A framework for parallelized efficient global optimization with application to vehicle crashworthiness optimization

This article presents a framework for simulation-based design optimization of computationally expensive problems, where economizing the generation of sample designs is highly desirable. One popular approach for such problems is efficient global optimization (EGO), where an initial set of design samples is used to construct a kriging model, which is then used to generate new ‘infill’ sample designs at regions of the search space where there is high expectancy of improvement. This article attempts to address one of the limitations of EGO, where generation of infill samples can become a difficult optimization problem in its own right, as well as allow the generation of multiple samples at a time in order to take advantage of parallel computing in the evaluation of the new samples. The proposed approach is tested on analytical functions, and then applied to the vehicle crashworthiness design of a full Geo Metro model undergoing frontal crash conditions.     

响应表面法在薄壁构件耐撞性优化设计中的应用研究

汽车结构的耐撞性及碰撞吸能优化是现代汽车工业重要的研究内容。耐撞性的优化涉及材料与结构的众多参数。传统的设计、碰撞仿真及试验往往只能在一定程度上改善结构的碰撞性能而无法达到限定条件下的最优状态。利用国际上近年来新发展起来的一种优化理论方法--响应表面法,结合传统的优化手段以及非线性有限元程序对薄壁构件的耐撞性问题进行了优化研究。耐撞性优化的结果表明,该方法具有较高的精确性和有效性。     

Topology optimization for crashworthiness of thin-walled structures under axial crash considering nonlinear plastic buckling and locations of plastic hinges

Although topology optimization is established for linear static problems, more effort is required for solving nonlinear plastic problems. A new topology optimization approach with equivalent static loads (ESLs) is suggested to find the optimum topologies and locations of plastic hinges of thin-walled crash boxes by considering crash-induced deformation, the main crash energy-absorbing mechanism. Together with finite element method crashworthiness analyses, considering all nonlinearities with rate-dependent plasticity, the method was developed using an appropriate time-incremental scheme of ESLs without removing any high values of loads. Analyses show that the crash boxes with optimum topologies have energy-absorbing capabilities equivalent to the original structure. The proposed method is evaluated for two crashes: a crash box at low speed and a double cell subjected to high-speed collision. The results indicate that this method captures nonlinear crushing behaviours and accurate locations of plastic hinges where, if proper reinforcements are made, energy absorption can be enhanced.     

An optimization design of vehicle axle system based on multiobjective cooperative optimization algorithm

ABSTRACT

To address multiobjective, multi constraint and time-consuming structural optimization problems in a vehicle axle system, a multiobjective cooperative optimization model of a vehicle axle structure is established. In light of the difficulty in the nondominated sorting of the NSGA-II algorithm caused by inconsistent effects of the uniformity objective function and physical objective function, this paper combines a multiobjective genetic algorithm with cooperative optimization and presents a strategy for handling the optimization of a vehicle axle structure. The uniformity objective function of the sub discipline is transformed to its self-constraint. Taking the multiobjective optimization of a vehicle axle system as an example, a multiobjective cooperative optimization design for the system is carried out in ISIGHT. The results show that the multiobjective cooperative optimization strategy can simplify the complexity of optimization problems and that the multiobjective cooperative optimization method based on an approximate model is favorable for accuracy and efficiency, thereby providing a theoretical basis for the optimization of similar complex structures in practical engineering.     

汽车前纵梁吸能盒结构耐撞性多目标优化

吸能盒上诱导槽的分布形式对汽车碰撞性能有很大影响.以汽车前纵梁吸能盒结构为对象,建立整车正面碰撞仿真模型,对吸能盒结构的耐撞性进行了多目标优化设计.以吸能盒上诱导槽之间的间距为设计变量,使用Hammersley方法采集样本点后,以车辆吸能盒的最大吸能量E、最大刚性墙反力F以及车身最大加速度a为目标函数,通过krigin...     

An adaptive response surface method for crashworthiness optimization

Response surface-based design optimization has been commonly used for optimizing large-scale design problems in the automotive industry. However, most response surface models are built by a limited number of design points without considering data uncertainty. In addition, the selection of a response surface in the literature is often arbitrary. This article uses a Bayesian metric to systematically select the best available response surface among several candidates in a library while considering data uncertainty. An adaptive, efficient response surface strategy, which minimizes the number of computationally intensive simulations, was developed for design optimization of large-scale complex problems. This methodology was demonstrated by a crashworthiness optimization example.     

Multi-objective robust optimization design of a front-end underframe structure for a high-speed train

A multi-objective robust design optimization of a front-end underframe structure for application in high-speed trains is proposed and the structural parameter uncertainty is considered. A finite element model of the structure is developed and verified by dynamic impact experiments. The sensitivity analysis demonstrates that the thicknesses of the centre sill have significant influences on structural crushing behaviours. The specific energy absorption and the initial peak crushing force (F_p) are taken as optimization objectives. Compared with the baseline structure, the 6-sigma robust design shows that the F_p and the structural mass are reduced by 54.86% and 13.06%, respectively, and the robust optimum is more reliable. The 6-sigma robust optimal solution has an efficient energy-absorbing capacity while satisfying the design constraint. Thus, 6-sigma robust optimization can be applied for high-speed trains.     

Automated structural optimization system for integrated topology and shape optimization

Abstract

This paper combines previously developed techniques for image‐preprocessing and characteristic image‐interpreting together with a newly proposed automated shape‐optimization modeling technique into an integrated topology‐optimization and shape‐optimization system. As a result, structure designers are provided with an efficient and reliable automated structural optimization system (ASOS). The automated shape‐optimization modeling technique, the key technique in ASOS, uses hole‐expanding strategy, interference analysis, and hole shape‐adjusting strategy to automatically define the design variables and side constraints needed for shape optimization. This technique not only eliminates the need to manually define design variables and side constraints for shape optimization, but during the process of shape optimization also prevents interference between the interior holes and the exterior boundary. The ASOS is tested in three different structural configuration design examples.     

Lightweight and crashworthiness design of an electric vehicle using a six-sigma robust design optimization method

Design optimization plays an important role in electric vehicle (EV) design. However, fluctuations in design variables and noise factors during the forming process affect the stability of optimization results. This study uses six-sigma robust design optimization to explore the lightweight design and crashworthiness of EVs with uncertainty. A full-scale finite element model of an EV is established. Then, multi-objective design optimization is performed by integrating optimal Latin hypercube sampling, radial basis functions and non-dominated sorting genetic algorithm-II to achieve minimum peak acceleration and mass. Finally, six-sigma robust optimization designs are applied to improve the reliability and sigma level. Robust optimization using adaptive importance sampling is shown to be more efficient than that using Monte Carlo sampling. Moreover, deformation of the battery compartment and peak acceleration of the B-pillar are greatly decreased. The EV’s safety performance is improved and the lightweight effect is remarkable, validating the strong engineering practicability of the method.     

Sequential algorithms for structural design optimization under tolerance conditions

A deterministic optimization usually ignores the effects of uncertainties in design variables or design parameters on the constraints. In practical applications, it is required that the optimum solution can endure some tolerance so that the constraints are still satisfied when the solution undergoes variations within the tolerance range. An optimization problem under tolerance conditions is formulated in this article. It is a kind of robust design and a special case of a generalized semi-infinite programming (GSIP) problem. To overcome the deficiency of directly solving the double loop optimization, two sequential algorithms are then proposed for obtaining the solution, i.e. the double loop optimization is solved by a sequence of cycles. In each cycle a deterministic optimization and a worst case analysis are performed in succession. In sequential algorithm 1 (SA1), a shifting factor is introduced to adjust the feasible region in the next cycle, while in sequential algorithm 2 (SA2), the shifting factor is replaced by a shifting vector. Several examples are presented to demonstrate the efficiency of the proposed methods. An optimal design result based on the presented method can endure certain variation of design variables without violating the constraints. For GSIP, it is shown that SA1 can obtain a solution with equivalent accuracy and efficiency to a local reduction method (LRM). Nevertheless, the LRM is not applicable to the tolerance design problem studied in this article.     

A deterministic global approach for mixed-discrete structural optimization

This study proposes a novel approach for finding the exact global optimum of a mixed-discrete structural optimization problem. Although many approaches have been developed to solve the mixed-discrete structural optimization problem, they cannot guarantee finding a global solution or they adopt too many extra binary variables and constraints in reformulating the problem. The proposed deterministic method uses convexification strategies and linearization techniques to convert a structural optimization problem into a convex mixed-integer nonlinear programming problem solvable to obtain a global optimum. To enhance the computational efficiency in treating complicated problems, the range reduction technique is also applied to tighten variable bounds. Several numerical experiments drawn from practical structural design problems are presented to demonstrate the effectiveness of the proposed method.     

A robust optimization methodology for preliminary aircraft design

This article focuses on a robust optimization of an aircraft preliminary design under operational constraints. According to engineers' know-how, the aircraft preliminary design problem can be modelled as an uncertain optimization problem whose objective (the cost or the fuel consumption) is almost affine, and whose constraints are convex. It is shown that this uncertain optimization problem can be approximated in a conservative manner by an uncertain linear optimization program, which enables the use of the techniques of robust linear programming of Ben-Tal, El Ghaoui, and Nemirovski [Robust Optimization, Princeton University Press, 2009]. This methodology is then applied to two real cases of aircraft design and numerical results are presented.     

Finite mixture modeling for vehicle crash data with application to hotspot identification

The application of finite mixture regression models has recently gained an interest from highway safety researchers because of its considerable potential for addressing unobserved heterogeneity. Finite mixture models assume that the observations of a sample arise from two or more unobserved components with unknown proportions. Both fixed and varying weight parameter models have been shown to be useful for explaining the heterogeneity and the nature of the dispersion in crash data. Given the superior performance of the finite mixture model, this study, using observed and simulated data, investigated the relative performance of the finite mixture model and the traditional negative binomial (NB) model in terms of hotspot identification. For the observed data, rural multilane segment crash data for divided highways in California and Texas were used. The results showed that the difference measured by the percentage deviation in ranking orders was relatively small for this dataset. Nevertheless, the ranking results from the finite mixture model were considered more reliable than the NB model because of the better model specification. This finding was also supported by the simulation study which produced a high number of false positives and negatives when a mis-specified model was used for hotspot identification. Regarding an optimal threshold value for identifying hotspots, another simulation analysis indicated that there is a discrepancy between false discovery (increasing) and false negative rates (decreasing). Since the costs associated with false positives and false negatives are different, it is suggested that the selected optimal threshold value should be decided by considering the trade-offs between these two costs so that unnecessary expenses are minimized.     

用于结构优化设计的改进多目标群搜索算法

在多目标群搜索算法（multi-objective group search optimization, MGSO）基本原理的基础上,结合Pareto最优解理论,提出了基于约束改进的多目标群搜索算法（IMGSO）,并应用于多目标的结构优化设计．算法的改进主要有3个方面：第一,引入过渡可行域的概念来处理约束条件;第二,利用庄家法来构造非支配解集;最后,结合禁忌搜索算法和拥挤距离机制来选择发现者,以避免解集过早陷入局部最优,并提高收敛精度．采用IMGSO优化算法分别对平面和空间桁架结构进行了离散变量的截面优化设计,并与MGSO优化算法的计算结果进行了比较,结果表明改进的多目标群搜索优化算法IMGSO与MGSO算法相比具有更好的收敛精度.通过算例表明：IMGSO算法得到的解集中的解能大部分支配MGSO算法的解,在复杂高维结构中IMGSO算法的优越性更加明显,且收敛速度也有一定的提高,可有效应用于多目标的实际结构优化设计．     

A unified parametric design approach to structural shape optimization

This paper presents a general parametric design approach for 2-D shape optimization problems. This approach has been achieved by integrating practical design methodologies into numerical procedures. It is characterized by three features: (i) automatic selection of a minimum number of shape design variables based on the CAD geometric model; (ii) integration of sequential convex programming algorithms to solve equality constrained optimization problems; (iii) efficient sensitivity analysis by means of the improved semi-analytical method. It is shown that shape design variables can be either manually or systematically identified with the help of equality constraints describing the relationship between geometric entities. Numerical solutions are performed to demonstrate the applicability of the proposed approach. A discussion of the results is also given:     

A mathematical optimization methodology for the optimal design of a planar robotic manipulator

A general optimization methodology for the optimal design of robotic manipulators is presented and illustrated by its application to a realistic and practical three‐link revolute‐joint planar manipulator. The end‐effector carries out a prescribed vertical motion for which, respectively, the average torque requirement from electrical driving motors, and the electric input energy to the driving motors are minimized with respect to positional and dimensional design variables. In addition to simple physical bounds placed on the variables, the maximum deliverable torques of the driving motors and the allowable joint angles between successive links represent further constraints on the system. The optimization is carried out via a penalty function formulation of the constrained problem to which a proven robust unconstrained optimization method is applied. The problem of singularities (also known as degeneracy or lock‐up), which may occur for certain choices of design variables, is successfully dealt with by means of a specially proposed procedure in which a high artificial objective function value is computed for such ‘lock‐up trajectories’. Designs are obtained that are feasible and practical with reductions in the objective functions in comparison to that of arbitrarily chosen infeasible initial designs. Copyright © 1999 John Wiley & Sons, Ltd.     

Multiobjective optimization of multi-cell sections for the crashworthiness design

Plastic deformation of structures absorbs substantial kinetic energy when impact occurs. For this reason, energy-absorbing components have been extensively used in the structural design of vehicles to intentionally absorb a large portion of crash energy to reduce the severe injury of occupants. On the other hand, high peak crushing force may to a certain extent indicate the risk of structural integrity and biomechanical damage of occupants. For this reason, it is of great significance to maximize the energy absorption and minimize the peak force by seeking for optimal design of these components. This paper aims to design the multi-cell cross-sectional thin-walled columns with these two crashworthiness criteria. An explicit finite element analysis (FEA) is used to derive higher-order response surfaces for these two objectives. Both the single-objective and multi-objective optimizations are performed for the single, double, triple and quadruple cell sectional columns under longitudinal impact loading. A comparative analysis is consequently given to explore the relationship between these two design criteria with the different optimization formulations.     

Merits and limitations of optimality criteria method for structural optimization

The merits and limitations of the Optimality Criteria (OC) method for the minimum weight design of structures subjected to multiple load conditions under stress, displacement and frequency constraints were investigated by examining several numerical examples. The examples were solved utilizing the OC design code that was developed for this purpose at the NASA Lewis Research Center. This OC code incorporates OC methods available in the literature with generalizations for stress constraints, fully utilized design concepts, and hybrid methods that combine both techniques. It includes multiple choices for Lagrange multiplier and design variable update methods, design strategies for several constraint types, variable linking, displacement and integrated force method analysers, and analytical and numerical sensitivities. On the basis of the examples solved, the optimality criteria for general application were found to be satisfactory for problems with few active constraints or with small numbers of design variables. However, the OC method without stress constraints converged to optimum even for large structural systems. For problems with large numbers of behaviour constraints and design variables, the method appears to follow a subset of active constraints that can result in a heavier design. The computational efficiency of OC methods appears to be similar to some mathematical programming techniques.     

基于双向渐进结构优化的叶片锻件预成形设计

为了实现体积成形的预成形优化设计,基于双向渐进结构(BESO)优化的思想,提出了一种针对体积成形预成形设计的新方法——拓扑优化法,并详细给出了该方法的优化策略、单元增删准则、插值处理等关键技术.利用自行开发的优化程序,结合DEFORM-2D有限元模拟软件,以理想充填模腔、最小飞边状态为目标,以静水压力的大小作为单元的增删准则,从毛坯的欠填充状态出发,对二维叶片锻件的预成形结构进行了优化设计.优化结果表明:该方法算法原理清晰明确,实现方便,整个过程集成化后,从模拟到优化均可实现自动进行,运行效率高,并具有较高的优化精度.     

Convergence control of the iterative procedure for performance-measure-based probabilistic structural design optimization

The advanced mean value (AMV) iterative scheme is commonly used to evaluate probabilistic constraints in the performance measure approach (PMA) for probabilistic structural design optimization (PSDO). However, the iterative procedure of PSDO may fail to converge. In this article, the chaotic dynamics theory is suggested to investigate and attack the non-convergence difficulties of PMA-based PSDO. Essentially, the AMV iterative formula forms a discrete dynamical system with control parameters. If the AMV iterative sequences present the numerical instabilities of periodic oscillation, bifurcation, and even chaos in some control parameter interval, then the outer optimization loop in PSDO cannot converge and acquire the correct optimal design. Furthermore, the stability transformation method (STM) of chaos feedback control is applied to perform the convergence control of AMV, in order to capture the desired fixed points in the whole control parameter interval. Meanwhile, PSDO is solved by the approaches of PMA two-level and PMA with the sequential approximate programming (SAP)—PMA with SAP. Numerical results of several examples illustrate that STM can smoothly overcome the convergence failure of PSDO resulting from the periodic oscillation, bifurcation, and chaotic solutions of AMV iterative procedure for evaluating the probabilistic constraints. Moreover, the probabilistic optimization with uniform random variables, which is widely recognized as a highly nonlinear and fairly difficult problem, can be attacked through introducing the strategy of chaos control. In addition, the approach of PMA with SAP combining with STM is quite effective and efficient.