Reliability-based Design Optimization Method Research Articles

An adaptive local range sampling method for reliability-based design optimization using support vector machine and Kriging model

Kriging model is an effective method to overcome huge computational cost for reliability-based design optimization (RBDO) problems. However, the results of RBDO usually depend on constraint boundaries within the local range that contains the RBDO optimum. Determining this local range and building adaptive response surfaces within it can avoid selecting samples in unrelated areas. In this research, a new RBDO process is proposed. In the first phase, Kriging models of constraints are built based on Latin Hypercube sampling method, and updated by two new samples in each iteration. One of these two samples is selected based on SVM and mean squared error to make sure it is located near constraint boundaries. Another one is the deterministic optimum point (DOP) of current Kriging models, which is obtained based on the deterministic optimization and specifies the direction to the RBDO optimum. And the RBDO design point is obtained by SORA. When consecutive RBDO design points are close enough to each other, the local range is determined based on the current RBDO design point and the current DOP. In the second phase, new samples are located on constraint boundaries within the local range to refine Kriging models. The location and the size of the local range is adaptively defined by the RBDO design point and the DOP during each iteration. Several optimization examples are selected to test the computation capability of the proposed method. The results indicate that the new method is more efficient and more accurate.

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.

Multi-objective system reliability-based optimization method for design of a fully parametric concept car body

ABSTRACTThis article investigates multi-objective optimization under reliability constraints with applications in vehicle structural design. To improve computational efficiency, an improved multi-objective system reliability-based design optimization (MOSRBDO) method is developed, and used to explore the lightweight and high-performance design of a concept car body under uncertainty. A parametric model knowledge base is established, followed by the construction of a fully parametric concept car body of a multi-purpose vehicle (FPCCB-MPV) based on the knowledge base. The structural shape, gauge and topology optimization are then designed on the basis of FPCCB-MPV. The numerical implementation of MOSRBDO employs the double-loop method with design optimization in the outer loop and system reliability analysis in the inner loop. Multi-objective particle swarm optimization is used as the outer loop optimization solver. An improved multi-modal radial-based importance sampling (MRBIS) method is utilized as the system reliability solver for multi-constraint analysis in the inner loop. The accuracy and efficiency of the MRBIS method are demonstrated on three widely used test problems. In conclusion, MOSRBDO has been successfully applied for the design of a full parametric concept car body. The results show that the improved MOSRBDO method is more effective and efficient than the traditional MOSRBDO while achieving the same accuracy, and that the optimized body-in-white structure signifies a noticeable improvement from the baseline model.

Reliability-based design optimization applied to structures submitted to random fatigue loads

The reliability-based design optimization (RBDO) aims to find the most balanced design through a compromise between cost and safety when uncertainties affecting the system are considered. This strategy involves the evaluation of probabilistic constraints performed by the reliability analysis, leading to an expensive computational effort in order to solve the problem. For this reason, it is not trivial to take into account the high cycle fatigue in a RBDO method. This work deals with this problem, considering stationary Gaussian random load processes and the Sines multiaxial fatigue criterion. The efficiency is achieved through spectral tools related to the fatigue analysis. Therefore, the probabilistic constraints of the fatigue life in the RBDO problem are transformed in the frequency domain. The limit state function of the reliability analysis is defined by the mechanical component endurance submitted to random loads. The numerical application demonstrates the interest and the effectiveness of our proposal.

A Hybrid Loop Approach Using the Sufficient Descent Condition for Accurate, Robust, and Efficient Reliability-Based Design Optimization

For reliability-based design optimization (RBDO) problems, single loop approaches (SLA) are very efficient but prone to converge to inappropriate point for highly nonlinear constraint functions, and double loop approaches (DLA) are proven to be accurate but require more iterations to achieve stable results. In this paper, an adjusted advanced mean value (AAMV) method is firstly proposed to improve the robustness and efficiency of performance measure approach. The global convergence of the AAMV is guaranteed using sufficient descent condition for the reliability loop in RBDO. Then, a hybrid RBDO method is developed to improve the efficiency of DLA and accuracy of SLA, on the basis of sufficient descent condition and AAMV method, named as hybrid single and double loops (HSD) method. Three nonlinear concave and convex performance functions are used to illustrate the efficiency and robustness of the AAMV method; then the accuracy, robustness, and efficiency of the proposed HSD method are compared to current SLA and DLA through another three benchmark nonlinear RBDO examples. Results show that the AAMV is more robust and efficient than the existing reliability analysis methods. The HSD is more accurate than the SLA for highly nonlinear problems, and also exhibits a better performance than the DLA from the point of view of both robustness and efficiency.

Reliability-Based Design Optimization Concerning Objective Variation Under Mixed Probabilistic and Interval Uncertainties

Uncertainties, inevitable in nature, can be classified as probability based and interval based uncertainties in terms of its representations. Corresponding optimization strategies have been proposed to deal with these two types of uncertainties. It is more likely that both types of uncertainty can occur in one single problem, and thus, it is trivial to treat all uncertainties the same. A novel formulation for reliability-based design optimization (RBDO) under mixed probability and interval uncertainties is proposed in this paper, in which the objective variation is concerned. Furthermore, it is proposed to efficiently solve the worst-case parameter resulted from the interval uncertainty by utilizing the Utopian solution presented in a single-looped robust optimization (RO) approach where the inner optimization can be solved by matrix operations. The remaining problem can be solved utilizing any existing RBDO method. This work applies the performance measure approach to search for the most probable failure point (MPFP) and sequential quadratic programing (SQP) to solve the entire problem. One engineering example is given to demonstrate the applicability of the proposed approach and to illustrate the necessity to consider the objective robustness under certain circumstances.

Single loop single vector approach using the conjugate gradient in reliability based design optimization

Conventional reliability-based design optimization (RBDO) approaches require high computing costs. Among the existing RBDO methods, the single loop single vector approach (SLSV) converts the RBDO problem into a single loop deterministic optimization. Hence, it can efficiently reduce the design cost compared to other methods. However, this method has a weakness in that instability or inaccuracy in convergence can be increased according to the problem characteristics. It often happens when the performance function is highly nonlinear or concave. In this study, a novel method is proposed to overcome the problems. It is an SLSV method using the conjugate gradient that is calculated with the gradient directions at the most probable points (MPP) of the previous cycles. Mathematical examples and structural applications are solved to verify the proposed method. The numerical performances of the proposed method are compared with other RBDO methods such as the RIA, PMA, SORA and SLSV approaches. It is shown that the SLSV method using the conjugate gradient (SLSVCG) is not greatly influenced by problem characteristics and the convergence capability is quite superior. Also, the computational cost of the proposed method is significantly reduced and an excellent solution satisfying the specified reliability is obtained.

A hybrid self-adjusted mean value method for reliability-based design optimization using sufficient descent condition

Due to the efficiency and simplicity, advanced mean value (AMV) method is widely used to evaluate the probabilistic constraints in reliability-based design optimization (RBDO) problems. However, it may produce unstable results as periodic and chaos solutions for highly nonlinear performance functions. In this paper, the AMV is modified based on a self-adaptive step size, named as the self-adjusted mean value (SMV) method, where the step size for reliability analysis is adjusted based on a power function dynamically. Then, a hybrid self-adjusted mean value (HSMV) method is developed to enhance the robustness and efficiency of iterative scheme in the reliability loop, where the AMV is combined with the SMV on the basis of sufficient descent condition. Finally, the proposed methods (i.e. SMV and HSMV) are compared with other existing performance measure approaches through several nonlinear mathematical/structural examples. Results show that the SMV and HSMV are more efficient with enhanced robustness for both convex and concave performance functions.

Reliability-Based Design Optimization Method of Turbine Disk with Transformed Deterministic Constraints

AbstractTo improve the computational efficiency of the reliability-based design optimization (RBDO) of a complex structure with nonlinear and implicit limit-state function, the single-loop-single-vector (SLSV)-limit-state factor (LSF) (SLSV-LSF) method was developed by fully considering the advantages of the SLSV approach and the LSF method to transform uncertain constraints into deterministic constraints. The mathematical models of SLSV and LSF were established and the basic RBDO process of the SLSV-LSF method is presented. The shape optimization of an aeroengine turbine disk was completed based on the proposed method. From the reliability sensitivity analysis of the turbine disk, it is revealed that an uncertain constraint of average circumferential stress can be transformed into a deterministic constraint and material density can be regarded as a deterministic variable. Through the min-mass shape design of the turbine disk based on different approaches, it is demonstrated that the developed method main...

A new RBDO method using adaptive response surface and first-order score function for crashworthiness design

This study presents a new Reliability-based Design Optimization method using adaptive response surface and first-order score function analysis for complex system design optimization considering the variability of design variables. The adaptive response surface using Bayesian metric and Gaussian process based model bias correction method, is developed to represent the true performance functions and replace the true limit state function. First-order score function analysis is exploited to compute the sensitivities of probabilistic responses with respect to the design variables, which are the mean values of the random variables. Numerical results indicate that the proposed methods can produce the best response surface and estimate the sensitivities of probabilistic responses accurately. The proposed methodology is demonstrated by a vehicle crashworthiness design optimization problem with full frontal and offset frontal impacts.

An efficient method of solving design optimization problems with arbitrary random distributions in gradient-based design spaces

Reliability-Based Design Optimization (RBDO) algorithms, such as Reliability Index Approach (RIA) and Performance Measure Approach (PMA), have been developed to solve engineering optimization problems under design uncertainties. In some existing methods, the random design space is transformed to standard normal design space and the reliability assessment, such as reliability index from RIA or performance measure from PMA, is estimated in order to evaluate the failure probability. When the random variable is arbitrarily distributed and cannot be properly fitted to any known form of probability density function, the existing RBDO methods cannot perform reliability analysis in the original design space. This paper proposes a novel Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) to evaluate the failure probability of any arbitrarily distributed random variables in the original design space. The arbitrary distribution of the random variable is approximated by a merger of multiple Gaussian kernel functions in a single-variate coordinate that is directed toward the gradient of the constraint function. The failure probability is then estimated using the ensemble of each kernel reliability analysis. This paper further derives a linearly approximated probabilistic constraint at the design point with allowable reliability level in the original design space using the aforementioned fundamentals and techniques. Numerical examples with generated random distributions show that existing RBDO algorithms can improperly approximate the uncertainties as Gaussian distributions and provide solutions with poor assessments of reliabilities. On the other hand, the numerical results show EGTRA is capable of efficiently solving the RBDO problems with arbitrarily distributed uncertainties.

A reliability-based design optimization method for structures using evidence theory

A reliability-based design optimization method for structures using evidence theory

Conservative reliability-based design optimization method with insufficient input data

Reliability analysis and reliability-based design optimization (RBDO) require an exact input probabilistic model to obtain accurate probability of failure (PoF) and RBDO optimum design. However, often only limited input data is available to generate the input probabilistic model in practical engineering problems. The insufficient input data induces uncertainty in the input probabilistic model, and this uncertainty forces the PoF to be uncertain. Therefore, it is necessary to consider the PoF to follow a probability distribution. In this paper, the probability of the PoF is obtained with consecutive conditional probabilities of input distribution types and parameters using the Bayesian approach. The approximate conditional probabilities are obtained under reasonable assumptions, and Monte Carlo simulation is applied to calculate the probability of the PoF. The probability of the PoF at a user-specified target PoF is defined as the conservativeness level of the PoF. The conservativeness level, in addition to the target PoF, will be used as a probabilistic constraint in an RBDO process to obtain a conservative optimum design, for limited input data. Thus, the design sensitivity of the conservativeness level is derived to support an efficient optimization process. Using numerical examples, it is demonstrated that the conservativeness level should be involved in RBDO when input data is limited. The accuracy and efficiency of the proposed design sensitivity method is verified. Finally, conservative RBDO optimum designs are obtained using the developed methods for limited input data problems.

Structural design of high-performance capacitive accelerometers using parametric optimization with uncertainties

ABSTRACTElectrostatic or capacitive accelerometers are among the highest volume microelectromechanical systems (MEMS) products nowadays. The design of such devices is a complex task, since they depend on many performance requirements, which are often conflicting. Therefore, optimization techniques are often used in the design stage of these MEMS devices. Because of problems with reliability, the technology of MEMS is not yet well established. Thus, in this work, size optimization is combined with the reliability-based design optimization (RBDO) method to improve the performance of accelerometers. To account for uncertainties in the dimensions and material properties of these devices, the first order reliability method is applied to calculate the probabilities involved in the RBDO formulation. Practical examples of bulk-type capacitive accelerometer designs are presented and discussed to evaluate the potential of the implemented RBDO solver.

An approximate sequential optimization and reliability assessment method for reliability-based design optimization

Sequential optimization and reliability assessment (SORA) is one of the most popular decoupled approaches to solve reliability-based design optimization (RBDO) problem because of its efficiency and robustness. In SORA, the double loop structure is decoupled through a serial of cycles of deterministic optimization and reliability assessment. In each cycle, the deterministic optimization and reliability assessment are performed sequentially and the boundaries of violated constraints are shifted to the feasible direction according to the reliability information obtained in the previous cycle. In this paper, based on the concept of SORA, approximate most probable target point (MPTP) and approximate probabilistic performance measure (PPM) are adopted in reliability assessment. In each cycle, the approximate MPTP needs to be reserved, which will be used to obtain new approximate MPTP in the next cycle. There is no need to evaluate the performance function in the deterministic optimization since the approximate PPM and its sensitivity are used to formulate the linear Taylor expansion of the constraint function. One example is used to illustrate that the approximate MPTP will approach the accurate MPTP with the iteration. The design variables and the approximate MPTP converge simultaneously. Numerical results of several examples indicate the proposed method is robust and more efficient than SORA and other common RBDO methods.

A self-adaptive modified chaos control method for reliability-based design optimization

In the reliability-based design optimization (RBDO), the Advanced mean value (AMV) method sometimes yields unstable results such as chaotic and periodic solutions for highly nonlinear probabilistic constraints. The chaos control (CC), modified chaos control (MCC) and adaptive chaos control (ACC) methods are more robust than the AMV but inefficient for some moderately nonlinear performance functions. In this paper, a self-adaptive modified chaos control (SMCC) method is developed based on a dynamical control factor to improve the efficiency of MCC for reliability analysis and RBDO. The self-adaptive control factor is dynamically computed based on the new and previous results. The efficiency and robustness of the proposed SMCC are compared with the AMV, CC, MCC and ACC methods using several nonlinear structural/mathematical performance functions and RBDO problems. The results illustrate that the SMCC is more efficient than CC, MCC, and ACC methods, and also more robust than AMV method for highly nonlinear problems.

Enriched Serial-Loop Optimization Method for Efficient Reliability-Based Electromagnetic Designs

An enriched serial-loop method for reliability-based design optimization is presented to substantially improve computational efficiency as well as numerical accuracy when applied to electromagnetic design problems in the presence of uncertainties. Two numerical improvements are made over the original serial-loop optimization method, which employs a series of cycles of an equivalent deterministic optimization and reliability analysis. One is a feasibility check technique for the probabilistic constraints conducted only at the first design cycle. It identifies inactive constraints, which need not be considered after the next design cycles. The other is a reliability improvement scheme for slightly violated probabilistic constraints, which is performed at the end of iterative design cycles. When an obtained optimum is unsatisfactory to all probabilistic constraint conditions prescribed, the design point is slightly shifted toward a feasible design region by utilizing probabilistic information obtained. Finally, the TEAM Workshop Problem 22 is tested to compare the proposed method with the existing ones from a numerical efficiency and accuracy point of view.

Response surface single loop reliability-based design optimization with higher-order reliability assessment

Reliability-based design optimization (RBDO) aims at determination of the optimal design in the presence of uncertainty. The available Single-Loop approaches for RBDO are based on the First-Order Reliability Method (FORM) for the computation of the probability of failure, along with different approximations in order to avoid the expensive inner loop aiming at finding the Most Probable Point (MPP). However, the use of FORM in RBDO may not lead to sufficient accuracy depending on the degree of nonlinearity of the limit-state function. This is demonstrated for an extensively studied reliability-based design for vehicle crashworthiness problem solved in this paper, where all RBDO methods based on FORM strongly violates the probabilistic constraints. The Response Surface Single Loop (RSSL) method for RBDO is proposed based on the higher order probability computation for quadratic models previously presented by the authors. The RSSL-method bypasses the concept of an MPP and has high accuracy and efficiency. The method can solve problems with both constant and varying standard deviation of design variables and is particularly well suited for typical industrial applications where general quadratic response surface models can be used. If the quadratic response surface models of the deterministic constraints are valid in the whole region of interest, the method becomes a true single loop method with accuracy higher than traditional SORM. In other cases, quadratic response surface models are fitted to the deterministic constraints around the deterministic solution and the RBDO problem is solved using the proposed single loop method.

A semi-single-loop method using approximation of most probable point for reliability-based design optimization

A semi-single-loop structure for an efficient reliability-based design optimization (RBDO) is proposed in this study. This structure makes up for the weaknesses of the double-loop method and the single-loop method. In the double-loop method, a heavy computational cost could be required due to an inherent nature of a nested structure. The single-loop method lacks robustness, which means its performance is sensitively affected by several factors. A decoupled method has also been developed in an effort to improve the efficiency of RBDO. Although it separates the nested structure, it retains a double-loop structure. In the semi-single-loop method, a sensitivity analysis of the reliability analysis result is performed in order to approximate a most probable point (MPP). According to the validity of the approximation, the approximated MPP is used for the evaluation of probabilistic constraints. As long as the approximation of MPP is valid, the proposed method has a complete single-loop structure with great efficiency and assured accuracy. If the approximation of MPP loses its validity, a fast reliability analysis is carried out by selecting the approximated MPP as a starting point. The robustness and accuracy of the proposed method are fully ensured due to the concept of checking the validity of the approximated MPP. Numerical examples are tested to verify these advantages of the proposed method and to compare the results here to those of existing RBDO methods.

Design Sensitivity Method for Sampling-Based RBDO With Varying Standard Deviation

Conventional reliability-based design optimization (RBDO) uses the mean of input random variable as its design variable; and the standard deviation (STD) of the random variable is a fixed constant. However, the constant STD may not correctly represent certain RBDO problems well, especially when a specified tolerance of the input random variable is present as a percentage of the mean value. For this kind of design problem, the STD of the input random variable should vary as the corresponding design variable changes. In this paper, a method to calculate the design sensitivity of the probability of failure for RBDO with varying STD is developed. For sampling-based RBDO, which uses Monte Carlo simulation (MCS) for reliability analysis, the design sensitivity of the probability of failure is derived using a first-order score function. The score function contains the effect of the change in the STD in addition to the change in the mean. As copulas are used for the design sensitivity, correlated input random variables also can be used for RBDO with varying STD. Moreover, the design sensitivity can be calculated efficiently during the evaluation of the probability of failure. Using a mathematical example, the accuracy and efficiency of the developed design sensitivity method are verified. The RBDO result for mathematical and physical problems indicates that the developed method provides accurate design sensitivity in the optimization process.