Sensitivity importance‐based robust optimization of structures with incomplete probabilistic information

Robust design optimization (RDO) is usually performed by minimizing the nominal value of a performance function and its dispersion considering equal importance to each individual gradient of the performance function. However, it is well known that all gradients are not equally important. An efficient sensitivity importance‐based RDO technique is proposed in the present study for optimum design of structures characterized by bounded uncertain input parameters. The basic idea of the proposed RDO formulation is to improve the robustness of a performance function by using a new gradient index that utilizes the importance factors proportional to the importance of the gradients of the performance function. The same concept is also extended to the constraints. To enhance the robustness of the constraints, the constraint functions are also modified by using the importance factor proportional to the importance of the associated gradient of the constraint. Because all the variables are not equally important to capture the presence of uncertainty, an improved robust solution is obtained by the proposed approach compared with the conventional RDO approach. The present formulation is illustrated with the help of three informative examples. The results are compared with the conventional RDO results to study the effectiveness of the proposed RDO approach. Copyright © 2012 John Wiley & Sons, Ltd.     

模糊参数平面连续体结构的拓扑优化设计

研究了具有模糊参数的连续体结构在模糊载荷作用下的拓扑优化设计问题。利用信息熵将模糊变量转换为随机变量,构建了随机载荷作用下的随机参数的连续体结构的拓扑优化设计数学模型,以结构的形状拓扑信息为设计变量,结构总质量均值极小化为目标函数,满足单元应力可靠性为约束条件,利用分布函数法对应力可靠性约束进行了等价显式化处理。基于随机因子法,利用代数综合法导出了应力响应的数字特征的计算表达式。采用双方向渐进结构优化(BESO)方法求解。通过两个算例验证了该文模型及求解方法的合理性和有效性。     

概率及非概率不确定性条件下结构鲁棒设计方法

提出了在概率不确定性和非概率不确定性同时存在时的约束函数鲁棒性和目标函数鲁棒性的实现策略及结构鲁棒设计方法。将传统优化设计问题的约束条件改造成为能同时反映两类不确定性量波动变化影响的约束条件,以实现约束函数的鲁棒性;在传统优化设计问题目标函数中增加若干个关于目标函数灵敏度的新目标函数,构成一个多目标函数设计问题,以实现目标函数的鲁棒性。所提方法应用于一个10杆桁架结构设计,采用宽容排序法求解。计算结果表明,在相同的结构总质量限制条件下,目标函数鲁棒性程度随着变量不确定性程度的增加而降低;在相同的变量不确定性程度条件下,增加结构总质量能提高目标函数鲁棒性的程度。     

Robust design optimization of nonlinear energy sink under random system parameters

Generally, in designing nonlinear energy sink (NES), only uncertainties in the ground motion parameters are considered and the unconditional expected mean of the performance metric is minimized. However, such an approach has two major limitations. First, ignoring the uncertainties in the system parameters can result in an inefficient design of the NES. Second, only minimizing the unconditional mean of the performance metric may result in large variance of the response because of the uncertainties in the system parameters. To address these issues, we focus on robust design optimization (RDO) of NES under uncertain system and hazard parameters. The RDO is solved as a bi-objective optimization problem where the mean and the standard deviation of the performance metric are simultaneously minimized. This bi-objective optimization problem has been converted into a single objective problem by using the weighted sum method. However, solving an RDO problem can be computationally expensive. We thus used a novel machine learning technique, referred to as the hybrid polynomial correlated function expansion (H-PCFE), for solving the RDO problem in an efficient manner. Moreover, we adopt an adaptive framework where H-PCFE models trained at previous iterations are reused and hence, the computational cost is less. We illustrate that H-PCFE is computationally efficient and accurate as compared to other similar methods available in the literature. A numerical study showcasing the importance of incorporating the uncertain system parameters into the optimization procedure is shown. Using the same example, we also illustrate the importance of solving an RDO problem for NES design. Overall, considering the uncertainties in the parameters have resulted in a more efficient design. Determining NES parameters by solving an RDO problem results in a less sensitive design.     

Level-set topology optimization for robust design of structures under hybrid uncertainties

This paper will develop a new robust topology optimization (RTO) method based on level sets for structures subject to hybrid uncertainties, with a more efficient Karhunen-Loève hyperbolic Polynomial Chaos–Chebyshev Interval method to conduct the hybrid uncertain analysis. The loadings and material properties are considered hybrid uncertainties in structures. The parameters with sufficient information are regarded as random fields, while the parameters without sufficient information are treated as intervals. The Karhunen-Loève expansion is applied to discretize random fields into a finite number of random variables, and then, the original hybrid uncertainty analysis is transformed into a new process with random and interval parameters, to which the hyperbolic Polynomial Chaos–Chebyshev Interval is employed for the uncertainty analysis. RTO is formulated to minimize a weighted sum of the mean and standard variance of the structural objective function under the worst-case scenario. Several numerical examples are employed to demonstrate the effectiveness of the proposed RTO, and Monte Carlo simulation is used to validate the numerical accuracy of our proposed method.     

Uncertainty propagation in inverse reliability-based design of composite structures

An approach for the analysis of uncertainty propagation in reliability-based design optimization of composite laminate structures is presented. Using the Uniform Design Method (UDM), a set of design points is generated over a domain centered on the mean reference values of the random variables. A methodology based on inverse optimal design of composite structures to achieve a specified reliability level is proposed, and the corresponding maximum load is outlined as a function of ply angle. Using the generated UDM design points as input/output patterns, an Artificial Neural Network (ANN) is developed based on an evolutionary learning process. Then, a Monte Carlo simulation using ANN development is performed to simulate the behavior of the critical Tsai number, structural reliability index, and their relative sensitivities as a function of the ply angle of laminates. The results are generated for uniformly distributed random variables on a domain centered on mean values. The statistical analysis of the results enables the study of the variability of the reliability index and its sensitivity relative to the ply angle. Numerical examples showing the utility of the approach for robust design of angle-ply laminates are presented.     

Quantification of uncertainty modelling in stochastic analysis of FRP composites

The extensive use of FRP composite materials in a wide range of industries, and their inherent variability, has prompted many researchers to assess their performance from a probabilistic perspective. This paper attempts to quantify the uncertainty in FRP composites and to summarise the different stochastic modelling approaches suggested in the literature. Researchers have considered uncertainties starting at a constituent (fibre/matrix) level, at the ply level or at a coupon or component level. The constituent based approach could be further classified as a random variable based stochastic computational mechanics approach (whose usage is comparatively limited due to complex test data requirements and possible uncertainty propagation errors) and the more widely used morphology based random composite modelling which has been recommended for exploring local damage and failure characteristics. The ply level analysis using either stiffness/strength or fracture mechanics based models is suggested when the ply characteristics influence the composite properties significantly, or as a way to check the propagation of uncertainties across length scales. On the other hand, a coupon or component level based uncertainty modelling is suggested when global response characteristics govern the design objectives. Though relatively unexplored, appropriate cross-fertilisation between these approaches in a multi-scale modelling framework seems to be a promising avenue for stochastic analysis of composite structures. It is hoped that this review paper could facilitate and strengthen this process.     

Robust optimization of structures subjected to stochastic earthquake with limited information on system parameter uncertainty

The optimization of structures subjected to stochastic earthquake and characterized by uncertain parameters is usually posed in the form of non-linear programming with stochastic performance measures where the uncertain parameters are modelled as random variables. Such an approach, however, cannot be adopted in many real life situations when the limited information about uncertainty can be only modelled as of the uncertain but bounded (UBB) type. A robust optimization strategy for stochastic dynamic systems characterized by UBB parameters is proposed in the present study in the framework of the response surface method (RSM). In evaluating the stochastic constraints, repeated computations of the dynamic responses are avoided by applying an adaptive RSM based on the moving least squares method. Numerical results are presented to highlight the effectiveness of the proposed procedure. The effect of parameter uncertainty is also studied by comparing the results obtained from the proposed optimization approach with the conventional stochastic optimization results.     

Robust topology optimization for cellular composites with hybrid uncertainties

This paper will develop a new robust topology optimization method for the concurrent design of cellular composites with an array of identical microstructures subject to random‐interval hybrid uncertainties. A concurrent topology optimization framework is formulated to optimize both the composite macrostructure and the material microstructure. The robust objective function is defined based on the interval mean and interval variance of the corresponding objective function. A new uncertain propagation approach, termed as a hybrid univariate dimension reduction method, is proposed to estimate the interval mean and variance. The sensitivity information of the robust objective function can be obtained after the uncertainty analysis. Several numerical examples are used to validate the effectiveness of the proposed robust topology optimization method.     

Worst‐case robust design optimization under distributional assumptions

Presented in this paper is a novel robust design optimization (RDO) methodology. The problem is reformulated in order to relax, when required, the assumption of normality of objectives and constraints, which often underlies RDO. In the second place, taking into account engineering considerations concerning the risk associated with constraint violation, suitable estimates of tail conditional expectations are introduced in the set of robustness metrics. A computationally affordable yet accurate implementation of the proposed formulation is guaranteed by the adoption of a reduced quadrature technique to perform the uncertainty propagation. The methodology is successfully demonstrated with the aid of an industrial test case performing the sizing of a mid‐range passenger aircraft. Copyright © 2011 John Wiley & Sons, Ltd.     

Robust topology optimization for periodic structures by combining sensitivity averaging with a semianalytical method

Uncertainty factors play an important role in the design of periodic structures because structures with small periodic design spaces are extremely sensitive to loading uncertainty. Therefore, for the first time, this paper proposes a framework for robust topology optimization (RTO) of periodic structures assuming that load uncertainties follow a Gaussian distribution. In this framework, the expected value and variance of structural compliance can be easily computed using a semianalytical method combined with probability theory, which is important for RTO when uncertain variables follow probabilistic distributions. To obtain optimal topologies, the bidirectional evolutionary structural optimization method is used. Structural periodicity is calculated using a strategy of sensitivity averaging and consistency constraints. To eliminate the influence of numerical units when comparing the optimal results to deterministic and RTO solutions, a generic coefficient of variation is defined as the robust index, which contains both the expected value and variance. The proposed framework is verified through the optimization of both 2D and 3D structures with periodicity. Computational results demonstrate the feasibility and effectiveness of the proposed framework for designing robust periodic structures under loading uncertainties.     

参数不确定性细长火箭弹随机特征值问题研究

固有振动特性分析已成为大长径比火箭弹研制与动态设计的重要环节之一。系统的加工测量误差、几何、材料以及约束条件等的不确定性对火箭弹动态特性具有显著影响。该文基于传递矩阵法和摄动方法,建立了含不确定参数的细长火箭弹随机特征值问题分析方法,研究了参数不确定性对火箭弹振动特性的影响。该方法无需建立系统总体动力学方程,可大幅度提高随机特征值问题的计算效率、降低系统存储需求。分别应用该文方法与Monte Carlo 方法对某大长径比火箭弹随机特征值问题进行了分析,两种方法计算结果吻合较好,证明了该方法的有效性。     

Polymorphic uncertain nonlinear programming approach for maximizing the capacity of V-belt driving

In this paper, a polymorphic uncertain nonlinear programming (PUNP) approach is developed to formulate the problem of maximizing the capacity in a system of V-belt driving with uncertainties. The constructed optimization model is found to consist of a nonlinear objective function and some nonlinear constraints with some parameters which are of uncertain nature. These uncertain parameters are interval parameters, random interval parameters, fuzzy parameters or fuzzy interval parameters. To find a robust solution of the problem, a deterministic equivalent formulation (DEF) is established for the polymorphic uncertain nonlinear programming model. For a given satisfaction level, this DEF turns out to be a nonlinear programming involving only interval parameters. A solution method, called a sampling based interactive method, is developed such that a robust solution of the original model with polymorphic uncertainties is obtained by using standard smooth optimization techniques. The proposed method is applied into a real-world design of V-belt driving, and the results indicate that both the PUNP approach and the developed algorithm are useful to the optimization problem with polymorphic uncertainty.     

Robust design with imprecise random variables and its application in hydrokinetic turbine optimization

In robust design, uncertainty is commonly modelled with precise probability distributions. In reality, the distribution types and distribution parameters may not always be available owing to limited data. This research develops a robust design methodology to accommodate the mixture of both precise and imprecise random variables. By incorporating the Taguchi quality loss function and the minimax regret criterion, the methodology mitigates the effects of not only uncertain parameters but also uncertainties in the models of the uncertain parameters. Hydrokinetic turbine systems are a relatively new alternative energy technology, and both precise and imprecise random variables exist in the design of such systems. The developed methodology is applied to the robust design optimization of a hydrokinetic turbine system. The results demonstrate the effectiveness of the proposed methodology.     

汽车车身耐撞性设计参数的区间稳健性优化

基于区间分析,提出了一种考虑公差的汽车车身耐撞性稳健优化设计模型,可在有效降低耐撞性能对设计参数波动敏感性的同时实现公差范围的最大化。该模型首先利用对称公差来描述汽车碰撞模型中车身关键耐撞部件的主要尺寸、位置和形状等设计参数本身的不确定性,然后将参数设计和公差设计相结合,建立了以稳健性评价指标和公差评价指标为优化目标,设计变量名义值和公差同步优化的多目标优化模型。再次,利用区间可能度处理不确定约束,将该优化模型转换为确定性多目标优化模型。最后,将该模型应用于两个汽车耐撞性优化设计问题,并通过序列二次规划法和改进的非支配排序遗传算法进行求解,结果表明该方法及稳健优化设计模型可行且实用。     

Determination of robustness for a stiffened composite structure using stochastic analysis

It is a common practice to only consider the nominal means as input variables for both classical solid mechanics and finite element (FE) analysis problems. A single solution based on the mean values is then used in design. In reality all input variables are stochastic, existing within a range of possible values. Different combinations of these stochastic input variables will lead to differing output responses and the introduction of variability will cause each structure to have a response that deviates from the original specification, sometimes with catastrophic consequences. In this paper two variables, influence and sensitivity, have been identified as parameters affecting structural robustness. Variability and uncertainty in loads, geometry and lamina stiffness are introduced via a stochastic finite element analysis (SFEA) procedure. The procedure is applied to the design of composite yacht hulls comparing the robustness of designs aimed at satisfying a range of performance and cost requirements. It is shown that influence and sensitivity are useful in identifying designs that lead to imperfection tolerant structures.     

Minimum lifecycle cost design under multiple hazards

Design of structures against multiple hazards has become an important consideration for many important engineering facilities such as major structures and bridges. A central issue is proper consideration of the uncertainty in the demand and capacity and the balance of reliability against costs. In this study this problem is investigated based on minimization of expected lifecycle cost. The uncertainties in the loads and resistance are modeled by random processes and random variables. Costs of construction, consequences of structural limit states including damage, revenue loss, death and injury as well as discounting cost over time are considered. The importance of various design parameters is first examined by a parametric study. The method is then applied to design of a multistory office building against winds and earthquakes in Los Angeles, Seattle, and Charleston. The sensitivity of optimal design to important but uncertain design parameters such as structural life, discount rate and death and injury cost is investigated. The question of uniform reliability against different hazard is also examined. It is found that an optimization-based design is a viable approach to design against multiple hazards. The design is highly dependent on failure consequence and moderately sensitive to assumption of structural life span and discount rate. It may or may not be sensitive to death and injury cost assumption dependent on location and hazard risk characteristics. Uniform reliability against different hazards is not required. The design is often dominated, but not controlled, by the hazard that has large uncertainty and causes large consequences.     

区间参数平面连续体结构频率非概率可靠性拓扑优化

研究了具有区间参数的平面连续体结构在固有频率非概率基频约束和频率禁区约束下的拓扑优化设计问题。考虑结构弹性模量、质量密度和频率约束限具有区间不确定性,根据SIMP材料插值方法和区间变量运算法则,构建了基于频率非概率可靠性约束的弯曲薄板和平面应力薄板结构的拓扑优化数学模型表达式,并给出了进化优化准则。算例及其结果表明文中模型和方法的有效性。     

Performance optimization of multidisciplinary mechanical systems subject to uncertainties

This paper presents a methodology to solve a new class of stochastic optimization problems for multidisciplinary systems (multidisciplinary stochastic optimization or MSO) wherein the objective is to maximize system mechanical performance (e.g. aerodynamic efficiency) while satisfying reliability-based constraints (e.g. structural safety). Multidisciplinary problems require a different solution approach than those solved in earlier research in reliability-based structural optimization (single discipline) wherein the goal is usually to minimize weight (or cost) for a structural configuration subject to a limiting probability of failure or to minimize probability of failure subject to a limiting weight (or cost). For the problems solved herein, the objective is to maximize performance over the range of operating conditions, while satisfying constraints that ensure safe and reliable operation. Because the objective is performance based and because the constraints are reliability based, the random variables used in the objective must model variability in operating conditions, while the random variables used in the constraints must model uncertainty in extreme values (to ensure safety). Thus, the problem must be formulated to treat these two different types of variables at the same time, including the case when the same physical quantity (e.g. a particular load) appears in both the objective function and the constraints. In addition, the problem must be formulated to treat multiple load cases, which can again require modeling the same physical quantity with different random variables. Deterministic multidisciplinary optimization (MDO) problems have advanced to the stage where they are now commonly formulated with multiple load cases and multiple disciplines governing the objective and constraints. This advancement has enabled MDO to solve more realistic problems of much more practical interest. The formulation used herein solves stochastic optimization problems that are posed in this same way, enabling similar practical benefits but, in addition, producing optimum designs that are more robust than the deterministic optimum designs (since uncertainties are accounted for during the optimization process). The methodology has been implemented in the form of a baseline MSO shell that executes on both a massively parallel computer and a network of workstations. The MSO shell is demonstrated herein by a stochastic shape optimization of an axial compressor blade involving fully coupled aero-structural analysis.     

ROBUSTNESS OPTIMIZATION FOR CONSTRAINED NONLINEAR PROGRAMMING PROBLEMS∗

In realistic situations engineering designs should take into consideration random aberrations from the stipulated design variables arising from manufacturing variability. Moreover, many environmental parameters are often stochastic in nature. Traditional nonlinear optimization attempts to find a deterministic optimum of a cost function and does not take into account the effect of these random variations on the objective. This paper attempts to devise a technique for finding optima of constrained nonlinear functions that are robust with respect to such variations. The expectation of the function over a domain of aberrations in the parameters is taken as a measure of ‘robustness’ of the function value at a point. It is pointed out that robustness optimization is ideally an attempt to trade off between ‘optimality’ and ‘robustness’. A newly-developed multi-criteria optimization technique known as Normal-Boundary Intersection is used to find evenly-spaced points on the Pareto curve for the ‘optimality’ and ‘robustness’ criteria. This Pareto curve enables the user to make the trade-off decision explicitly, free of arbitrary ‘weighting’ parameters.

This paper also formulates a derivative-based approximation for evaluating the expected value of the objective function on the nonlinear manifold defined by the state equations for the system. Existing procedures for evaluating the expectation usually involve numerical integration techniques requiring many solutions of the state equations for one evaluation of the expectation. The procedure presented here bypasses the need for multiple solutions of the state equations and hence provides a cheaper and more easily optimizable approximation to the expectation. Finally, this paper discusses how nonlinear inequality constraints should be treated in the presence of random parameters in the design. Computational results are presented for finding a robust optimum of a nonlinear structural optimization problem.