Shape optimization of the initial blank in the sheet metal forming process using equivalent static loads

In the sheet metal forming process, forming the final desired shape is difficult to obtain due to wrinkling, tearing, failure of material, etc. Various conditions of the forming process should be controlled for the desired shape. These conditions are the velocity of the punch, the friction factor, the blank holding force, the initial shape of the blank and others. Many researchers have conducted studies to predetermine the initial blank shape. The structural optimization technique is one of them. Non‐linear response structural optimization is required because non‐linearities are involved in the analysis of the metal forming process. When the conventional method is utilized, the cost is extremely high due to repeated non‐linear analysis for function and sensitivity calculation. In this paper, the equivalent static loads (ESLs) method is used to determine the blank shape which leads to the final desired shape and reduced wrinkling. The ESLs method is a structural optimization method where non‐linear dynamic loads are transformed into ESLs, and these ESLs are utilized as external loads in linear response optimization. The design is updated in linear response optimization. Non‐linear analysis is performed with the updated design and the process proceeds in a cyclic manner. An optimization formulation is defined for the examples, the formulated problems are solved to verify the proposed method and the results are discussed. Non‐linear analysis is performed using the commercial software LS‐DYNA, NASTRAN is used for calculating the ESLs and linear response optimization, and an interface program for LS‐DYNA and NASTRAN is developed. Copyright © 2010 John Wiley & Sons, Ltd.     

Transformation of dynamic loads into equivalent static loads based on modal analysis

All the forces in the real‐world act dynamically on structures. Since dynamic loads are extremely difficult to handle in analysis and design, static loads are usually utilized with dynamic factors. Static loads are especially exploited well in structural optimization where many analyses are carried out. However, the dynamic factors are not determined logically. Therefore, structural engineers often come up with unreliable solutions. An analytical method based on modal analysis is proposed for the transformation of dynamic loads into Equivalent Static Loads (ESLs). The ESLs are calculated to generate an identical displacement field with that from dynamic loads at a certain time. The process is derived and evaluated mathematically by using the modal analysis. Since the exact solution is extremely expensive, some approximation methods are proposed. Error analyses have been conducted for the approximation methods. Standard examples for structural design are selected and solved by the proposed method. Applications of the method to structural optimization are discussed. Copyright © John Wiley & Sons, Ltd.     

Isogeometric shape optimization for quasi‐static processes

A framework to solve shape optimization problems for quasi‐static processes is developed and implemented numerically within the context of isogeometric analysis (IGA). Recent contributions in shape optimization within IGA have been limited to static or steady‐state loading conditions. In the present contribution, the formulation of shape optimization is extended to include time‐dependent loads and responses. A general objective functional is used to accommodate both structural shape optimization and passive control for mechanical problems. An adjoint sensitivity analysis is performed at the continuous level and subsequently discretized within the context of IGA. The methodology and its numerical implementation are tested using benchmark static problems of optimal shapes of orifices in plates under remote bi‐axial tension and pure shear. Under quasi‐static loading conditions, the method is validated using a passive control approach with an a priori known solution. Several applications of time‐dependent mechanical problems are shown to illustrate the capabilities of this approach. In particular, a problem is considered where an external load is allowed to move along the surface of a structure. The shape of the structure is modified in order to control the time‐dependent displacement of the point where the load is applied according to a pre‐specified target. Copyright © 2015 John Wiley & Sons, Ltd.     

移动荷载作用下三塔悬索桥的强迫振动研究

针对泰州大桥开展结构动力特性计算,分析了不同移动速度以及不同移动荷载大小对于三塔两跨悬索桥结构动力响应的影响。结果表明,三塔两跨悬索桥的一阶竖弯振动基频约为0.08 Hz,远低于载重汽车的固有频率,不会形成共振条件。简单的移动荷载作用下三塔两跨连续支承悬索桥的内力、位移响应与同等静力荷载作用效果比较接近。移动荷载的质量越大、速度越大,结构的内力、位移响应就越大,并与车重呈近似正比例关系。结构内力、位移的荷载放大系数不随移动荷载质量的大小而改变。     

Optimum design criteria for elastic structures subject to random dynamic loads

A new stochastic optimization method, which makes use of a constraint on structural reliability, is proposed for structures subject to dynamic random loads. A minimum weight problem is posed, in which a constraint condition imposes that the failure probability must be smaller than a given admissible level. The failure is determined by the first crossing outside the safe domain of a suitable structural response vector. The method is used to find the optimal shape of an elastic vertical column supporting a fixed mass positioned on the top, subject to a Gaussian filtered stationary stochastic horizontal acceleration process. The column, with variable annular cross-section, is described by a deterministic elastic multi-degree-of-freedom system. It is assumed that failure is reached when its lateral displacement exceeds an acceptable threshold value. Under this constraint, the structural weight is minimized and the optimal shape is determined for different structural conditions.     

Nonlinear response topology optimization using equivalent static loads—case studies

Nonlinear structural optimization is fairly expensive and difficult, because a large number of nonlinear analyses is required due to the large number of design variables involved in topology optimization. In element density based topology optimization, the low density elements create mesh distortion and the updating of finite element material with low density elements has a severe effect on the optimization results in the next cycles. In order to overcome these difficulties, the equivalent static loads method for nonlinear response structural optimization (ESLSO) primarily used for size and shape optimization has been applied to topology optimization. The nonlinear analysis is performed with the given loading conditions to calculate equivalent static loads (ESLs) and these ESLs are used to perform linear response optimization. In this paper, the authors have presented the results of five case studies with material, geometric and contact nonlinearities showing good agreement and providing justification of the proposed method.     

Bidirectional evolutionary structural optimization for nonlinear structures under dynamic loads

This study aims to develop efficient numerical optimization methods for finding the optimal topology of nonlinear structures under dynamic loads. The numerical models are developed using the bidirectional evolutionary structural optimization method for stiffness maximization problems with mass constraints. The mathematical formulation of topology optimization approach is developed based on the element virtual strain energy as the design variable and minimization of compliance as the objective function. The suitability of the proposed method for topology optimization of nonlinear structures is demonstrated through a series of two- and three-dimensional benchmark designs. Several issues relating to the nonlinear structures subjected to dynamic loads such as material, geometric, and contact nonlinearities are addressed in the examples. It is shown that the proposed approach generates more reliable designs for nonlinear structures.     

复合材料层合板屈曲稳定性优化设计及其灵敏度计算方法

笔者在有限元分析基础上研究了以屈曲稳定性作为约束条件或优化目标的复合材料层合板结构优化设计及其灵敏度分析方法,重点讨论了屈曲临界荷载灵敏度对内力场和载荷的依赖关系及其在铺层优化、尺寸优化和形状优化问题中的不同计算方法,并在JIFEX软件中实现了复杂结构复合材料层合板优化设计方法。数值算例验证了本文算法和程序的有效性。     

冲击荷载下一端简支一端固支高桥墩的动力屈曲

考虑剪切变形和大位移的影响,建立一端简支一端固支高桥墩在冲击荷载作用下的非线性动力学基本方程式;通过位移形函数假设,采用伽辽金积分方法得到了时间变率的动力学控制方程;利用四阶Runge-Kutta法进行数值求解,得到冲击荷载作用下高桥墩的临界荷载、位移响应曲线以及各种荷载和几何参数对临界荷载的影响规律。通过数值算例比较了三角形冲击荷载和矩形冲击荷载作用下的荷载-位移幅值响应曲线;给出了矩形冲击荷载作用下,不同峰值的位移响应曲线、临界荷载随高桥墩柔度的变化曲线、不同冲击持续时间对临界荷载的影响     

Optimization of large structures subjected to dynamic loads with the multiplier method

The multiplier method for optimization of large-scale mechanical and structural systems subjected to dynamic loads is investigated. A large-scale dynamic response optimization problem is formulated and solved in several alternate ways using the first- and second-order forms of the equations of motion. Results are compared with those obtained with the sequential quadratic programming algorithm—a primal method. In all the cases investigated, the multiplier algorithm is more efficient than the primal method. Therefore, it is concluded that the multiplier method is more appropriate for dynamic response optimization of large-scale problems.     

冲击荷载下两端简支高桥墩的动力屈曲

考虑剪切变形和大位移的影响,建立高桥墩在冲击荷载作用下的非线性动力学基本方程式;通过位移形函数假设,采用伽辽金积分方法得到了时间变率的动力学控制方程;利用四阶Runge-Kutta法进行数值求解,得到冲击荷载作用下高桥墩的临界荷载、位移响应曲线以及各种荷载和几何参数对临界荷载的影响规律。通过数值算例比较了三角形冲击荷载和矩形冲击荷载作用下的荷载-位移幅值响应曲线;给出了矩形冲击荷载作用下,不同峰值的位移响应曲线、临界荷载随高桥墩柔度的变化曲线、不同冲击持续时间对临界荷载的影响。     

Optimal structural design under dynamic loads

This paper presents an algorithm for optimal design of elastic structures, subjected to dynamic loads. Finite element, modal analysis and a generalized steepest descent method are employed in developing a computational algorithm. Structural weight is minimized subject to constraints on displacement, stress, structural frequency, and member size. Optimum results for several example problems are presented and compared with those available in the literature.     

窄带随机激励下的结构动力学拓扑优化设计

基于模态截断法和一些近似处理,导出了一套窄带随机均方动响应及其灵敏度公式,再根据工程实际要求,研究了静载荷和窄带随机载荷作用下结构的动力学拓扑优化设计方法和算法,提供的算例显示了本方法的合理性和有效性。     

考虑自振周期的高层混凝土建筑结构动力优化设计

风载对高层建筑结构会产生一定的动力效应，控制这种动力效应的有效方法之一是限制结构的自振周期。本文提出了一种以有限元分析和严格推导的优化准则方法为基础的动力优化设计方法，将结构构件的截面尺寸作为设计变量，在构件单元的强度、构件单元尺寸和自振周期的限制等约束条件下，求解以最小结构重量为目标的最优解。算例表明，此算法十分有效、可靠，可适用于大型高层混凝土建筑结构的优化设计。     

A topological optimization approach for structural design of a high‐speed low‐load mechanism using the equivalent static loads method

In high‐speed low‐load mechanisms, the principal loads are the inertial forces caused by the high accelerations and velocities. Hence, mechanical design should consider lightweight structures to minimize such loads. In this paper, a topological optimization method is presented on the basis of the equivalent static loads method. Finite element (FE) models of the mechanism in different positions are constructed, and the equivalent loads are obtained using flexible multibody dynamics simulation. Kinetic DOFs are used to simulate the motion joints, and a quasi‐static analysis is performed to obtain the structural responses. The element sensitivity is calculated according to the static‐load‐equivalent equilibrium, in such a way that the influence on the inertial force is considered. A dimensionless component sensitivity factor (strain energy caused by unit load divided by kinetic energy from unit velocity) is used, which quantifies the significance of each element. Finally, the topological optimization approach is presented on the basis of the evolutionary structural optimization method, where the objective is to find the maximum ratio of strain energy to kinetic energy. In order to show the efficiency of the presented method, we presented two numerical cases. The results of these analyses show that the presented method is more efficient and can be easily implemented in commercial FE analysis software. Copyright © 2011 John Wiley & Sons, Ltd.     

Arrays of dynamic circular loads moving on an infinite plate

Steady‐state dynamic responses of a Kirchhoff plate resting on a viscoelastic Winkler foundation subject to arrays of moving constant and harmonic loads with uniform circular distribution is studied in this paper using Fourier transform and generalized Duhamel's integral. A computational schema is constructed based on a two‐fold fast Fourier transform in a moving co‐ordinate system. Numerical case studies using a single moving load as well as four moving loads mimicking four wheel‐loads of a moving vehicle are conducted. Parameters of the plate are typical values of structural and material properties of highway pavements. Displacement responses of the plate to these five loading speeds seem to be very close to each other, though the vertical velocity and acceleration responses show considerable differences. The effect of load frequency on amplitude of response of plate is insignificant for parameters used in this study. Copyright © 2006 John Wiley & Sons, Ltd.     

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。     

Comparison of robust,reliability-based and non-probabilistic topology optimization under uncertain loads and stress constraints

It is nowadays widely acknowledged that optimal structural design should be robust with respect to the uncertainties in loads and material parameters. However, there are several alternatives to consider such uncertainties in structural optimization problems. This paper presents a comprehensive comparison between the results of three different approaches to topology optimization under uncertain loading, considering stress constraints: (1) the robust formulation, which requires only the mean and standard deviation of stresses at each element; (2) the reliability-based formulation, which imposes a reliability constraint on computed stresses; (3) the non-probabilistic formulation, which considers a worst-case scenario for the stresses caused by uncertain loads. The information required by each method, regarding the uncertain loads, and the uncertainty propagation approach used in each case is quite different. The robust formulation requires only mean and standard deviation of uncertain loads; stresses are computed via a first-order perturbation approach. The reliability-based formulation requires full probability distributions of random loads, reliability constraints are computed via a first-order performance measure approach. The non-probabilistic formulation is applicable for bounded uncertain loads; only lower and upper bounds are used, and worst-case stresses are computed via a nested optimization with anti-optimization. The three approaches are quite different in the handling of uncertainties; however, the basic topology optimization framework is the same: the traditional density approach is employed for material parameterization, while the augmented Lagrangian method is employed to solve the resulting problem, in order to handle the large number of stress constraints. Results are computed for two reference problems: similarities and differences between optimized topologies obtained with the three formulations are exploited and discussed.     

一种改进的广义遗传算法及其在结构动力优化问题中的应用

该文提出了一种改进的广义遗传算法。算法中引入了异种机制以提高种群的多样性,在保证收敛速度的同时防止早熟收敛。该方法应用于随机风载荷作用下有应力约束的多参数结构动力响应优化问题,数值算例表明:异种机制能够有效地提高广义遗传算法收敛于全局最优解的概率并加快收敛速度;带有异种机制的广义遗传算法能够有效地求解复杂的结构动力优化问题。