An efficient strategy for reliability-based multidisciplinary design optimization of twin-web disk with non-probabilistic model

The twin-web disk holds big promise for increasing efficiency of the aircraft engine. Its reliability-based multidisciplinary design optimization involves several disciplines including fluid mechanics, heat transfer, structural strength, and vibration. The solution to this optimization problem requires three-loop calculations including loops for optimization, reliability, and interdisciplinary consistence often making its computational cost unacceptably high. The lack of sufficient amount of probabilistic data, especially for this brand-new turbine disk, makes matters worse. In this paper, the non-probabilistic uncertain variables are described by an evidence theory-based fuzzy set method, which we extend to general structure of uncertain data. We also propose two modifications of the active learning kriging model: one of them for the purpose of optimization with respect to the distance from the optimum point and another one for the purpose of assessing reliability by introducing the importance concept. Applications of these two modifications are demonstrated in this paper. Finally, a multi-adaptive learning kriging strategy for non-probabilistic reliability-based multidisciplinary design optimization of twin-web disk is proposed to improve its power efficiency and reliability in a computationally effective way.     

A semi-analytic method for calculating non-probabilistic reliability index based on interval models

This paper proves: (1) non-probabilistic reliability index of a structure exists merely at one of intersection points at which normalized failure surfaces of the structure intersects the straight lines passing not only through origin of an normalized infinite space but also through vertices of a symmetric convex polyhedron with its sym-center at the origin, and (2) the non-probabilistic reliability index equals to absolute value of the coordinate components of a particular intersection point. Based on a reduction of the feasible region, a semi-analytical method for calculating the reliability index is developed. The method proves to be simple and of practical significance, and has several advantages over the existing unconstrained multivariate nonlinear optimization approach.     

Structural design optimization based on hybrid time-variant reliability measure under non-probabilistic convex uncertainties

Structural safety assessment issue, considering the influence of uncertain factors, is widely concerned currently. However, uncertain parameters present time-variant characteristics during the entire structural design procedure. Considering materials aging, loads varying and damage accumulation, the current reliability-based design optimization (RBDO) strategy that combines the static/time-invariant assumption with the random theory will be inapplicable when tackling with the optimal design issues for lifecycle mechanical problems. In light of this, a new study on non-probabilistic time-dependent reliability assessment and design under time-variant and time-invariant convex mixed variables is investigated in this paper. The hybrid reliability measure is first given by the first-passage methodology, and the solution aspects should depend on the regulation treatment and the convex theorem. To guarantee the rationality and efficiency of the optimization task, the improved GA algorithm is involved. Two numerical examples are discussed to demonstrate the validity and usage of the presented methodology.     

Discussions on non-probabilistic convex modelling for uncertain problems

Non-probabilistic convex model utilizes a convex set to quantify the uncertainty domain of uncertain-but-bounded parameters, which is very effective for structural uncertainty analysis with limited or poor-quality experimental data. To overcome the complexity and diversity of the formulations of current convex models, in this paper, a unified framework for construction of the non-probabilistic convex models is proposed. By introducing the correlation analysis technique, the mathematical expression of a convex model can be conveniently formulated once the correlation matrix of the uncertain parameters is created. More importantly, from the theoretic analysis level, an evaluation criterion for convex modelling methods is proposed, which can be regarded as a test standard for validity verification of subsequent newly proposed convex modelling methods. And from the practical application level, two model assessment indexes are proposed, by which the adaptabilities of different convex models to a specific uncertain problem with given experimental samples can be estimated. Four numerical examples are investigated to demonstrate the effectiveness of the present study.     

Efficient strategy for reliability-based optimization design of multidisciplinary coupled system with interval parameters

Non-probabilistic reliability based multidisciplinary design optimization has been widely acknowledged as an advanced methodology for complex system design when the data is insufficient. In this work, the uncertainty propagation analysis method in multidisciplinary system based on subinterval theory is firstly studied to obtain the uncertain responses. Then, based on the non-probabilistic set theory, the interval reliability based multidisciplinary design optimization model is established. Considering that the gradient information of interval reliability cannot be acquired in the whole design domain, which causes convergence difficulties and prohibitive computation, an interval reliability displacement based multidisciplinary design optimization method is proposed to address the issue. In the proposed method, the interval reliability displacement is introduced to measure the degree of interval reliability. By doing so, not only the connotation of the interval reliability is guaranteed, but more importantly, the partial gradient region for interval reliability is equivalently converted into full gradient region for reliability displacement. Consequently, the gradient information can be acquired under any circumstances and thus the convergence process is highly accelerated by utilizing the gradient optimization algorithms.     

A novel method for non-probabilistic convex modelling based on data from practical engineering

In this paper, a novel method for non-probabilistic convex modelling with the bounds to precisely encircle all the data of uncertain parameters extracted from practical engineering is developed. The method is based on the traditional statistical method and the correlation analysis technique. Mean values and correlation coefficients of uncertain parameters are first calculated by utilizing the information of all the given data. Then, a simple yet effective optimization procedure is first introduced in the mathematical modelling process for uncertain parameters to obtain their precise bounds. This procedure works by optimizing the area of the convex model, at the same time, covering all the given data. Thus, the effective mathematical expression of the convex models are finally formulated. To test the prediction capability and generalization ability of the proposed convex modelling method, evaluation criteria, i.e. volume ratio, standard volume ratio, and prediction accuracy are established. The performance of the proposed method is systematically studied and compared with other existing competitive methods through test standards. The results demonstrate the effectiveness and efficiency of the present method.     

Non-Probabilistic Structural Reliability Analysis Integrating the PSO and the Kriging Model北大核心CSCD

针对复杂结构可靠性分析中面临的隐式功能函数和小样本问题,提出了一种粒子群优化和Kriging模型相结合的结构非概率可靠性分析方法。采用多维椭球描述结构不确定参数,运用粒子群优化对模型相关参数进行求解,并构建隐式功能函数的Kriging模型进行可靠性分析。三个算例结果表明所提方法有效可行,精度和效率均优于基于Kriging模型的非概率可靠性分析方法。     

铁路应急设施非概率可靠性选址优化研究

为解决小样本、贫信息下铁路应急资源储备点的可靠性选址问题,创新性地将选址-路径问题与区间非概率可靠性方法结合起来,考虑灾情发生后应急设施点在可接受的时间范围内响应受灾点的需求能力及其稳定程度,采用区间值度量路段阻抗,基于区间非概率可靠性理论及区间运算规则,提出路径的非概率可靠性度量及可靠最短路径选择方法;建立基于区间时间阻抗下可靠最短路径的无容量设施选址模型,提出约束条件限制的Monte Carlo改进算法,确定了铁路资源储备点选址的最优方案。实例表明,本文的优化方案能更好地保证救援的时间可靠性,改进的求解算法具有更小的时间复杂度,有效地缩短了运算时间,改善了解的质量。本文的方法与模型体系对于实现铁路应急设施可靠性选址,为决策者提供决策支持,提高铁路应急响应能力具有重要的指导意义。     

A novel single-loop procedure for time-variant reliability analysis based on Kriging model

This paper proposes a novel single-loop procedure for time-variant reliability analysis based on a Kriging model. A new strategy is presented to decouple the double-loop Kriging model for time-variant reliability analysis, in which the extreme value response in double-loop procedure is replaced by the best value in the current sampled points to avoid the inner optimization loop. Consequently, the extreme value response surface for time-variant reliability analysis can be directly established through a single-loop Kriging surrogate model. To further improve the accuracy of the proposed Kriging model, two methods are provided to adaptively choose a new sample point for updating the model. One method is to apply two commonly used learning functions to select the new sample point that resides as close to the extreme value response surface as possible, and the other is to apply a new learning function to select the new point. Synchronously, the corresponding different stopping criteria are also provided. It is worth nothing that the proposed single-loop Kriging model for time-variant reliability analysis is for a single time-variant performance function. To verify the proposed method, it is applied to four examples, two of which have with random process and others have not. Other popular methods for time-variant reliability analysis including the existing single-loop Kriging model are also used for the comparative analysis and their results testify the effectiveness of the proposed method.     

A robustification approach in unconstrained quadratic optimization

Unconstrained convex quadratic optimization problems subject to parameter perturbations are considered. A robustification approach is proposed and analyzed which reduces the sensitivity of the optimal function value with respect to the parameter. Since reducing the sensitivity and maintaining a small objective value are competing goals, strategies for balancing these two objectives are discussed. Numerical examples illustrate the approach.     

Novel reliability-based optimization method for thermal structure with hybrid random,interval and fuzzy parameters

In this paper, novel reliability-based optimization model and method are proposed for thermal structure design with random, interval and fuzzy uncertainties in material properties, external loads and boundary conditions. Random variables are used to quantify the probabilistic uncertainty with sufficient sample data; whereas, interval variables and fuzzy variables are adopted to model the non-probabilistic uncertainty associated with objective limited information and subjective expert opinions, respectively. Using the interval ranking strategy, the level-cut limit state function is precisely quantified to represent the safety state. The eventual safety possibility is derived based on multiple integral, where the cut levels of different fuzzy variables are considered to be independent. Then a hybrid reliability-based optimization model is established with considerable computational cost caused by three-layer nested loop. To improve the computational efficiency, a subinterval vertex method is presented to replace the inner-loop and middle-loop. Comparing numerical results with traditional reliability model, a mono-objective example and a multi-objective example are provided to demonstrate the feasibility of proposed method for hybrid reliability analysis and optimization in practical engineering.     

Correlation propagation for uncertainty analysis of structures based on a non-probabilistic ellipsoidal model

Traditional non-probabilistic methods for uncertainty propagation problems evaluate only the lower and upper bounds of structural responses, lacking any analysis of the correlations among the structural multi-responses. In this paper, a new non-probabilistic correlation propagation method is proposed to effectively evaluate the intervals and non-probabilistic correlation matrix of the structural responses. The uncertainty propagation process with correlated parameters is first decomposed into an interval propagation problem and a correlation propagation problem. The ellipsoidal model is then utilized to describe the uncertainty domain of the correlated parameters. For the interval propagation problem, a subinterval decomposition analysis method is developed based on the ellipsoidal model to efficiently evaluate the intervals of responses with a low computational cost. More importantly, the non-probabilistic correlation propagation equations are newly derived for theoretically predicting the correlations among the uncertain responses. Finally, the multi-dimensional ellipsoidal model is adopted again to represent both uncertainties and correlations of multi-responses. Three examples are presented to examine the accuracy and effectiveness of the proposed method both numerically and experimentally.     

Optimal Error Correction and Methods of Feasible Directions

The main objective of this study is to discuss the optimum correction of linear inequality systems and absolute value equations (AVE). In this work, a simple and efficient feasible direction method will be provided for solving two fractional nonconvex minimization problems that result from the optimal correction of a linear system. We will show that, in some special-but frequently encountered-cases, we can solve convex optimization problems instead of not-necessarily-convex fractional problems. And, by using the method of feasible directions, we solve the optimal correction problem. Some examples are provided to illustrate the efficiency and validity of the proposed method.     

Two basic problems in reliability-based structural optimization

Optimization of structures with respect to performance, weight or cost is a well-known application of mathematical optimization theory. However optimization of structures with respect to weight or cost under probabilistic reliability constraints or optimization with respect to reliability under cost/weight constraints has been subject of only very few studies. The difficulty in using probabilistic constraints or reliability targets lies in the fact that modern reliability methods themselves are formulated as a problem of optimization. In this paper two special formulations based on the so-called first-order reliability method (FORM) are presented. It is demonstrated that both problems can be solved by a one-level optimization problem, at least for problems in which structural failure is characterized by a single failure criterion. Three examples demonstrate the algorithm indicating that the proposed formulations are comparable in numerical effort with an approach based on semi-infinite programming but are definitely superior to a two-level formulation.     

LMI Optimization Approach to Synchronization of Stochastic Delayed Discrete-Time Complex Networks

Complex networks are widespread in real-world systems of engineering, physics, biology, and sociology. This paper is concerned with the problem of synchronization for stochastic discrete-time drive-response networks. A dynamic feedback controller has been proposed to achieve the goal of the paper. Then, based on the Lyapunov second method and LMI (linear matrix inequality) optimization approach, a delay-independent stability criterion is established that guarantees the asymptotical mean-square synchronization of two identical delayed networks with stochastic disturbances. The criterion is expressed in terms of LMIs, which can be easily solved by various convex optimization algorithms. Finally, two numerical examples are given to illustrate the proposed method.     

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.     

Non-probabilistic Bayesian update method for model validation

Model validation is the principal strategy to evaluate the accuracy and reliability of computational simulations. A systematic model validation procedure including uncertainty quantification, model update and prediction is described based on a non-probabilistic interval model. The crucial technical challenge in model validation is limited data, thus the non-probabilistic interval model is adopted to describe uncertain parameters. To establish the model update formula, the concepts of the interval escape rate and interval coverage rate are first described. Then, not only can the possibility of failure be estimated but also the credibility of the possibility of failure based on the proposed model validation method. The data in the validation experiment are used to update the credibility of each interval model, while the data from the accreditation experiment are used to conduct a final check of the validated models. To demonstrate that the proposed method can be applied to model validation problems successfully, a validation benchmark, the static frame challenge problem, is implemented. In addition, a practical aviation structure engineering validation problem is described. The results of these two validation problems show the feasibility and effectiveness of the proposed model validation method. The theoretical framework proposed in this paper is also suitable for model validation of computational simulations in other research fields.     

Maximum Divert for Planetary Landing Using Convex Optimization

This paper presents a real-time solution method of the maximum divert trajectory optimization problem for planetary landing. In mid-course, the vehicle is to abort and retarget to a landing site as far from the nominal as physically possible. The divert trajectory must satisfy velocity constraints in the range and cross range directions and a total speed constraint. The thrust magnitude is bounded above and below so that once on, the engine cannot be turned off. Because this constraint is not convex, it is relaxed to a convex constraint and lossless convexification is proved. A transformation of variables is introduced in the nonlinear dynamics and an approximation is made so that the problem becomes a second-order cone problem, which can be solved to global optimality in polynomial time whenever a feasible solution exists. A number of examples are solved to illustrate the effectiveness and efficiency of the solution method.     

Charged ball method for solving some computational geometry problems

The concept of replacement of the initial stationary optimization problem with some nonstationary mechanical system tending with time to the position of equilibrium, which coincides with the solution of the initial problem, makes it possible to construct effective numerical algorithms. First, differential equations of the movement should be derived. Then we pass to the difference scheme and define the iteration algorithm. There is a wide class of optimization methods constructed in such a way. One of the most known representatives of this class is the heavy ball method. As a rule, such type of algorithms includes parameters that highly affect the convergence rate. In this paper, the charged ball method, belonging to this class, is proposed and investigated. It is a new effective optimization method that allows solving some computational geometry problems. A problem of orthogonal projection of a point onto a convex closed set with a smooth boundary and the problem of finding the minimum distance between two such sets are considered in detail. The convergence theorems are proved, and the estimates for the convergence rate are found. Numerical examples illustrating the work of the proposed algorithms are given.     

A new method for reliability analysis of structures with mixed random and convex variables

This paper proposes a method combining projection-outline-based active learning strategy with Kriging metamodel for reliability analysis of structures with mixed random and convex variables. In this method, it is determined that the approximation accuracy of projection outlines on the limit-state surface is crucial for estimation of failure probability instead of the whole limit-state surface. To efficiently improve the approximation accuracy of projection outlines, a new projection-outline-based active learning strategy is developed to sequentially obtain update points located around the projection outlines. Taking into account the influence of metamodel uncertainty on the estimation of failure probability, a quantification function of metamodel uncertainty is developed and introduced in the stopping condition of Kriging metamodel update. Finally, Monte Carlo simulation is employed to calculate the failure probability based on the refined Kriging metamodel. Four examples including the Burro Creek Bridge and a piezoelectric energy harvester are tested to validate the performance of the proposed method. Results indicate that the proposed method is accurate and efficient for reliability analysis of structures with mixed random and convex variables.