Adaptive improved response surface method for reliability-based design optimization

Reliability-based design optimization (RBDO) requires the evaluation of probabilistic constraints (or reliability), which can be very time consuming. Therefore, a practical solution for efficient reliability analysis is needed. The response surface method (RSM) and dimension reduction (DR) are two well-known approximation methods that construct the probabilistic limit state functions for reliability analysis. This article proposes a new RSM-based approximation approach, named the adaptive improved response surface method (AIRSM), which uses the moving least-squares method in conjunction with a new weight function. AIRSM is tested with two simplified designs of experiments: saturated design and central composite design. Its performance on reliability analysis is compared with DR in terms of efficiency and accuracy in multiple RBDO test problems.     

Enhanced modified reliability index approach for efficient and robust reliability-based design optimization

The reliability index approach (RIA) is one of the effective tools for solving the reliability-based design optimization (RBDO) probabilistic model, which models the uncertainties with probability constraints. However, its wide application in engineering is limited due to low efficiency and convergence problems. The RIA-based modified reliability index approach (MRIA) appears to be very robust and accurate than RIA but yields inefficient for the most probable point (MPP) search with highly nonlinear probabilistic constraints. In this study, an enhanced modified reliability index approach (EMRIA) is developed to improve the efficiency and robustness of searching for MPP and is utilized for RBDO. In the EMRIA, an innovative active set using rigorous inequality is applied to construct the region of exploring for MPP, where the unnecessary probabilistic constraint could be eliminated adaptively during the iterative process. Moreover, the double loop strategy (DLS) is integrated into the EMRIA to strengthen the efficiency and robustness of large-scale RBDO problems. Two numerical examples demonstrated that the EMRIA is an efficient and robust method for MPP search in comparison with current first-order reliability methods. Six RBDO problems quoted also indicate that DLS-based EMRIA has good performance to solve complex RBDO problems.     

A reliability-based robust optimization design for the drum brake using adaptive Kriging surrogate model

To decrease random parameters’ influence on the drum brake reliability, the reliability-based robust optimization design (RBROD) of the electric vehicle brake is proposed. Based on the assumption that the maximum temperature of the brake cannot exceed the allowable temperature, a performance function model of thermal–mechanical coupling reliability of drum brakes is established by the adaptive Kriging method, and the analysis of reliability sensitivity and RBROD are conducted. The accuracy of the proposed model is verified by temperature measurement experiment under emergency braking condition. The robust optimization design improves the drum brake reliability to 0.99998 and reduce the influence of the design parameters on the reliability, with the absolute values of the reliability sensitivity and the weight of the drum brake are significantly smaller. Therefore, the objectives of reliability design, robustness design, and optimization design are simultaneously achieved by the proposed methods. Besides, the relative error of the proposed method is 0.373%, the number of function evaluations is 39, and the comparison with four meta-model methods show that the proposed method holds high-accuracy and high-efficiency. This study provides a high-precision theoretical explanation for the robust optimization design of drum brake.     

考虑可靠度可信区间的结构可靠性优化设计

 在贝叶斯统计理论和结构可靠性优化设计方法的基础上,研究了结构在小样本情况下考虑可靠度可信区间的结构可靠性优化设计问题.将结构可靠度作为随机变量,根据先验信息和样本信息,采用贝叶斯推断技术获得结构可靠度的概率分布,给出了可靠度的点估计及区间估计.建立了考虑可靠度可信区间的结构可靠性优化设计模型,提出了考虑可靠度可信区间的结构可靠性优化设计方法.所提出的方法为解决小样本情况下的结构可靠性优化设计问题提供了新的解决方案.数值算例验证了所提出的结构可靠性优化设计方法的有效性和正确性.     

An efficient approach for reliability-based topology optimization

This article presents an efficient approach for reliability-based topology optimization (RBTO) in which the computational effort involved in solving the RBTO problem is equivalent to that of solving a deterministic topology optimization (DTO) problem. The methodology presented is built upon the bidirectional evolutionary structural optimization (BESO) method used for solving the deterministic optimization problem. The proposed method is suitable for linear elastic problems with independent and normally distributed loads, subjected to deflection and reliability constraints. The linear relationship between the deflection and stiffness matrices along with the principle of superposition are exploited to handle reliability constraints to develop an efficient algorithm for solving RBTO problems. Four example problems with various random variables and single or multiple applied loads are presented to demonstrate the applicability of the proposed approach in solving RBTO problems. The major contribution of this article comes from the improved efficiency of the proposed algorithm when measured in terms of the computational effort involved in the finite element analysis runs required to compute the optimum solution. For the examples presented with a single applied load, it is shown that the CPU time required in computing the optimum solution for the RBTO problem is 15–30% less than the time required to solve the DTO problems. The improved computational efficiency allows for incorporation of reliability considerations in topology optimization without an increase in the computational time needed to solve the DTO problem.     

A reliability-based fatigue design for mechanical components under material variability

The mechanical components subjected cyclic load unusually fail due to fatigue. The traditional deterministic design method still has the risk of failure while the safety factor method sometimes is overconservative and uneconomic. In this study, reliability-based design optimization is applied in structural design of components under low cycle fatigue. A constitutive model (Jiang and Sehitoglu model) was written into user-defined material subroutine of finite element software to make simulation more accurate. In addition, an adaptive least squares support vector machines (LS-SVM)-based response surface method is employed to improve the efficiency of design process. After constructing the implicit life model, a hybrid directional step method is employed to implement the performance measure approach. Finally, a simple case (thickness optimization for cantilever tube) is used to demonstrate the whole procedure of proposed design procedure.     

An inverse reliability analysis method for reliability-based design optimization with random and dependent interval variables constrained within ellipsoids

In practical design problems, interval variables exist. Many existing methods can handle only independent interval variables. Some interval variables, however, are dependent. In this work, dependent interval variables constrained within a multi-ellipsoid convex set are considered and incorporated into reliability-based design optimization (RBDO). An efficient RBDO method is proposed by employing the sequential single-loop procedure, which separates the coupled reliability analysis procedure from the deterministic optimization procedure. In the reliability analysis procedure, a single-loop optimization for the inverse reliability analysis is performed, and an efficient inverse reliability analysis method for searching for the worst-case most probable point (WMPP) is developed. The search method contains two stages. The first stage deals the situation where the WMPP is on the boundary of the feasible region, while the second stage accommodates the situation where the WMPP is inside the feasible region by interpolation. Three examples are used for a demonstration.     

A new hybrid reliability-based design optimization method under random and interval uncertainties

This article proposes a new method for hybrid reliability-based design optimization under random and interval uncertainties (HRBDO-RI). In this method, Monte Carlo simulation (MCS) is employed to estimate the upper bound of failure probability, and stochastic sensitivity analysis (SSA) is extended to calculate the sensitivity information of failure probability in HRBDO-RI. Due to a large number of samples involved in MCS and SSA, Kriging metamodels are constructed to substitute true constraints. To avoid unnecessary computational cost on Kriging metamodel construction, a new screening criterion based on the coefficient of variation of failure probability is developed to judge active constraints in HRBDO-RI. Then a projection-outline-based active learning Kriging is achieved by sequentially select update points around the projection outlines on the limit-state surfaces of active constraints. Furthermore, the prediction uncertainty of Kriging metamodel is quantified and considered in the termination of Kriging update. Several examples, including a piezoelectric energy harvester design, are presented to test the accuracy and efficiency of the proposed method for HRBDO-RI.     

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

This 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 analysis and reliability-based design optimization of roadway horizontal curves using a first-order reliability method

This article presents reliability analysis and reliability-based optimization of roadway minimum radius design based on vehicle dynamics, mainly focusing on exit ramps and interchanges. The performance functions are formulated as failure modes of vehicle rollover and sideslip. To accurately describe the failure modes, analytical models for rollover and sideslip are derived considering nonlinear characteristics of vehicle behaviour using the commercial software TruckSim. The probability of an accident is evaluated using the first-order reliability method and numerical studies are conducted using a single-unit truck model. To propose a practical application for the study, the reliability analysis for the minimum radius recommended by American Association of State Highway and Transportation Officials is conducted. The results show that, even if there are deviations from assumed design conditions of the current design guideline, the proposed design method can guarantee given target margins of safety against rollover and sideslip. Based on the reliability analysis, reliability-based design optimization is carried out and the results indicate new recommendations for minimum radii satisfying given target reliability levels.     

多失效模式机械系统可靠性稳健设计方法研究

将机械系统可靠性设计理论和稳健设计方法相结合,讨论了多失效模式机械系统可靠性稳健设计问题,提出了多失效模式机械系统可靠性稳健设计的计算方法．把可靠性灵敏度融入可靠性优化设计模型之中,将机械系统可靠性稳健设计归结为满足可靠性要求的多目标优化问题．在基本随机参数的前二阶矩已知的情况下,可以迅速准确地得到机械系统可靠性稳健设计信息．     

Dynamic reliability-based robust design optimization with time-variant probabilistic constraints

With the increasing complexity of engineering systems, ensuring high system reliability and system performance robustness throughout a product life cycle is of vital importance in practical engineering design. Dynamic reliability analysis, which is generally encountered due to time-variant system random inputs, becomes a primary challenge in reliability-based robust design optimization (RBRDO). This article presents a new approach to efficiently carry out dynamic reliability analysis for RBRDO. The key idea of the proposed approach is to convert time-variant probabilistic constraints to time-invariant ones by efficiently constructing a nested extreme response surface (NERS) and then carry out dynamic reliability analysis using NERS in an iterative RBRDO process. The NERS employs an efficient global optimization technique to identify the extreme time responses that correspond to the worst case scenario of system time-variant limit state functions. With these extreme time samples, a kriging-based time prediction model is built and used to estimate extreme responses for any given arbitrary design in the design space. An adaptive response prediction and model maturation mechanism is developed to guarantee the accuracy and efficiency of the proposed NERS approach. The NERS is integrated with RBRDO with time-variant probabilistic constraints to achieve optimum designs of engineered systems with desired reliability and performance robustness. Two case studies are used to demonstrate the efficacy of the proposed approach.     

Gradient projection for reliability-based design optimization using evidence theory

Uncertainty quantification and risk assessment in the optimal design of structural systems has always been a critical consideration for engineers. When new technologies are developed or implemented and budgets are limited for full-scale testing, the result is insufficient datasets for construction of probability distributions. Making assumptions about these probability distributions can potentially introduce more uncertainty to the system than it quantifies. Evidence theory represents a method to handle epistemic uncertainty that represents a lack of knowledge or information in the numerical optimization process. Therefore, it is a natural tool to use for uncertainty quantification and risk assessment especially in the optimization design cycle for future aerospace structures where new technologies are being applied. For evidence theory to be recognized as a useful tool, it must be efficiently applied in a robust design optimization scheme. This article demonstrates a new method for projecting the reliability gradient, based on the measures of belief and plausibility, without gathering any excess information other than what is required to determine these measures. This represents a huge saving in computational time over other methods available in the current literature. The technique developed in this article is demonstrated with three optimization examples.     

Tolerance synthesis by a new method for system reliability-based optimization

The performance–cost ratio of a product may be significantly improved by a better tolerance assignment. Current practice guided by experience usually leads to a conservative and costly design. Because yield in mechanical engineering is analogous to system reliability in structural engineering, this article applies a system reliability-based optimization formulation to tolerance synthesis and presents a sequential approximate programming method. This method is an extension of the approach for the optimization with component reliability constraints. Two often-quoted examples are analyzed in detail to illustrate its effectiveness and a strategy for tolerance assignment is recommended.     

A global single-loop deterministic approach for reliability-based design optimization of truss structures with continuous and discrete design variables

A single-loop deterministic method (SLDM) has previously been proposed for solving reliability-based design optimization (RBDO) problems. In SLDM, probabilistic constraints are converted to approximate deterministic constraints. Consequently, RBDO problems can be transformed into approximate deterministic optimization problems, and hence the computational cost of solving such problems is reduced significantly. However, SLDM is limited to continuous design variables, and the obtained solutions are often trapped into local extrema. To overcome these two disadvantages, a global single-loop deterministic approach is developed in this article, and then it is applied to solve the RBDO problems of truss structures with both continuous and discrete design variables. The proposed approach is a combination of SLDM and improved differential evolution (IDE). The IDE algorithm is an improved version of the original differential evolution (DE) algorithm with two improvements: a roulette wheel selection with stochastic acceptance and an elitist selection technique. These improvements are applied to the mutation and selection phases of DE to enhance its convergence rate and accuracy. To demonstrate the reliability, efficiency and applicability of the proposed method, three numerical examples are executed, and the obtained results are compared with those available in the literature.     

Reliability-based design optimization with frequency constraints using a new safest point approach

In the reliability-based design optimization (RBDO) model, the mean values of uncertain 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, an RBDO solution that reduces the structural weight in non-critical regions provides not only an improved design, but also a higher level of confidence in the design. Solving such nested optimization problems is extremely expensive for large-scale multidisciplinary systems that are likewise computationally intensive. This article focuses on the study of a particular problem representing the failure mode of structural vibration analysis. A new method is proposed, called safest point, that can efficiently give the reliability-based optimum solution under frequency constraints, and then several probability distributions are developed, which are mathematically nonlinear functions, for the proposed method. Finally, the efficiency of the extended approach is demonstrated for probability distributions such as log-normal and uniform distributions, and its applicability to the design of structures undergoing fluid–structure interaction phenomena, especially the design process of aeroelastic structures, is also demonstrated.     

Collaborative optimization with inverse reliability for multidisciplinary systems uncertainty analysis

This article introduces a method which combines the collaborative optimization framework and the inverse reliability strategy to assess the uncertainty encountered in the multidisciplinary design process. This method conducts the sub-system analysis and optimization concurrently and then improves the process of searching for the most probable point (MPP). It reduces the load of the system-level optimizer significantly. This advantage is specifically more prominent for large-scale engineering system design. Meanwhile, because the disciplinary analyses are treated as the equality constraints in the disciplinary optimization, the computation load can be further reduced. Examples are used to illustrate the accuracy and efficiency of the proposed method.     

A unified parametric design approach to structural shape optimization

This paper presents a general parametric design approach for 2-D shape optimization problems. This approach has been achieved by integrating practical design methodologies into numerical procedures. It is characterized by three features: (i) automatic selection of a minimum number of shape design variables based on the CAD geometric model; (ii) integration of sequential convex programming algorithms to solve equality constrained optimization problems; (iii) efficient sensitivity analysis by means of the improved semi-analytical method. It is shown that shape design variables can be either manually or systematically identified with the help of equality constraints describing the relationship between geometric entities. Numerical solutions are performed to demonstrate the applicability of the proposed approach. A discussion of the results is also given:     

Multi-mode reliability-based design of horizontal curves

Recently, reliability analysis has been advocated as an effective approach to account for uncertainty in the geometric design process and to evaluate the risk associated with a particular design. In this approach, a risk measure (e.g. probability of noncompliance) is calculated to represent the probability that a specific design would not meet standard requirements. The majority of previous applications of reliability analysis in geometric design focused on evaluating the probability of noncompliance for only one mode of noncompliance such as insufficient sight distance. However, in many design situations, more than one mode of noncompliance may be present (e.g. insufficient sight distance and vehicle skidding at horizontal curves). In these situations, utilizing a multi-mode reliability approach that considers more than one failure (noncompliance) mode is required. The main objective of this paper is to demonstrate the application of multi-mode (system) reliability analysis to the design of horizontal curves. The process is demonstrated by a case study of Sea-to-Sky Highway located between Vancouver and Whistler, in southern British Columbia, Canada. Two noncompliance modes were considered: insufficient sight distance and vehicle skidding. The results show the importance of accounting for several noncompliance modes in the reliability model. The system reliability concept could be used in future studies to calibrate the design of various design elements in order to achieve consistent safety levels based on all possible modes of noncompliance.