Reliability-based Design Optimization Model Research Articles

Single-Loop Sampling for Estimating Failure-Probability Upper-Bound Function

Under random-interval mixed uncertainties of structures, failure-probability upper-bound function (FPUBF), which varies with the distribution parameters of random inputs, can not only provide the influence of distribution parameters on the failure-probability upper bound (FPUB), but also contribute to decoupling a reliability-based design optimization model. Although FPUBF can be estimated by repeatedly evaluating FPUBs at different distribution parameter realizations, it suffers from unaffordable computational cost resulting from this double-loop framework. To address this issue, this paper proposes a single-loop sampling strategy (SL) to estimate FPUBF at arbitrary realizations in the interested distribution parameter region. Instead of the huge computational cost of a double-loop framework, the SL estimates the entire FPUBF only by one simulation analysis. Moreover, importance sampling (IS) variance reduction technique is introduced, and a single-loop IS probability density function (PDF), or SL-IS-PDF, is constructed to more efficiently estimate FPUBF by reducing the required size of the candidate sample pool. For approximating the optimal SL-IS-PDF and identifying the states of candidate samples efficiently, the double-loop adaptive Kriging model of performance function is introduced to further reduce the number of performance function evaluations. A numerical example and two composite structure examples are employed to verify the accuracy, efficiency, and feasibility of the proposed methods.

Meta model-based and cross entropy-based importance sampling algorithms for efficiently solving system failure probability function

The multi-mode system failure probability function (SFPF) can quantify how the distribution parameters of the random input vector affect the system safety and decouple the system reliability-based design optimization model. However, for a problem with a time-consuming implicit performance function and rare failure domain, efficiently solving the SFPF remains significantly challenging. Therefore, in this study, two efficient algorithms are proposed, namely, the meta model-based importance sampling and cross entropy-based importance sampling. The contributions of this study are twofold. The first is constructing a single-loop optimal importance sampling density (SL-OISD) method to decouple the double-loop framework for analyzing the SFPF. The second is establishing two methods to efficiently approximate the SL-OISD and complete the SFPF estimation. The first method is based on the meta model of the system performance function, which is abbreviated as SL-Meta-IS. The second method is based on minimizing the cross entropy between the Gaussian mixture density model and SL-OISD, which is abbreviated as SL-CE-IS. To reduce the number of evaluating the system performance function when approximating the SL-OISD, sampling the SL-OISD, and identifying the state of the samples for completing the SFPF estimation, an adaptive Kriging model of the system performance function is introduced into SL-Meta-IS and SL-CE-IS. Owing to decoupling the double-loop framework into a single-loop framework, replacing the time-consuming system performance function with the economic Kriging model, and employing importance sampling variance reduction techniques to address issues related to the rare failure domain, the proposed SL-Meta-IS and SL-CE-IS methods greatly enhance the efficiency of SFPF estimations. The numerical and practical examples demonstrate that the two proposed methods are superior to the existing algorithms; moreover, the efficiency of SL-CE-IS is higher than that of SL-Meta-IS.

An equivalent RBDO method under the hybrid of fuzzy and interval variables in the worst case

AbstractIn the reliability‐based design optimization (RBDO) problems of structures or systems, only a single type of variable involving is rare, and most of time are hybrid variables. Random variables are often used to describe uncertainty, and these random variables can usually be described by certain probability distributions. However, in practice, the probability distributions of some random variables are difficult or even impossible to be obtained, or the obtained probability distributions may be inaccurate, and can only be roughly described by fuzzy values or intervals. Therefore, this paper is committed to solving RBDO problems under the hybrid of fuzzy variables and interval variables. Firstly, the RBDO model of the structure under the hybrid variables is established. Secondly, the fuzzy variables are transformed to random variables according to entropy invariance. Finally, RBDO is carried out based on sequential single‐loop method in the worst case. A numerical example is presented to illustrate the feasibility of the proposed method.

An Extended SORA Method for Hybrid Reliability-Based Design Optimization

In many practical engineering problems, distributions of some random variables may not be precisely known or even only the variation ranges can be given, due to which the Reliability-Based Design Optimization (RBDO) methods cannot be applied directly. In this paper, a hybrid RBDO model is established to deal with hybrid uncertainties including random variables with interval distribution parameters and coexistence of random and interval variables. The hybrid uncertainties lead to an interval of reliability, thus giving rise to a triple-loop optimization problem, which burdens the computational effort significantly. Therefore, an extended Sequential Optimization and Reliability Assessment (SORA) method is proposed, where the worst-case point over the bounded domain of interval uncertainty is employed to replace reliability constraint with deterministic optimization. Therefore, the hybrid RBDO problem can be efficiently solved by a series of deterministic optimization procedures. The efficiency and accuracy of the proposed method are illustrated through several numerical examples.

A single-loop method for reliability-based design optimization with interval distribution parameters

The Reliability-Based Design Optimization (RBDO) provides an effective way to obtain the optimum design in the presence of random uncertainties which follow the precise probability distribution function in the structural optimization design. However, in many practical engineering problems, the probability distribution which describes the stochastic nature of the uncertainties cannot be precisely obtained due to limited information. To quantify this kind of imprecise uncertainties, a probability-interval hybrid model emerged, where all uncertain variables are treated as random variables while some distribution parameters can only be given variation intervals. For such kind of uncertainties, this paper establishes a hybrid reliability-based design optimization model and proposes a single-loop solution algorithm. The interval parameters lead to an interval of reliability for each constraint function, thus giving rise to a triple-loop optimization problem for the hybrid RBDO, which is difficult to solve due to the unaffordable computational effort and the hinder of convergency. In this paper, the Karush–Kuhn–Tucker (KKT) optimality conditions of the inner loops are imposed as equivalent deterministic equality constraints. The original triple-loop optimization is thus converted into an equivalent single-loop problem, which alleviates the computational demand significantly. The efficiency and accuracy of the proposed Single-Loop Method (SLM) is verified through several practical engineering problems.

Reliability based design optimization applied to the high electron mobility transistor (HEMT)

Among the most important components in complex and high-power mechatronic systems is the transistor. The High Electron Mobility Transistor (HEMT) is a technology under development. This paper presents the hybrid Reliability Based Design Optimization (RBDO) method applied to the HEMT technology in order to improve its performance and reliability. The execution of RBDO processes requires the development and coupling of two models: the finite element model using Comsol multiphysics® software and the RBDO model by Matlab® software. The first model is used to simulate the electro-thermomechanical behavior of components. The second model is used to solve the optimization problem by coupling with the first model. After the application of this process, it was possible to determine the optimal values of the design variables which allows optimizing the multiphysics behavior of the structure and the reliability level of the HEMT.

Efficient method using Whale Optimization Algorithm for reliability-based design optimization of labyrinth spillway

Spillways are essential parts of dams, in which the main task of these structures is to allow the passing of excess water and floods from the upstream to the downstream. In this regards, the main goal of this paper is proposing a novel framework for the probabilistic design of labyrinth spillway structures using a new developed reliability-based design optimization (RBDO) approach. In this RBDO approach, the total volume of the spillway is considered as the objective function of the optimization problem under uncertainties, while the labyrinth spillway parameters are considered as the design variables. Hereafter, to solve the formulated RBDO problem of the labyrinth spillway design, a new proposed model that consist of coupling the Monte Carlo Simulation (MCS) with a hybrid Artificial Neural Network (ANN) based Whale Optimization Algorithm (WOA) model is developed. The hybrid ANN-WOA is utilized to approximate the labyrinth spillway response in order to reduce the computational cost during the RBDO analysis. The proposed MCS-ANN-WOA model was implemented on the Ute dam labyrinth spillway at Logan, New Mexico (USA). The obtained results showed that the proposed RBDO model performance is more accurate and robust compared to the deterministic optimization (DO) approaches for an optimal design of the labyrinth spillway shape with the consideration of the safety levels.

Reliability-based design optimization for vehicle body crashworthiness based on copula functions

ABSTRACT Vehicle body crashworthiness is crucial in a traffic accident. It can protect passengers from serious injury or death and reduce vehicle damage. To provide a more effective design framework for ensuring crash safety, a reliability-based design optimization (RBDO) method for vehicle body crashworthiness considering complex parametric correlation is developed by integrating copula functions. An RBDO model is formulated for the vehicle body crashworthiness problem based on a finite element model of a vehicle crash. Taking into account the correlation of random parameters, the optimal copula function is selected by the Bayesian method to model parametric correlation and build the joint probability distribution function. An efficient solution strategy is provided to perform the reliability analysis and optimization design of crashworthiness. The results demonstrate that the proposed method is reliable and has a high accuracy for crashworthiness design.

Fuzzy Multi-SVR Learning Model for Reliability-Based Design Optimization of Turbine Blades.

The effectiveness of a model is the key factor of influencing the reliability-based design optimization (RBDO) of multi-failure turbine blades in the power system. A machine learning-based RBDO approach, called fuzzy multi-SVR learning method, was proposed by absorbing the strengths of fuzzy theory, support vector machine of regression (SVR), and multi-response surface method. The model of fuzzy multi-SVR learning method was established by adopting artificial bee colony algorithm to optimize the parameters of SVR models and considering the fuzziness of constraints based on fuzzy theory, in respect of the basic thought of multi-response surface method. The RBDO model and procedure with fuzzy multi-SVR learning method were then resolved and designed by multi-objective genetic algorithm. Lastly, the fuzzy RBDO of a turbine blade with multi-failure modes was performed regarding the design parameters of rotor speed, temperature, and aerodynamic pressure, and the design objectives of blade stress, strain, and deformation, and the fuzzy constraints of reliability degree and boundary conditions, as well. It is revealed (1) the stress and deformation of turbine blade are reduced by 92.38 MPa and 0.09838 mm, respectively. (2) The comprehensive reliability degree of the blade was improved by 3.45% from 95.4% to 98.85%. (3) It is verified that the fuzzy multi-SVR learning method is workable for the fuzzy RBDO of complex structures just like a multi-failure blade with high modeling precision, as well as high optimization, efficiency, and accuracy. The efforts of this study open a new research way, i.e., machine learning-based RBDO, for the RBDO of multi-failure structures, which expands the application of machine learning methods, and enriches the mechanical reliability design method and theory as well.

Direct reliability-based design optimization of uncertain structures with interval parameters

In order to enhance the reliability of an uncertain structure with interval parameters and reduce its chance of function failure under potentially critical conditions, an interval reliability-based design optimization model is constructed. With the introduction of a unified formula for efficiently computing interval reliability, a new concept of the degree of interval reliability violation (DIRV) and the DIRV-based preferential guidelines are put forward for the direct ranking of various design vectors. A direct interval optimization algorithm integrating a nested genetic algorithm (GA) and the Kriging technique is proposed for solving the interval reliability-based design model, which avoids the complicated model transformation process in indirect ones and yields an interval solution that provides more insights into the optimization problem. The effectiveness of the proposed algorithm is demonstrated by a numeric example. Finally, the proposed direct reliability-based design optimization method is applied to the optimization of a press upper beam with interval uncertain parameters, the results of which demonstrate its feasibility and effectiveness in engineering.

Reliability-based design optimization for flexible mechanism with particle swarm optimization and advanced extremum response surface method

To improve the computational efficiency of the reliability-based design optimization (RBDO) of flexible mechanism, particle swarm optimization-advanced extremum response surface method (PSO-AERSM) was proposed by integrating particle swarm optimization (PSO) algorithm and advanced extremum response surface method (AERSM). Firstly, the AERSM was developed and its mathematical model was established based on artificial neural network, and the PSO algorithm was investigated. And then the RBDO model of flexible mechanism was presented based on AERSM and PSO. Finally, regarding cross-sectional area as design variable, the reliability optimization of flexible mechanism was implemented subject to reliability degree and uncertainties based on the proposed approach. The optimization results show that the cross-section sizes obviously reduce by 22.96 mm2 while keeping reliability degree. Through the comparison of methods, it is demonstrated that the AERSM holds high computational efficiency while keeping computational precision for the RBDO of flexible mechanism, and PSO algorithm minimizes the response of the objective function. The efforts of this work provide a useful sight for the reliability optimization of flexible mechanism, and enrich and develop the reliability theory as well.

Reliability based design optimization of concrete mix proportions using generalized ridge regression model

This paper presents Reliability Based Design Optimization (RBDO) model to deal with uncertainties involved in concrete mix design process. The optimization problem is formulated in such a way that probabilistic concrete mix input parameters showing random characteristics are determined by minimizing the cost of concrete subjected to concrete compressive strength constraint for a given target reliability. Linear and quadratic models based on Ordinary Least Square Regression (OLSR), Traditional Ridge Regression (TRR) and Generalized Ridge Regression (GRR) techniques have been explored to select the best model to explicitly represent compressive strength of concrete. The RBDO model is solved by Sequential Optimization and Reliability Assessment (SORA) method using fully quadratic GRR model. Optimization results for a wide range of target compressive strength and reliability levels of 0.90, 0.95 and 0.99 have been reported. Also, safety factor based Deterministic Design Optimization (DDO) designs for each case are obtained. It has been observed that deterministic optimal designs are cost effective but proposed RBDO model gives improved design performance.

Reliability-Based Design Optimization for Crane Metallic Structure Using ACO and AFOSM Based on China Standards

The design optimization of crane metallic structures is of great significance in reducing their weight and cost. Although it is known that uncertainties in the loads, geometry, dimensions, and materials of crane metallic structures are inherent and inevitable and that deterministic structural optimization can lead to an unreliable structure in practical applications, little amount of research on these factors has been reported. This paper considers a sensitivity analysis of uncertain variables and constructs a reliability-based design optimization model of an overhead traveling crane metallic structure. An advanced first-order second-moment method is used to calculate the reliability indices of probabilistic constraints at each design point. An effective ant colony optimization with a mutation local search is developed to achieve the global optimal solution. By applying our reliability-based design optimization to a realistic crane structure, we demonstrate that, compared with the practical design and the deterministic design optimization, the proposed method could find the lighter structure weight while satisfying the deterministic and probabilistic stress, deflection, and stiffness constraints and is therefore both feasible and effective.

Discussion of “Reliability‐based design optimization with dependent interval variables” by Xiaoping Du, International Journal for Numerical Methods in Engineering 2012; 91:218–228

SUMMARYIn the subject paper, a reliability‐based design optimization (RBDO) model with both random and dependent interval uncertainties was proposed based on the First Order Reliability Method. The lower bound of reliability defined in Equation (9) of the subject paper was utilized as the constraint in this RBDO model. The author claimed that it is the minimum reliability with both random and interval variables. However, we prove that it is not the minimum value. It is therefore suggested that the minimum reliability should be used in the RBDO model. Copyright © 2014 John Wiley & Sons, Ltd.

Reliability Simulation and Design Optimization for Mechanical Maintenance

Reliability model of a mechanical product system will be newly reconstructed and maintenance cost will increase because failed parts can be replaced with new components during service,which should be accounted for in system design. In this paper,a reliability model and reliability-based design optimization methodology for maintenance are presented. First,based on the time-to-failure density function of the part of the system,the age distributions of all parts of the system during service are investigated,a reliability model of the mechanical system for maintenance is developed. Then,reliability simulations of the systems with Weibull probability density functions are performed,the system minimum reliability and steady reliability for maintenance are defined based on reliability simulation during the life cycle of the system. Thirdly,a maintenance cost model is developed based on replacement rates of the parts,a reliability-based design optimization model for maintenance is presented,in which total life cycle cost is considered as design objective and system reliability as design constrain. Finally,the reliability-based design optimization methodology for maintenance is used to design of a link ring for the chain conveyor,which shows that optimal design with the lowest maintenance cost can be obtained,and minimum reliability and steady reliability of the system can satisfy requirement of system reliability during service of the chain conveyor.

Recent methodologies for reliability-based design optimization

In the field of Deterministic Design Optimization (DDO), the designer reduces the structural cost without taking into account uncertainties concerning materials, geometry and loading. This way the resulting optimum solution may represent a lower level of reliability and thus a higher risk of failure. But in the Reliability-Based Design Optimization (RBDO) model, the mean values of uncertain system variables are usually used as design variables, and the cost is optimized subject to prescribed probabilistic constraints as defined by a nonlinear mathematical programming problem. Therefore, a RBDO solution that reduces the structural weight in uncritical regions does not only provide an improved design but also a higher level of confidence in the design. In this paper, we present the advantage of the DDO and RBDO models and next some recent methodologies in order to show that the RBDO model is a practical tool for structural engineers.

Extension of optimum safety factor method to nonlinear reliability-based design optimization

In the reliability-based design optimization (RBDO) model, the mean values of uncertain system variables are usually applied as design variables, and the cost is optimized subject to prescribed probabilistic constraints as defined by a nonlinear mathematical programming problem. Therefore, a RBDO solution that reduces the structural weight in uncritical regions does not only provide an improved design but also a higher level of confidence in the design. In this paper, we present recent developments for the RBDO model relative to two points of view: reliability and optimization. Next, we develop several distributions for the hybrid method and the optimum safety factor methods (linear and nonlinear RBDO). Finally, we demonstrate the efficiency of our safety factor approach extended to nonlinear RBDO with application to a tri-material structure.

Optimum values of structural safety factors for a predefined reliability level with extension to multiple limit states

In the field of deterministic structural optimization, the designer reduces the structural cost without taking into account uncertainties concerning materials, geometry and loading. This way, the resulting optimum solution may represent a lower level of reliability and thus a higher risk of failure. It is the objective of reliability-based design optimization (RBDO) to design structures that should be both economic and reliable. The coupling between mechanical modeling, reliability analyses and optimization methods leads to very high computational costs and weak convergence stability. Since the traditional RBDO solution is achieved by alternating between reliability and optimization iterations, the structural designers performing deterministic optimization do not consider the RBDO model as a practical tool for the design of real structures. Fortunately, a hybrid method based on simultaneous solution of the reliability and the optimization problem, has successfully reduced the computational time problem. The hybrid method allows us to satisfy a required reliability level, but the vector of variables here contains both deterministic and random variables. The hybrid RBDO problem is thus more complex than that of deterministic design. The major difficulty lies in the evaluation of the structural reliability, which is carried out by a special optimization procedure. In this paper a new methodology is presented with the aim of finding a global solution to RBDO problems without additional computing cost for the reliability evaluation. The safety factor formulation for a single limit state case has been used to efficiently reduce the computational time . This technique is fundamentally based on a study of the sensitivity of the limit state function with respect to the design variables. In order to demonstrate analytically the efficiency of this methodology, the optimality condition is then used. The efficiency of this technique is also extended to multiple limit state cases. Two numerical examples are presented at the end of the paper to demonstrate the applicability of the new methodology.

Efficient reliability-based design optimization using a hybrid space with application to finite element analysis

The design of high technology structures aims to define the best compromise between cost and safety. The Reliability-Based Design Optimization (RBDO) allows us to design structures which satisfy economical and safety requirements. However, in practical applications, the coupling between the mechanical modelling, the reliability analyses and the optimization methods leads to very high computational time and weak convergence stability. Traditionally, the solution of the RBDO model is achieved by alternating reliability and optimization iterations. This approach leads to low numerical efficiency, which is disadvantageous for engineering applications on real structures. In order to avoid this difficulty, we propose herein a very efficient method based on the simultaneous solution of the reliability and optimization problems. The procedure leads to parallel convergence for both problems in a Hybrid Design Space (HDS). The efficiency of the proposed methodology is demonstrated on the design of a steel hook, where the RBDO is combined with Finite Element Analysis (FEA).