Reliability sensitivity analysis for structural systems in interval probability form

Reliability sensitivity analysis is used to find the rate of change in the probability of failure (or reliability) due to the changes in distribution parameters such as the means and standard deviations. Most of the existing reliability sensitivity analysis methods assume that all the probabilities and distribution parameters are precisely known. That is, every statistical parameter involved is perfectly determined. However, there are two types of uncertainties, epistemic and aleatory uncertainties that may not be perfectly determined in engineering practices. In this paper, both epistemic and aleatory uncertainties are considered in reliability sensitivity analysis and modeled using P-boxes. The proposed method is based on Monte Carlo simulation (MCS), weighted regression, interval algorithm and first order reliability method (FORM). We linearize original non-linear limit-state function by MCS rather than by expansion as a first order Taylor series at most probable point (MPP) because the MPP search is an iterative optimization process. Finally, we introduce an optimization model for sensitivity analysis under both aleatory and epistemic uncertainties. Four numerical examples are presented to demonstrate the proposed method.     

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.     

System reliability analysis with saddlepoint approximation

System reliability is usually estimated through component reliability, which is commonly computed by the First Order Reliability Method (FORM). The FORM is computationally efficient, but may not be accurate for nonlinear limit-state functions. An alternative system reliability analysis method is proposed based on saddlepoint approximation. Unlike the FORM that linearizes limit-state functions in a transformed random space, the proposed method linearizes the limited-state functions without any transformation. After the linearization, the joint probability density of limit-state functions is estimated by the multivariate saddlepoint approximation. Without the nonnormal-to-normal transformation, the present method is more accurate than the FORM when the transformation increases the nonlinearity of limit-state functions. As demonstrated in the two examples, the new method is also as efficient as the FORM.     

Large scale structural optimization: Computational methods and optimization algorithms

Summary The objective of this paper is to investigate the efficiency of various optimization methods based on mathematical programming and evolutionary algorithms for solving structural optimization problems under static and seismic loading conditions. Particular emphasis is given on modified versions of the basic evolutionary algorithms aiming at improving the performance of the optimization procedure. Modified versions of both genetic algorithms and evolution strategies combined with mathematical programming methods to form hybrid methodologies are also tested and compared and proved particularly promising. Furthermore, the structural analysis phase is replaced by a neural network prediction for the computation of the necessary data required by the evolutionary algorithms. Advanced domain decomposition techniques particularly tailored for parallel solution of large-scale sensitivity analysis problems are also implemented. The efficiency of a rigorous approach for treating seismic loading is investigated and compared with a simplified dynamic analysis adopted by seismic codes in the framework of finding the optimum design of structures with minimum weight. In this context a number of accelerograms are produced from the elastic design response spectrum of the region. These accelerograms constitute the multiple loading conditions under which the structures are optimally designed. The numerical tests presented demonstrate the computational advantages of the discussed methods, which become more pronounced in large-scale optimization problems.     

Efficient computational system reliability analysis of reinforced soil-retaining structures under seismic conditions including the effect of simulated noise

This article presents a computational reliability analysis of reinforced soil-retaining structures (RSRS) under seismic conditions. The internal stability of RSRS is evaluated using the horizontal slice method (HSM) with modified pseudo-dynamic seismic forces. Two different failure modes of RSRS are identified and their reliability indices are computed using the first-order reliability method (FORM). The critical probabilistic failure surface is identified using a three-tier optimization scheme. Reliability index of the system is computed by considering the modes of failure to be connected in series. The tension mode is found to be the most critical mode of failure. The present study identifies that the wall height (H), shear wave velocity of the soil (V_s), and predominant frequency of the input motion (ω) govern the response of RSRS. Reliability indices depend on a parameter termed as the normalized frequency (ωH/V_s) and their values decrease with an increase in the value of ωH/V_s. Increase in the damping ratio of soil, increases the value of reliability indices, especially for ωH/V_s values, which are close to π/2. The FORM suffers from few critical shortcomings such as linear assumption of limit state surface at the most probable point of failure and its ability to consider only the statistical uncertainties excluding the effect of epistemic uncertainties. This calls for sampling-based numerical techniques such as Monte-Carlo simulation (MCS) which gives more comprehensive understanding of the problem under consideration in a probabilistic framework. Thus, a computationally efficient surrogate-assisted MCS is carried out to validate the present formulation and provide numerical insights by capturing the system dynamics over the entire design domain. Adoption of the efficient surrogate-assisted approach allowed us to quantify the epistemic uncertainty associated with the system using Gaussian white noise (GWN). Subsequently, its effects on the system reliability index and probabilistic behavior of the critical parameters are presented. The numerical results clearly indicate that it is imperative to take into account the probabilistic deviations of the critical performance parameters for RSRS to ensure adequate safety and serviceability under operational condition while quantifying the reliability of such systems.

     

Exponential penalty function formulation for multilevel optimization using the analytical target cascading framework

An exponential penalty function (EPF) formulation based on method of multipliers is presented for solving multilevel optimization problems within the framework of analytical target cascading. The original all-at-once constrained optimization problem is decomposed into a hierarchical system with consistency constraints enforcing the target-response coupling in the connected elements. The objective function is combined with the consistency constraints in each element to formulate an augmented Lagrangian with EPF. The EPF formulation is implemented using double-loop (EPF I) and single-loop (EPF II) coordination strategies and two penalty-parameter-updating schemes. Four benchmark problems representing nonlinear convex and non-convex optimization problems with different number of design variables and design constraints are used to evaluate the computational characteristics of the proposed approaches. The same problems are also solved using four other approaches suggested in the literature, and the overall computational efficiency characteristics are compared and discussed.     

A general RBDO decoupling approach for different reliability analysis methods

There are available in the literature several papers on the development of methods to decouple the reliability analysis and the structural optimization to solve RBDO problems. Most of them focused on strategies that employ the First Order Reliability Method (FORM) to approximate the reliability constraints. Despite of all these developments, one limitation prevailed: the lack of accuracy in the approximation of the reliability constraints due to the use of FORM. Thus, in this paper, a novel approach for RBDO is presented in order to overcome such a limitation. In this approach, we use the concept of shifting vectors, originally developed in the context of the Sequential Optimization and Reliability Assessment (SORA). However, the shifting vectors are found and updated based on a novel strategy. The resulting framework is able to use any technique for the reliability analysis stage, such as Monte Carlo simulation, second order reliability methods, stochastic polynomials, among others. Thus, the proposed approach overcomes the aforementioned limitation of most of RBDO decoupling techniques, which required the use of FORM for reliability analysis. Several examples are analyzed in order to show the effectiveness of the methodology. Focus is given on examples that are poorly solved or even cannot be tackled by FORM based approaches, such as highly nonlinear limit state functions comprised by a maximum operator or problems with discrete random variables. It should be remarked that the proposed approach was not developed to be more computationally efficient than RBDO decoupling strategies based FORM, but to allow the utilization of any, including more accurate, reliability analysis method.     

Comparison of global and local response surface techniques in reliability-based optimization of composite structures

This paper discusses the development and application of two alternative strategies, in the form of global and sequential local response surface (RS) techniques, for the solution of reliability-based optimization (RBO) problems. The problem of a thin-walled composite circular cylinder under axial buckling instability is used as a demonstrative example. In this case, the global technique uses a single second-order RS model to estimate the axial buckling load over the entire feasible design space (FDS), whereas the local technique uses multiple first-order RS models, with each applied to a small subregion of the FDS. Alternative methods for the calculation of unknown coefficients in each RS model are explored prior to the solution of the optimization problem. The example RBO problem is formulated as a function of 23 uncorrelated random variables that include material properties, the thickness and orientation angle of each ply, the diameter and length of the cylinder, as well as the applied load. The mean values of the 8 ply thicknesses are treated as independent design variables. While the coefficients of variation of all random variables are held fixed, the standard deviations of the ply thicknesses can vary during the optimization process as a result of changes in the design variables. The structural reliability analysis is based on the first-order reliability method with the reliability index treated as the design constraint. In addition to the probabilistic sensitivity analysis of the reliability index, the results of the RBO problem are presented for different combinations of cylinder length and diameter and laminate ply patterns. The two strategies are found to produce similar results in terms of accuracy, with the sequential local RS technique having a considerably better computational efficiency.     

Study on discrete optimization techniques in reliability-based optimization of truss structures

The paper deals with application of two optimization techniques to solve mixed (discrete–continuous) reliability-based optimization (RBO) problem of truss structures. The mixed RBO problem is formulated as the minimization of structural volume subjected to the constraints on the values of componental reliability indices determined by FORM approach. The cross-sectional areas of truss bars and coordinates of the specified truss nodes are considered as discrete and continuous design variables, respectively. The specified allowable reliability indices are associated with limit states in the form of the admissible displacements of the chosen truss nodes, admissible stress or local buckling of the elements as well as a global loss of stability. Two optimization techniques, namely: transformation and controlled enumeration methods are employed to solve the optimization problem. The transformation method allows to transform the mixed optimization problem into the continuous one. Two numerical examples: 10 bar planar truss and spatial truss dome are used to illustrate the proposed methodology of solution. Results obtained by both methods are compared and appropriate conclusions are drawn.     

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.     

A new directional stability transformation method of chaos control for first order reliability analysis

The HL-RF iterative algorithm of the first order reliability method (FORM) is popularly applied to evaluate reliability index in structural reliability analysis and reliability-based design optimization. However, it sometimes suffers from non-convergence problems, such as bifurcation, periodic oscillation, and chaos for nonlinear limit state functions. This paper derives the formulation of the Lyapunov exponents for the HL-RF iterative algorithm in order to identify these complicated numerical instability phenomena of discrete chaotic dynamic systems. Moreover, the essential cause of low efficiency for the stability transform method (STM) of convergence control of FORM is revealed. Then, a novel method, directional stability transformation method (DSTM), is proposed to reduce the number of function evaluations of original STM as a chaos feedback control approach. The efficiency and convergence of different reliability evaluation methods, including the HL-RF algorithm, STM and DSTM, are analyzed and compared by several numerical examples. It is indicated that the proposed DSTM method is versatile, efficient and robust, and the bifurcation, periodic oscillation, and chaos of FORM is controlled effectively.     

多分类器融合系统设计与应用

设计高性能的分类系统是模式识别研究领域追求的目标,多分类器系统MCS是实现该目标的一个有效途径。在对比MCS与单分类器系统SCS的基础上,阐述了MCS的设计与优化,并对当前的优化技术进行了分类和比较,指出了存在的问题及未来的研究方向。给出了一个用MCS实现空战目标识别的应用实例,该实例以目标的战术性能参数为分类特征,通过和规则融合多个BP网络分类器求得系统决策。实验结果表明,MCS能显著提高系统的识别率和可信度。     

Sequential approach to reliability analysis of multidisciplinary analysis systems

The goal of this study is to present an efficient strategy for reliability analysis of multidisciplinary analysis systems. Existing methods have performed the reliability analysis using nonlinear optimization techniques. This is mainly due to the fact that they directly apply multidisciplinary design optimization (MDO) frameworks to the reliability analysis formulation. Accordingly, the reliability analysis and the multidisciplinary analysis (MDA) are tightly coupled in a single optimizer, which hampers the use of recursive and function-approximation-based reliability analysis methods such as the first-order reliability method (FORM). In order to implement an efficient reliability analysis method for multidisciplinary analysis systems, we propose a new strategy named sequential approach to reliability analysis for multidisciplinary analysis systems (SARAM). In this approach, the reliability analysis and MDA are decomposed and arranged in a sequential manner, making a recursive loop. The key features are as follows. First, by the nature of the recursive loop, it can utilize the efficient advanced first-order reliability method (AFORM). It is known that AFORM converges fast in many cases and requires only the value and the gradient of the limit-state function. Second, the decomposed architecture makes it possible to execute concurrent subsystem analyses for both the reliability analysis and MDA. The concurrent subsystem analyses are conducted by using the global sensitivity equation (GSE). The efficiency of the SARAM method was verified using two illustrative examples taken from the literatures. Compared with existing methods, it showed the least number of subsystem analyses over the other methods while maintaining accuracy.     

Penalty guided genetic search for reliability design optimization

Reliability optimization has been studied in the literature for decades, usually using a mathematical programming approach. Because of these solution methodologies, restrictions on the type of allowable design have been made, however heuristic optimization approaches are free of such binding restrictions. One difficulty in applying heuristic approaches to reliability design is the highly constrained nature of the problems, both in terms of number of constraints and the difficulty of satisfying constraints. This paper presents a penalty guided genetic algorithm which efficiently and effectively searches over promising feasible and infeasible regions to identify a final, feasible optimal, or near optimal, solution. The penalty function is adaptive and responds to the search history. Results obtained on 33 test problems from the literature dominate previous solution techniques.     

Adaptive finite element methods for shape optimization of linearly elastic structures

Grid adaptive methods combined with means for automatic remeshing are applied to problems in shape optimal design of linearly elastic structures. The quantitative effect of element distortion near the design boundaries is identified in terms of interpolation error associated with the finite element discretization. The grid adaptation is itself formulated as a structural optimization problem, with an objective function that reflects the discretization error. A ‘necessary condition’ from this formulation provides the basis for a computational procedure to predict the modified grid.To avoid the sometimes drastic distortion of the FEM grid that might otherwise occur in conjunction with design change, remeshing must be performed at intermediate stages of the overall solution process. In order to produce results for the optimal shape design without interruption in this process, the computer program combines numerical grid generation and automatic remeshing with the grid adaptation and design change. Results for several shape design problems obtained with the use of grid adaptation are compared to computational results predicted from a fixed grid. Both ‘r-’ and ‘h-adaptation’ are tested.     

Quantile-based optimization under uncertainties using adaptive Kriging surrogate models

Uncertainties are inherent to real-world systems. Taking them into account is crucial in industrial design problems and this might be achieved through reliability-based design optimization (RBDO) techniques. In this paper, we propose a quantile-based approach to solve RBDO problems. We first transform the safety constraints usually formulated as admissible probabilities of failure into constraints on quantiles of the performance criteria. In this formulation, the quantile level controls the degree of conservatism of the design. Starting with the premise that industrial applications often involve high-fidelity and time-consuming computational models, the proposed approach makes use of Kriging surrogate models (a.k.a. Gaussian process modeling). Thanks to the Kriging variance (a measure of the local accuracy of the surrogate), we derive a procedure with two stages of enrichment of the design of computer experiments (DoE) used to construct the surrogate model. The first stage globally reduces the Kriging epistemic uncertainty and adds points in the vicinity of the limit-state surfaces describing the system performance to be attained. The second stage locally checks, and if necessary, improves the accuracy of the quantiles estimated along the optimization iterations. Applications to three analytical examples and to the optimal design of a car body subsystem (minimal mass under mechanical safety constraints) show the accuracy and the remarkable efficiency brought by the proposed procedure.     

A variable-complexity approach to second-order reliability-based optimization

This paper outlines a new methodology for second-order reliability-based optimization (RBO). A variable-complexity (VC) approach is used to implement a computationally efficient VCRBO algorithm, which reduces the number of costly second-order reliability analyses by using a lower fidelity, scaled mean-value technique during the majority of the constraint assessments. Two numerical examples are presented, which provide a comparison of several standard RBO approaches with the proposed algorithm. The examples include both Gaussian and non-Gaussian uncertainty to introduce significant nonlinearities in the limit state functions (LSFs). The design spaces and LSFs for both examples are presented, along with a discussion of the computational cost associated with the different RBO approaches.     

A multi-objective hyper-heuristic based on choice function

Hyper-heuristics are emerging methodologies that perform a search over the space of heuristics in an attempt to solve difficult computational optimization problems. We present a learning selection choice function based hyper-heuristic to solve multi-objective optimization problems. This high level approach controls and combines the strengths of three well-known multi-objective evolutionary algorithms (i.e. NSGAII, SPEA2 and MOGA), utilizing them as the low level heuristics. The performance of the proposed learning hyper-heuristic is investigated on the Walking Fish Group test suite which is a common benchmark for multi-objective optimization. Additionally, the proposed hyper-heuristic is applied to the vehicle crashworthiness design problem as a real-world multi-objective problem. The experimental results demonstrate the effectiveness of the hyper-heuristic approach when compared to the performance of each low level heuristic run on its own, as well as being compared to other approaches including an adaptive multi-method search, namely AMALGAM.     

An inverse-measure-based unilevel architecture for reliability-based design optimization

Reliability-based design optimization (RBDO) is a methodology for finding optimized designs that are characterized with a low probability of failure. Primarily, RBDO consists of optimizing a merit function while satisfying reliability constraints. The reliability constraints are constraints on the probability of failure corresponding to each of the failure modes of the system or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. During the last few years, a variety of different formulations have been developed for RBDO. Traditionally, these have been formulated as a double-loop (nested) optimization problem. The upper level optimization loop generally involves optimizing a merit function subject to reliability constraints, and the lower level optimization loop(s) compute(s) the probabilities of failure corresponding to the failure mode(s) that govern(s) the system failure. This formulation is, by nature, computationally intensive. Researchers have provided sequential strategies to address this issue, where the deterministic optimization and reliability analysis are decoupled, and the process is performed iteratively until convergence is achieved. These methods, though attractive in terms of obtaining a workable reliable design at considerably reduced computational costs, often lead to premature convergence and therefore yield spurious optimal designs. In this paper, a novel unilevel formulation for RBDO is developed. In the proposed formulation, the lower level optimization (evaluation of reliability constraints in the double-loop formulation) is replaced by its corresponding first-order Karush–Kuhn–Tucker (KKT) necessary optimality conditions at the upper level optimization. Such a replacement is computationally equivalent to solving the original nested optimization if the lower level optimization problem is solved by numerically satisfying the KKT conditions (which is typically the case). It is shown through the use of test problems that the proposed formulation is numerically robust (stable) and computationally efficient compared to the existing approaches for RBDO.     

Reliability-based structural optimization with probability and convex set hybrid models

For structural systems exhibiting both probabilistic and bounded uncertainties, it may be suitable to describe these uncertainties with probability and convex set models respectively in the design optimization problem. Based on the probabilistic and multi-ellipsoid convex set hybrid model, this paper presents a mathematical definition of reliability index for measuring the safety of structures in presence of parameter or load uncertainties. The optimization problem incorporating such reliability constraints is then mathematically formulated. By using the performance measure approach, the optimization problem is reformulated into a more tractable one. Moreover, the nested double-loop optimization problem is transformed into an approximate single-loop minimization problem by considering the optimality conditions and linearization of the limit-state function, which further facilitates efficient solution of the design problem. Numerical examples demonstrate the validity of the proposed formulation as well as the efficiency of the presented numerical techniques.