Optimisation of tensile membrane structures under uncertain wind loads using PCE and kriging based metamodels

Tensile membrane structures (TMS) are light-weight flexible structures that are designed to span long distances with structural efficiency. The stability of a TMS is jeopardised under heavy wind forces due to its inherent flexibility and inability to carry out-of-plane moment and shear. A stable TMS under uncertain wind loads (without any tearing failure) can only be achieved by a proper choice of the initial prestress. In this work, a double-loop reliability-based design optimisation (RBDO) of TMS under uncertain wind load is proposed. Using a sequential polynomial chaos expansion (PCE) and kriging based metamodel, this RBDO reduces the cost of inner-loop reliability analysis involving an intensive finite element solver. The proposed general approach is applied to the RBDO of two benchmark TMS and its computational efficiency is demonstrated through these case studies. The method developed here is suggested for RBDO of large and complex engineering systems requiring costly numerical solution.     

Application of reliability-based optimization to earth-moving machine: hydraulic cylinder components design process

Reliability-based design optimization (RBDO) incorporates probabilistic analysis into optimization process so that an optimum design has a great chance of staying in the feasible design space when the inevitable variability in design variables/parameters is considered. One of the biggest drawbacks of applying RBDO to practical problem is its high computational cost that is often impractical to industries. In search of the most suitable RBDO method for industrial applications, we first evaluated several existing RBDO approaches in details such as the double-loop RBDO, the sequential optimization and reliability assessment, and the response surface method. Then, based on industry needs, a platform incorporating/integrating the existing algorithm of optimization and reliability analysis is built for a practical RBDO problem. Effectiveness of the proposed RBDO approach is demonstrated using a simple cantilever beam problem and a more complicated industry problem.     

基于Adams的风力发电机齿轮故障分析

在风力发电机的各种故障中,齿轮箱失效是导致风力发电机故障和维修的主要原因之一,对于齿轮故障的诊断,准确地提取各种典型故障的特征是进行齿轮故障诊断的关键.基于Adams分别对正常和故障齿轮模型进行动态啮合仿真,在验证转速的基础上对断齿这一典型齿轮故障特性进行分析,研究结果可以为行星齿轮箱的齿轮故障诊断和监测提供理论依据.     

Single-loop system reliability-based topology optimization considering statistical dependence between limit-states

This paper presents a single-loop algorithm for system reliability-based topology optimization (SRBTO) that can account for statistical dependence between multiple limit-states, and its applications to computationally demanding topology optimization (TO) problems. A single-loop reliability-based design optimization (RBDO) algorithm replaces the inner-loop iterations to evaluate probabilistic constraints by a non-iterative approximation. The proposed single-loop SRBTO algorithm accounts for the statistical dependence between the limit-states by using the matrix-based system reliability (MSR) method to compute the system failure probability and its parameter sensitivities. The SRBTO/MSR approach is applicable to general system events including series, parallel, cut-set and link-set systems and provides the gradients of the system failure probability to facilitate gradient-based optimization. In most RBTO applications, probabilistic constraints are evaluated by use of the first-order reliability method for efficiency. In order to improve the accuracy of the reliability calculations for RBDO or RBTO problems with high nonlinearity, we introduce a new single-loop RBDO scheme utilizing the second-order reliability method and implement it to the proposed SRBTO algorithm. Moreover, in order to overcome challenges in applying the proposed algorithm to computationally demanding topology optimization problems, we utilize the multiresolution topology optimization (MTOP) method, which achieves computational efficiency in topology optimization by assigning different levels of resolutions to three meshes representing finite element analysis, design variables and material density distribution respectively. The paper provides numerical examples of two- and three-dimensional topology optimization problems to demonstrate the proposed SRBTO algorithm and its applications. The optimal topologies from deterministic, component and system RBTOs are compared with one another to investigate the impact of optimization schemes on final topologies. Monte Carlo simulations are also performed to verify the accuracy of the failure probabilities computed by the proposed approach.     

Reliability-based and deterministic design optimization of a FSAE brake pedal: a risk allocation analysis

While probabilistic designs can translate into significant weight savings through better risk allocation, deterministic design optimization remains widely used in industry. To promote the use of probabilistic designs among engineering students and practitioners, this work solves reliability based design optimization (RBDO) and deterministic design optimization (DDO) models of a FSAE brake pedal with multiple failure modes (stress and buckling) with their relative performance evaluated through a risk allocation analysis. The problems of interest were systematically solved through the following steps: i) topology optimization to specify the brake-pedal shape, ii) numerical 3D brake-pedal modeling under uncertainty for stress and buckling analysis, iii) mass (M), maximum von Mises stress (S_max) and buckling load factor (f_buck) surrogate modeling, iv) global sensitivity analysis and surrogate model selection, and v) surrogate-based RBDO and DDO with risk allocation analysis. Results show that when compared to DDO with alternative safety factors, for the same probability of system failure, the RBDO brake pedal designs were significantly lighter and more robust (less mass variability).     

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 cell evolution method for reliability-based design optimization

The reliability-based design optimization (RBDO) presents to be a systematic and powerful approach for process designs under uncertainties. The traditional double-loop methods for solving RBDO problems can be computationally inefficient because the inner reliability analysis loop has to be iteratively performed for each probabilistic constraint. To solve RBDOs in an alternative and more effective way, Deb et al. [1] proposed recently the use of evolutionary algorithms with an incorporated fastPMA. Since the imbedded fastPMA needs the gradient calculations and the initial guesses of the most probable points (MPPs), their proposed algorithm would encounter difficulties in dealing with non-differentiable constraints and the effectiveness could be degraded significantly as the initial guesses are far from the true MPPs. In this paper, a novel population-based evolutionary algorithm, named cell evolution method, is proposed to improve the computational efficiency and effectiveness of solving the RBDO problems. By using the proposed cell evolution method, a family of test cells is generated based on the target reliability index and with these reliability test cells the determination of the MPPs for probabilistic constraints becomes a simple parallel calculation task, without the needs of gradient calculations and any initial guesses. Having determined the MPPs, a modified real-coded genetic algorithm is applied to evolve these cells into a final one that satisfies all the constraints and has the best objective function value for the RBDO. Especially, the nucleus of the final cell contains the reliable solution to the RBDO problem. Illustrative examples are provided to demonstrate the effectiveness and applicability of the proposed cell evolution method in solving RBDOs. Simulation results reveal that the proposed cell evolution method outperforms comparative methods in both the computational efficiency and solution accuracy, especially for multi-modal RBDO problems.     

An efficient strategy for reliability-based multidisciplinary design optimization using BLISS

This paper presents an efficient reliability-based multidisciplinary design optimization (RBMDO) strategy. The conventional RBMDO has tri-level loops: the first level is an optimization in the deterministic space, the second one is a reliability analysis in the probabilistic space, and the third one is the multidisciplinary analysis. Since it is computationally inefficient when high-fidelity simulation methods are involved, an efficient strategy is proposed. The strategy [named probabilistic bi-level integrated system synthesis (ProBLISS)] utilizes a single-level reliability-based design optimization (RBDO) approach, in which the reliability analysis and optimization are conducted in a sequential manner by approximating limit state functions. The single-level RBDO is associated with the BLISS formulation to solve RBMDO problems. Since both the single-level RBDO and BLISS are mainly driven by approximate models, the accuracy of models can be a critical issue for convergence. The convergence of the strategy is guaranteed by employing the trust region–sequential quadratic programming framework, which validates approximation models in the trust region radius. Two multidisciplinary problems are tested to verify the strategy. ProBLISS significantly reduces the computational cost and shows stable convergence while maintaining accuracy.     

Extensions of design potential concept for reliability-based design optimization to nonsmooth and extreme cases

The reliability-based design optimization (RBDO) can be described by the design potential concept in a unified system space, where the probabilistic constraint is identified by the design potential surface of the reliability target that is obtained analytically from the first-order reliability method (FORM). This paper extends the design potential concept to treat nonsmooth probabilistic constraints and extreme case design in RBDO. In addition, refinement of the design potential surface, which yields better optimum design, can be obtained using more accurate second-order reliability method (SORM). By integrating performance probability analysis into the iterative design optimization process, the design potential concept leads to a very effective design potential method (DPM) for robust system parameter design. It can also be applied effectively to extreme case design (ECD) by directly representing a probabilistic constraint in terms of the system performance function. Received July 25, 2000     

Reliability-based design optimization of composite stiffened panels in post-buckling regime

This paper focuses on Deterministic and Reliability Based Design Optimization (DO and RBDO) of composite stiffened panels considering post-buckling regime and progressive failure analysis. The ultimate load that a post-buckled panel can hold is to be maximised by changing the stacking sequence of both skin and stringers composite layups. The RBDO problem looks for a design that collapses beyond the shortening of failure obtained in the DO phase with a target reliability while considering uncertainty in the elastic properties of the composite material. The RBDO algorithm proposed is decoupled and hence separates the Reliability Analysis (RA) from the deterministic optimization. The main code to drive both the DO and RBDO approaches is written in MATLAB and employs Genetic Algorithms (GA) to solve the DO loops because discrete design variables and highly nonlinear response functions are expected. The code is linked with Abaqus to perform parallel explicit nonlinear finite element analyses in order to obtain the structural responses at each generation. The RA is solved through an inverse Most Probable failure Point (MPP) search algorithm that benefits from a Polynomial Chaos Expansion with Latin Hypercube Sampling (PCE-LHS) metamodel when the structural responses are required. The results led to small reductions in the maximum load that the panels can bear but otherwise assure that they will collapse beyond the shortening of failure imposed with a high reliability.

     

A local adaptive sampling method for reliability-based design optimization using Kriging model

Reliability-based design optimization (RBDO) in practical applications is hindered by its huge computational cost during structure reliability evaluating process. Kriging-model-based RBDO is an effective method to overcome this difficulty. However, the accuracy of Kriging model depends directly on how to select the sample points. In this paper, the local adaptive sampling (LAS) is proposed to enhance the efficiency of constructing Kriging models for RBDO problems. In LAS, after initialization, new samples for probabilistic constraints are mainly selected within the local region around the current design point from each optimization iteration, and in the local sampling region, sample points are first considered to be located on the limit state constraint boundaries. The size of the LAS region is adaptively defined according to the nonlinearity of the performance functions. The computation capability of the proposed method is demonstrated using three mathematical RBDO problems and a honeycomb crash-worthiness design application. The comparison results show that the proposed method is very efficient.     

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.     

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.     

Reliability based design optimization of wire bonding in power microelectronic devices

This paper presents a numerical investigation of the probabilistic approach in estimating the reliability of wire bonding, and develops a reliability-based design optimization Methodology (RBDO) for microelectronic device structures. The objective of the RBDO method is to design structures which should be both economical and reliable where the solution reduces the structural weight in uncritical regions. It does not only provide an improved design, but also a higher level of confidence in the design. The Finite element simulation model intends to analyze the sequence of the failure events in power microelectronic devices. This numerical model is used to estimate the probability of failure of power module regarding the wire bonding connection. However, due to time-consuming of the multiphysics finite element simulation, a response surface method is used to approximate the response output of the limit state, in this way the reliability analysis is performed directly to the response surface by using the First and the Second Order Reliability Methods FORM/SORM. Subsequently the reliability analysis is integrated in the optimization process to improve the performance and reliability of structural design of wire bonding. The sequential RBDO algorithm is used to solve this problem and to find the best structural designs which realize the best compromise between cost and safety.     

An enhanced simulation-based design method coupled with meta-heuristic search algorithm for accurate reliability-based design optimization

The enhanced weighted simulation-based design method in conjunction with particle swarm optimization (PSO) is developed as a pseudo double-loop algorithm for accurate reliability-based design optimization (RBDO). According to this hybrid method, generated samples of weighed simulation method (WSM) are considered as initial population of the PSO. The proposed population is then employed to evaluate the safety level of each PSO swarm (design candidates) during movement. Using this strategy, there is no required to conduct new sampling for reliability assessment of design candidates (PSO swarms). Employing PSO as the search engine of RBDO and WSM as the reliability analyzer provide more accurate results with few samples and also increase the application range of traditional WSM. Besides, a shift strategy is also introduced to increase the capability of the WSM to investigate general RBDO problems including both deterministic and random design variables. Several examples are investigated to demonstrate the accuracy and robustness of the method. Results demonstrate the computational efficiency and superiority of the proposed method for practical engineering problems with highly nonlinear and implicit probabilistic constrains.     

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

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

兆瓦级风力发电机组轮毂强度数值分析

基于GL规范,采用有限元法对某兆瓦级风力发电机组轮毂进行极限强度分析.对轮毂S N曲线进行修正,分析疲劳计算需要考虑的载荷工况,使用GH Bladed软件仿真得到疲劳计算所需的载荷时间序列.使用ANSYS/nCode软件对轮毂进行疲劳寿命分析,结果表明轮毂的极限强度和疲劳强度均满足设计规范的要求.分析结果可以为兆瓦级风力发电机组轮毂的结构设计提供参考.     

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.     

Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems

There are two commonly used analytical reliability analysis methods: linear approximation - first-order reliability method (FORM), and quadratic approximation - second-order reliability method (SORM), of the performance function. The reliability analysis using FORM could be acceptable in accuracy for mildly nonlinear performance functions, whereas the reliability analysis using SORM may be necessary for accuracy of nonlinear and multi-dimensional performance functions. Even though the reliability analysis using SORM may be accurate, it is not as much used for probability of failure calculation since SORM requires the second-order sensitivities. Moreover, the SORM-based inverse reliability analysis is rather difficult to develop.This paper proposes an inverse reliability analysis method that can be used to obtain accurate probability of failure calculation without requiring the second-order sensitivities for reliability-based design optimization (RBDO) of nonlinear and multi-dimensional systems. For the inverse reliability analysis, the most probable point (MPP)-based dimension reduction method (DRM) is developed. Since the FORM-based reliability index (β) is inaccurate for the MPP search of the nonlinear performance function, a three-step computational procedure is proposed to improve accuracy of the inverse reliability analysis: probability of failure calculation using constraint shift, reliability index update, and MPP update. Using the three steps, a new DRM-based MPP is obtained, which estimates the probability of failure of the performance function more accurately than FORM and more efficiently than SORM. The DRM-based MPP is then used for the next design iteration of RBDO to obtain an accurate optimum design even for nonlinear and/or multi-dimensional system. Since the DRM-based RBDO requires more function evaluations, the enriched performance measure approach (PMA+) with new tolerances for constraint activeness and reduced rotation matrix is used to reduce the number of function evaluations.     

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

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