabilistic Analysis and Decision Under Uncertainty

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.
Technical Paper

Multi-Level Decoupled Optimization of Wind Turbine Structures

This paper proposes a multi-level decoupled method for optimizing the structural design of a wind turbine blade. The proposed method reduces the design space by employing a two-level optimization process. At the high-level, the structural properties of each section are approximated by an exponential function of the distance of that section from the blade root. High-level design variables are the coefficients of this approximating function. Target values for the structural properties of the blade are determined at that level. At the low-level, sections are divided into small decoupled groups. For each section, the low-level optimizer finds the thickness of laminate layers with a minimum mass, whose structural properties meet the targets determined by the high-level optimizer. In the proposed method, each low-level optimizer only considers a small number of design variables for a particular section, while traditional, single-level methods consider all design variables simultaneously.