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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.

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.

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.

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. 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.

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.

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.

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.

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.