Multi-objective robust design of energy-absorbing components using coupled process–performance simulations

The stochastic uncertainties associated with the material, process and product are represented and propagated to process and performance responses. A finite element-based sequential coupled process–performance framework is used to simulate the forming and energy absorption responses of a thin-walled tube in a manner that both material properties and component geometry can evolve from one stage to the next for better prediction of the structural performance measures. Metamodelling techniques are used to develop surrogate models for manufacturing and performance responses. One set of metamodels relates the responses to the random variables whereas the other relates the mean and standard deviation of the responses to the selected design variables. A multi-objective robust design optimization problem is formulated and solved to illustrate the methodology and the influence of uncertainties on manufacturability and energy absorption of a metallic double-hat tube. The results are compared with those of deterministic and augmented robust optimization problems.     

A particle swarm optimizer for constrained multi-objective engineering design problems

This article presents a particle swarm optimizer (PSO) capable of handling constrained multi-objective optimization problems. The latter occur frequently in engineering design, especially when cost and performance are simultaneously optimized. The proposed algorithm combines the swarm intelligence fundamentals with elements from bio-inspired algorithms. A distinctive feature of the algorithm is the utilization of an arithmetic recombination operator, which allows interaction between non-dominated particles. Furthermore, there is no utilization of an external archive to store optimal solutions. The PSO algorithm is applied to multi-objective optimization benchmark problems and also to constrained multi-objective engineering design problems. The algorithmic effectiveness is demonstrated through comparisons of the PSO results with those obtained from other evolutionary optimization algorithms. The proposed particle swarm optimizer was able to perform in a very satisfactory manner in problems with multiple constraints and/or high dimensionality. Promising results were also obtained for a multi-objective engineering design problem with mixed variables.     

飞机总体设计中的一种多目标鲁棒优化方法

 提出一种基于灵敏度的多目标鲁棒优化方法。针对各维设计变量存在扰动的情况,在原约束多目标优化模型上,附加偏差目标函数,并采用最差估计法对约束条件进行鲁棒可行性调整。采用全局敏度方程方法来计算目标函数和约束函数对设计变量的敏度,进而采用Pareto遗传算法搜索约束多目标优化问题的非劣解集,设计者可以根据不同的设计准则从中选择合适的设计点。将上述方法用于飞机总体参数优化设计,并与采用常规优化方法所得的优化结果进行了分析和比较。     

SIMULATED ANNEALING AND PARALLEL PROCESSING: AN IMPLEMENTATION FOR CONSTRAINED GLOBAL DESIGN OPTIMIZATION

Global optimization becomes important as more and more complex designs are evaluated and optimized for superior performance. Often parametric designs are highly constrained, adding complexity to the design problem. In this work simulated annealing (SA), a stochastic global optimization technique, is implemented by augmenting it with a feasibility improvement scheme (FIS) that makes it possible to formulate and solve a constrained optimization problem without resorting to artificially modifying the objective function. The FIS is also found to help recover from the infeasible design space rapidly. The effectiveness of the improved algorithm is demonstrated by solving a welded beam design problem and a two part stamping optimization problem. Large scale practical design problems may prohibit the efficient use of computationally intensive iterative algorithms such as SA. Hence the FIS augmented SA algorithm is implemented on an Intel iPSC/860 parallel super-computer using a data parallel structure of the algorithm for the solution of large scale optimization problems. The numerical results demonstrate the effectiveness of the FIS as well as the parallel version of the SA algorithm. Expressions are developed for the estimation of the speedup of iterative algorithms running on a parallel computer with hyper-cube interconnection topology. Computational speedup in excess of 8 is achieved using 16 processors. The timing results given for the example problems provide guidelines to designers in the use of parallel computers for iterative processes.     

Variation-aware clock network buffer sizing using robust multi-objective optimization

Many engineering optimization problems include unavoidable uncertainties in parameters or variables. Ignoring such uncertainties when solving the optimization problems may lead to inferior solutions that may even violate problem constraints. Another challenge in most engineering optimization problems is having different conflicting objectives that cannot be minimized simultaneously. Finding a balanced trade-off between these objectives is a complex and time-consuming task. In this paper, an optimization framework is proposed to address both of these challenges. First, we exploit a self-calibrating multi-objective framework to achieve a balanced trade-off between the conflicting objectives. Then, we develop the robust counterpart of the uncertainty-aware self-calibrating multi-objective optimization framework. The significance of this framework is that it does not need any manual tuning by the designer. We also develop a mathematical demonstration of the objective scale invariance property of the proposed framework. The engineering problem considered in this paper to illustrate the effectiveness of the proposed framework is a popular sizing problem in digital integrated circuit design. However, the proposed framework can be applied to any uncertain multi-objective optimization problem that can be formulated in the geometric programming format. We propose to consider variations in the sizes of circuit elements during the optimization process by employing ellipsoidal uncertainty model. For validation, several industrial clock networks are sized by the proposed framework. The results show a significant reduction in one objective (power, on average 38 %) as well as significant increase in the robustness of solutions to the variations. This is achieved with no significant degradation in the other objective (timing metrics of the circuit) or reduction in its standard deviation which demonstrates a more robust solution.     

Constrained multi-objective optimization algorithm with an ensemble of constraint handling methods

Different constraint handling techniques have been used with multi-objective evolutionary algorithms (MOEA) to solve constrained multi-objective optimization problems. It is impossible for a single constraint handling technique to outperform all other constraint handling techniques always on every problem irrespective of the exhaustiveness of the parameter tuning. To overcome this selection problem, an ensemble of constraint handling methods (ECHM) is used to tackle constrained multi-objective optimization problems. The ECHM is integrated with a multi-objective differential evolution (MODE) algorithm. The performance is compared between the ECHM and the same single constraint handling methods using the same MODE (using codes available from http://www3.ntu.edu.sg/home/EPNSugan/index.htm). The results show that ECHM overall outperforms the single constraint handling methods.     

结构动力鲁棒优化设计方法综述

如何提高结构动力学性能的鲁棒性,以减小各种不确定性因素对设计结果的影响是当前学术界和工程界研究和关注的热点问题之一。该文阐述了结构动力鲁棒优化设计的基本概念,从基于Taguchi的方法、基于多目标优化的方法和基于响应面建模的方法三个方面对结构动力鲁棒优化设计的研究进行了综述。以双转子为例,从结构的动力响应要求出发,采用响应面建模、多目标优化的方法进行了设计并与采用Taguchi方法得到的结果进行比较。结果表明,基于响应面建模、多目标优化的方法能够获得多个具有鲁棒性的设计方案,在处理具有不确定性的结构动力学问题时有着很大的应用潜力。最后,对当前方法和后续研究内容作了简要总结和展望。     

计及隔振率的发动机悬置系统稳健优化设计

发动机悬置系统的隔振率和解耦率是汽车动力总成设计的两个主要性能指标,通过ISIGHT软件集成MATLAB与ADAMS软件,建立计及隔振率的悬置系统稳健优化数学模型。考虑到橡胶悬置的生产工艺,将悬置的各向刚度值处理成耦合变量,以整车下悬置系统的频率合理分配为约束,并利用PSO优化算法对发动机悬置系统参数进行多目标优化求解。实车试验验证该方法的有效性,提高发动机悬置系统设计的实用性。     

Evolutionary reliable regional Kriging surrogate for expensive optimization

This work proposes a sequential optimization algorithm, EORKS, combining a Kriging surrogate from an adaptive sampling and an iterative constrained search in the dynamic reliable regions to reduce the sampling size in expensive optimization. A surrogate established from small samples is liable to limited generality, which leads to a false prediction of optimum. EORKS applies Kriging variance to establish the reliable region neighbouring the learning samples to constrain the evolutionary searches of the surrogate. The verified quasi-optimum is used as an additional sample to dynamically update the regional model according to the prediction accuracy. A hybrid infilling strategy switches between the iterative quasi-optima and the maximum expected improvement from Kriging to prevent early convergence of local optimum. EORKS provides superior optima in several benchmark functions and an engineering design problem, using much smaller samples compared with the literature results, which demonstrates the sampling efficiency and searching robustness.     

Root cause analysis strategy for robust design domain recognition

Design domain identification with desirable attributes (e.g. feasibility, robustness and reliability) provides advantages when tackling large-scale engineering optimization problems. For the purpose of dealing with feasibility robustness design problems, this article proposes a root cause analysis (RCA) strategy to identify desirable design domains by investigating the root causes of performance indicator variation for the starting sampling initiation of evolutionary algorithms. The iterative dichotomizer 3 method using a decision tree technique is applied to identify reduced feasible design domain sets. The robustness of candidate domains is then evaluated through a probabilistic principal component analysis-based criterion. The identified robust design domains enable optimal designs to be obtained that are relatively insensitive to input variations. An analytical example and an automotive structural optimization problem are demonstrated to show the validity of the proposed RCA strategy.     

An enhanced genetic algorithm for structural topology optimization

Genetic algorithms (GAs) have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures. In this paper, a versatile, robust and enhanced GA is proposed for structural topology optimization by using problem‐specific knowledge. The original discrete black‐and‐white (0–1) problem is directly solved by using a bit‐array representation method. To address the related pronounced connectivity issue effectively, the four‐neighbourhood connectivity is used to suppress the occurrence of checkerboard patterns. A simpler version of the perimeter control approach is developed to obtain a well‐posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge‐free design. To handle the problem of representation degeneracy effectively, a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner. An efficient FEM‐based function evaluation method is developed to reduce the computational cost. A dynamic penalty method is presented for the GA to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy. With all these enhancements and appropriate choice of the GA operators, the present GA can achieve significant improvements in evolving into near‐optimum solutions and viable topologies with checkerboard free, mesh independent and hinge‐free characteristics. Numerical results show that the present GA can be more efficient and robust than the conventional GAs in solving the structural topology optimization problems of minimum compliance design, minimum weight design and optimal compliant mechanisms design. It is suggested that the present enhanced GA using problem‐specific knowledge can be a powerful global search tool for structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.     

Expected improvement based infill sampling for global robust optimization of constrained problems

A novel adaptive sampling scheme for efficient global robust optimization of constrained problems is proposed. The method addresses expensive to simulate black-box constrained problems affected by uncertainties for which only the bounds are known, while the probability distribution is not available. An iterative strategy for global robust optimization that adaptively samples the Kriging metamodel of the computationally expensive problem is proposed. The presented approach is tested on several benchmark problems and the average performance based on 100 runs is evaluated. The applicability of the method to engineering problems is also illustrated by applying robust optimization on an integrated photonic device affected by manufacturing uncertainties. The numerical results show consistent convergence to the global robust optimum using a limited number of expensive simulations.     

Reliability-based robust multi-objective optimization applied to engineering system design

ABSTRACT

Probabilistic and non-probabilistic methods have been proposed to deal with design problems under uncertainties. Reliability-based design and robust design are probabilistic strategies traditionally used for this purpose. In the present contribution, reliability-based robust design optimization (RBRDO) is formulated as a multi-objective problem considering the interaction of both approaches. The proposed methodology is based on the differential evolution algorithm associated with two strategies to deal with reliability and robustness, respectively, namely inverse reliability analysis and the effective mean concept. This multi-objective optimization problem considers the maximization of reliability and robustness coefficients as additional objective functions. The effectiveness of the methodology is illustrated by two classical test cases and a rotor-dynamics application. The results demonstrate that the proposed methodology is an alternative method to solve RBRDO problems.     

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。     

Multi-objective optimization by a new hybridized method: applications to random mechanical systems

In this article two linear problems with random Gaussian loading are transformed into multi-objective optimization problems. The first problem is the design of a pillar geometry with respect to a compressive random load process. The second problem is the design of a truss structure with respect to a vertical random load process for several frequency bands. A new algorithm, motivated by the Pincus representation formula hybridized with the Nelder–Mead algorithm, is proposed to solve the two multi-objective optimization problems. To generate the Pareto curve, the normal boundary intersection method is used to produce a series of constrained single-objective optimizations. The second problem, depending on the frequency band of excitation, can have as Pareto curve a single point, a standard Pareto curve, or a discontinuous Pareto curve, a fact that has been reported here for the first time in the literature, to the best of the authors’ knowledge.     

Multi-objective design optimization of electrostatically actuated microbeam resonators with and without parameter uncertainty

Electrostatically actuated microbeam resonators are widely used components in microelectromechanical systems for sensing and signal filtering purposes. Due to the uncertainties resulting from manufacturing processes, material properties, and modeling assumptions, microbeam resonators may exhibit significant variations in their performance compared to nominal designs. There has been limited research on the performance prediction and the design optimization of such microsystems while accounting for relevant uncertainties. In this study, such uncertainties are considered in terms of the variability of parameters that define the dimensions, the material properties, and the operating conditions of the device. In addition, uncertainties with respect to a two-dimensional model of a microbeam resonator subject to electrostatic actuation are considered. A finite element model consisting of both the microbeam and the substrate is developed. The actuation forces are predicted by a reduced order electrostatic model, which accounts for the electromechanical interaction. A computationally efficient procedure is presented for simulating the steady-state dynamic response under electrostatic forces. The probabilistic performance of the microresonator is investigated using Monte Carlo simulation. A genetic algorithm is used to optimize the stochastic behavior of the microbeam resonator. The design is posed as combinatorial multi-objective optimization problem. Two design criteria describing the filter performance in terms of the shape of the frequency–response curve are simultaneously considered. The numerical results demonstrate the effectiveness of this procedure for the multi-objective optimization design of microbeam resonators and the importance of considering parameter uncertainty in the design of these devices.     

A framework for design optimization using surrogates

Design optimization is a computationally expensive process as it requires the assessment of numerous designs and each of such assessments may be based on expensive analyses (e.g. computational fluid dynamics method or finite element based method). One way to contain the computational time within affordable limits is to use computationally cheaper approximations (surrogates) in lieu of the actual analyses during the course of optimization. This article introduces a framework for design optimization using surrogates. The framework is built upon a stochastic, zero-order, population-based optimization algorithm, which is embedded with a modified elitism scheme, to ensure convergence in the actual function space. The accuracy of the surrogate model is maintained via periodic retraining and the number of data points required to create the surrogate model is identified by a k-means clustering algorithm. A comparison is provided between different surrogate models (Kriging, radial basis functions (Exact and Fixed) and Cokriging) using a number of mathematical test functions and engineering design optimization problems. The results clearly indicate that for a given fixed number of actual function evaluations, the surrogate assisted optimization model consistently performs better than a pure optimization model using actual function evaluations.     

Multiobjective robust optimization method for drawbead design in sheet metal forming

It is recognized that fracture and wrinkling in sheet metal forming can be eliminated via an appropriate drawbead design. Although deterministic multiobjective optimization algorithms and finite element analysis (FEA) have been applied in this respect to improve formability and shorten design cycle, the design could become less meaningful or even unacceptable when considering practical variation in design variables and noises of system parameters. To tackle this problem, we present a multiobjective robust optimization methodology to address the effects of parametric uncertainties on drawbead design, where the six sigma principle is adopted to measure the variations, a dual response surface method is used to construct surrogate model and a multiobjective particle swarm optimization is developed to generate robust Pareto solutions. In this paper, the procedure of drawbead design is divided into two stages: firstly, equivalent drawbead restraining forces (DBRF) are obtained by developing a multiobjective robust particle swarm optimization, and secondly the DBRF model is integrated into a single-objective particle swarm optimization (PSO) to optimize geometric parameters of drawbead. The optimal design showed a good agreement with the physical drawbead geometry and remarkably improve the formability and robust. Thus, the presented method provides an effective solution to geometric design of drawbead for improving product quality.     

Uncertainty‐based robust aerodynamic optimization of rotor blades

The aerodynamic performance of a compressor is highly sensitive to uncertain working conditions. This paper presents an efficient robust aerodynamic optimization method on the basis of nondeterministic computational fluid dynamic (CFD) simulation and multi‐objective genetic algorithm (MOGA). A nonintrusive polynomial chaos method is used in conjunction with an existing well‐verified CFD module to quantify the uncertainty propagation in the flow field. This method is validated by comparing with a Monte Carlo method through full 3D CFD simulations on an axial compressor (National Aeronautics and Space Administration rotor 37). On the basis of the validation, the nondeterministic CFD is coupled with a surrogate‐based MOGA to search for the Pareto front. A practical engineering application is implemented to the robust aerodynamic optimization of rotor 37 under random outlet static pressure. Two curve angles and two sweep angles at tip and hub are used as design variables. Convergence analysis shows that the surrogate‐based MOGA can obtain the Pareto front properly. Significant improvements of both mean and variance of the efficiency are achieved by the robust optimization. The comparison of the robust optimization results with that of the initial design, and a deterministic optimization demonstrate that the proposed method can be applied to turbomachinery successfully. Copyright © 2012 John Wiley & Sons, Ltd.     

Robust design of tuned mass dampers installed on multi-degree-of-freedom structures subjected to seismic action

This study deals with robust optimum design of tuned mass dampers installed on multi-degree-of-freedom systems subjected to stochastic seismic actions, assuming the structural and seismic model parameters to be uncertain. A new global performance index for evaluating the efficiency of protection systems is proposed, as an alternative to commonly used local performance indices such as the maximum interstorey drift. The latter can be considered a good estimator of seismic damage, but it does not measure the whole structural integrity. The direct perturbation method based on first order approximation is adopted to evaluate the effects of uncertainties on the response. The robust design is formulated as a multi-objective optimization problem, in which both the mean and the standard deviation of the performance index are simultaneously minimized. A comparison of the effectiveness and robustness of tuned mass dampers designed using local or global performance indices is carried out, considering different levels of uncertainty.