Minimum cost design of framed structures by the mini-max dual method

Minimum cost design of a framed structure is considered by using the mini-mas dual method. Stress and/or displacement constraints are imposed as behavioural constraints. The minimum cost design problem has a discrete objective function and discrete design variables. A sequence of approximate optimization problems in created by using the first-order Taylor series expansion for displacements with respect to the reciprocals of cross-sectional areas and moments of inertia. Each approximate problem is solved in the dual space. Two simple structural examples are given to show the appropriateness and efficiency of the proposed procedure. Approximate solutions are obtained within five structural analysis.     

区间参数结构动力优化的改进方法

针对区间参数结构,提出一种改进的动力响应的区间优化方法。由于区间优化问题一般要比确定性优化问题的求解复杂得多,因此,通过优化结构动力响应区间值的上界,将区间优化问题转化为近似的确定性优化问题。为了得到结构动力响应更加准确的区间值,把结构动力响应Taylor展开式中的一阶导数也看成区间的,这样得到的区间值能近似包含精确值。在区间优化方法中,设计变量的中值和半径都被选为优化变量,可以得到比传统确定性优化方法更多的优化信息。把该方法应用于典型刚架结构,优化结果表明,区间优化方法不仅能得到与传统优化方法大致相当的设计变量最优值,还能得到实际问题中当设计变量取不到最优值而有微小变化时,目标函数值的一个变化范围。     

Dual methods for discrete structural optimization problems

The purpose of this paper is to present a mathematical programming method developed to solve structural optimization problems involving discrete variables. We work in the following context: the structural responses are computed by the finite elements method and convex and separable approximation schemes are used to generate a sequence of explicit approximate subproblems.Each of them is solved in the dual space with a subgradient‐based algorithm (or with a variant of it) specially developed to maximize the not everywhere differentiable dual function. To show that the application field is large, the presented applications are issued from different domains of structural design, such as sizing of thin‐walled structures, geometrical configuration of trusses, topology optimization of membrane or 3‐D structures and welding points numbering in car bodies. The main drawback of using the dual approach is that the obtained solution is generally not the global optimum. This is linked to the presence of a duality gap, due to the non‐convexity of the primal discrete subproblems. Fortunately, this gap can be quantified: a maximum bound on its value can be computed. Moreover, it turns out that the duality gap is decreasing for higher number of variables; the maximum bound on the duality gap is generally negligible in the treated applications. The developed algorithms are very efficient for 2‐D and 3‐D topology optimization, where applications involving thousands of binary design variables are solved in a very short time. Copyright © 2000 John Wiley & Sons, Ltd.     

Efficient approximation concepts using second order information

The primary goal of this paper is to show how second derivative information can be used in an effective way in structural optimization problems. The basic idea is to generate such an information at the expense of only one more ‘virtual load case’ in the sensitivity analysis part of the finite element code. To achieve this goal a primal–dual approach is employed, that can also be interpreted as a sequential quadratic programming method. Another objective is to relate the proposed method to the well known family of approximation concepts techniques, where the primary optimization problem is transformed into a sequence of non-linear explicit subproblems. When restricted to diagonal second derivatives, the new approach can be viewed as a recursive convex programming method, similar to the ‘Convex Linearization’ method (CONLIN), and to its recent generalization, the ‘Method of Moving Asymptotes’ (MMA). This new method has been successfully tested on simple problems that can be solved in closed form, as well as on sizing optimization of trusses. In all cases the method converges faster than CONLIN, MMA or other approximation techniques based on reciprocal variables.     

基于遗传算法的离散型结构拓扑优化设计

采用遗传算法求解包括桁架结构和框架结构的离散型结构拓扑优化问题。在遗传算法的基础上,通过引入拓扑变量并修改被删除杆件的材料弹性模量,提出了一个受多工况荷载作用,能同时考虑应力、稳定及位移等约束的离散型结构拓扑优化问题统一数学模型。该模型不但能同时适用于桁架结构和框架结构等离散型结构拓扑优化问题,而且还能解决奇异最优解问题。结合上述统一数学模型和遗传算法,给出了求解离散型结构拓扑优化问题的优化方法。算例结果表明,采用该文提出的拓扑优化方法可有效、方便地对桁架结构、框架结构等离散型结构进行拓扑优化设计。     

Implementation of Discrete Capability into the Enhanced Multipoint Approximation Method for Solving Mixed Integer-Continuous Optimization Problems

Multipoint approximation method (MAM) focuses on the development of metamodels for the objective and constraint functions in solving a mid-range optimization problem within a trust region. To develop an optimization technique applicable to mixed integer-continuous design optimization problems in which the objective and constraint functions are computationally expensive and could be impossible to evaluate at some combinations of design variables, a simple and efficient algorithm, coordinate search, is implemented in the MAM. This discrete optimization capability is examined by the well established benchmark problem and its effectiveness is also evaluated as the discreteness interval for discrete design variables is increased from 0.2 to 1. Furthermore, an application to the optimization of a lattice composite fuselage structure where one of design variables (number of helical ribs) is integer is also presented to demonstrate the efficiency of this capability.     

An engineering method for complex structural optimization involving both size and topology design variables

This work presents an engineering method for optimizing structures made of bars, beams, plates, or a combination of those components. Corresponding problems involve both continuous (size) and discrete (topology) variables. Using a branched multipoint approximate function, which involves such mixed variables, a series of sequential approximate problems are constructed to make the primal problem explicit. To solve the approximate problems, genetic algorithm (GA) is utilized to optimize discrete variables, and when calculating individual fitness values in GA, a second-level approximate problem only involving retained continuous variables is built to optimize continuous variables. The solution to the second-level approximate problem can be easily obtained with dual methods. Structural analyses are only needed before improving the branched approximate functions in the iteration cycles. The method aims at optimal design of discrete structures consisting of bars, beams, plates, or other components. Numerical examples are given to illustrate its effectiveness, including frame topology optimization, layout optimization of stiffeners modeled with beams or shells, concurrent layout optimization of beam and shell components, and an application in a microsatellite structure. Optimization results show that the number of structural analyses is dramatically decreased when compared with pure GA while even comparable to pure sizing optimization.     

Constrained minimization of vibrational magnitudes using a modal reduction method

A numerically effective method is suggested and applied for evaluating objective and constraint functions when so-called vibrational magnitudes of a mechanical structure are minimized. General damped linear structures under external harmonic loading are considered. The magnitude functions studied can relate to displacements, velocities and accelerations and also to sectional and reactive forces. Both magnitudes at a specific frequency and peak magnitudes and averaged magnitudes over a frequency range are investigated. An arbitrary set of magnitude functions can be used in the constraints. Design variables are masses, dampings and stiffnesses of discrete and discretized continuous elements contained in the structure. The objective and constraint functions are expressed by use of the modal parameters (generally complex-valued) of the structural system. A reduced modal model is established and updated during the optimization process. Approximate derivatives (sensitivities) of the objective and constraint functions with respect to changes in design variables are calculated employing perturbed modal parameters. The optimization problem is solved by use of a primal method. Numerical examples demonstrate applications to the classical damped vibration absorber with two design variables and to a beam system used in a light-weight machine foundation with 14 design variables.     

A new harmony search algorithm for solving mixed–discrete engineering optimization problems

In general design optimization problems, it is usually assumed that the design variables are continuous. However, many practical problems in engineering design require considering the design variables as integer or discrete values. The presence of discrete and integer variables along with continuous variables adds to the complexity of the optimization problem. Very few of the existing methods can yield a globally optimal solution when the objective functions are non-convex and non-differentiable. This article presents a mixed–discrete harmony search approach for solving these nonlinear optimization problems which contain integer, discrete and continuous variables. Some engineering design examples are also presented to demonstrate the effectiveness of the proposed method.     

基于响应面法的无轨伸缩式门式起重机轻量化设计

依赖于传统经验方法的无轨伸缩式门式起重机结构设计往往趋于保守,材料浪费严重,为了充分发挥材料的承载性能,有必要对它进行优化分析。对无轨伸缩式门式起重机结构进行优化设计时,针对非线性接触模型计算不收敛与计算效率低等问题,提出以节点耦合模型代替接触模型来构建响应面近似模型,并将近似模型与多岛遗传算法相结合对结构参数进行优化。以某无轨伸缩式门式起重机各构件截面尺寸为初始设计变量,以结构强度、刚度、自重为模型响应,通过调用Isight平台中最优拉丁超立方设计方法产生设计变量样本点,再由试验设计（design of experiment,DOE）模块调用有限元分析软件ANSYS完成样本点模型仿真以筛选出对响应影响较大的设计变量。利用优选后的设计变量构建响应面近似模型,以结构强度和刚度为约束条件,门式起重机结构质量最轻为优化目标,采用多岛遗传算法对响应面模型进行优化。结果表明：在保证结构性能的前提下,优化后门式起重机质量减轻23.4%,轻量化效果明显。所提出的优化策略可较快获得全局最优解,减少了计算量,为无轨伸缩式门式起重机结构改进提供了理论依据。     

Structure optimization of connection frames based on frequency sensitivity in macro-micro motion platforms

High-performance connection frames are of great significance for ultra-high acceleration and ultra-precision positioning in macro-micro motion platforms. This paper first takes the connection frame as a research object,builds a finite element model(FEM) of the natural frequency of the frame, and then verifies the correctness of this model. The frequency sensitivity method is then used to perturb the structural parameters of the FEM of the connection frame, and the sensitivities of the first-order natural frequency and mass of the corresponding structural parameters are obtained by calculation and analysis. The design variables are also determined. The natural frequency is used as the optimization objective, and the design parameters and mass of the connection frame are constrained. The structural parameters of the connecting frame are obtained through optimization, and the model is built and verified by experiments. The results show that the first-order natural frequency of the connecting frame is effectively improved by the frequency sensitivity method, avoids resonance between the connecting frame and the voice coil motor, and realizes the lightweight design of the connection frame. This research provides a reliable basis for the stable operation and ultra-precision positioning of ultra-high acceleration macro-motion platforms.     

Optimum design of concrete cable-stayed bridges

The design of cable-stayed bridges involves a significant number of design variables and design objectives. The concrete cable-stayed bridge optimization is formulated here as a multi-objective optimization problem with objectives of minimum cost, minimum deflections and minimum stresses. A numerical method is developed to obtain the optimum design of such structures. This numerical method includes: structural analysis, sensitivity analysis and optimization. The structural analysis accounts for all the relevant effects (concrete time-dependent effects, construction stages and geometrical nonlinear effects). The structural response to changes in the design variables is achieved by a discrete direct sensitivity analysis procedure, and an entropy-based approach was used for structural optimization. The features and applicability of the proposed method are demonstrated by numerical examples concerning the optimization of a real-sized concrete cable-stayed bridge.     

An efficient algorithm for the optimum design of trusses with discrete variables

A method to efficiently solve the problem of minimum weight design of plane and space trusses with discrete or mixed variables is developed. The method can also be applied to continuous variables. The original formulation leads to a non-linear constrained minimization problem with inequality constraints, which is solved by means of a sequence of approximate problems using dual techniques. In the dual space, the objective function is to be maximized, depends on continuous variables, is concave and has first and second order discontinuities. In addition, the constraints deal simply with restricting the dual variables to be non-negative. To solve the problem an ad hoc algorithm from mathematical programming has been adapted. Some examples have been developed to show the effectiveness of the method.     

Single and multiobjective structural optimization in discrete-continuous variables using simulated annealing

A multivariable optimization technique based on the Monte-Carlo method used in statistical mechanics studies of condensed systems is adapted for solving single and multiobjective structural optimization problems. This procedure, known as simulated annealing, draws an analogy between energy minimization in physical systems and objective function minimization in structural systems. The search for a minimum is simulated by a relaxation of the statistical mechanical system where a probabilistic acceptance criterion is used to accept or reject candidate designs. To model the multiple objective functions in the problem formulation, a cooperative game theoretic approach is used. Numerical results obtained using three different annealing strategies for the single and multiobjective design of structures with discrete-continuous variables are presented. The influence of cooling schedule parameters on the optimum solutions obtained is discussed. Simulation results indicate that, in several instances, the optimum solutions obtained using simulated annealing outperform the optimum solutions obtained using some gradient-based and discrete optimization techniques. The results also indicate that simulated annealing has substantial potential for additional applications in optimization, especially for problems with mixed discrete-continuous variables.     

Optimal placement of actuators and sensors in control-augmented structural optimization

A control-augmented structural synthesis methodology is presented in which actuator and sensor placement is treated in terms of (0, 1) variables. Structural member sizes and control variables are treated simultaneously as design variables. A multiobjective utopian approach is used to obtain a compromise solution for inherently conflicting objective functions such as structural mass, control effort and number of actuators. Constraints are imposed on natural frequencies, peak transient displacements and accelerations, peak actuator forces and dynamic stability as well as controllability and observability of the system. The combinatorial aspects of the mixed (0, 1)-continuous variable design optimization problem are made tractable by combining approximation concepts with branch and bound techniques. Some numerical results for example problems are presented to illustrate the efficacy of the design procedure set forth.     

基于变频率区间约束的结构材料优化设计

针对频率约束的结构材料优化问题,基于结构拓扑优化思想,提出变频率区间约束的结构材料优化方法。借鉴均匀化及ICM(独立、连续、映射)方法,以微观单元拓扑变量倒数为设计变量,导出宏观单元等效质量矩阵及导数,进而获得频率一阶近似展开式。结合变频率区间约束思想,获得以结构质量为目标函数、频率为约束条件的连续体微结构拓扑优化近似模型;采用对偶方法求解。通过算例验证该方法的有效性及可行性,表明考虑质量矩阵变化影响所得优化结果更合理。     

Design optimization of cold-formed steel portal frames taking into account the effect of building topology

Cold-formed steel portal frames are a popular form of construction for low-rise commercial, light industrial and agricultural buildings with spans of up to 20 m. In this article, a real-coded genetic algorithm is described that is used to minimize the cost of the main frame of such buildings. The key decision variables considered in this proposed algorithm consist of both the spacing and pitch of the frame as continuous variables, as well as the discrete section sizes. A routine taking the structural analysis and frame design for cold-formed steel sections is embedded into a genetic algorithm. The results show that the real-coded genetic algorithm handles effectively the mixture of design variables, with high robustness and consistency in achieving the optimum solution. All wind load combinations according to Australian code are considered in this research. Results for frames with knee braces are also included, for which the optimization achieved even larger savings in cost.     

Continuum structural topological optimizations with stress constraints based on an active constraint technique

Stress‐related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. Continuum structural topological optimization with stress constraints is of wide engineering application prospect, in which there still are many problems to solve, such as the stress concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization method of continuum structures with the stress constraints and the objective function being the structural volume has been presented in this paper. To solve the stress concentration issue, an approximate stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient constraint is constructed for the optimized structure under the r?th load case. To obtain stable convergent series solutions and enhance the control on the stress level, two p‐norm global stress constraint functions with different indexes are adopted, and some weighting p‐norm global stress constraint functions are introduced for any load case. And an equivalent topological optimization model with reduced stress constraints is constructed,being incorporated with the rational approximation for material properties, an active constraint technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed, based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level optimization problem with artificial variables and many possible non‐active design variables is proposed by adopting an inequality constrained nonlinear programming method with simple trust regions, based on the primal‐dual theory, in which the non‐smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a two‐level optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed to deal with the aggregation constraints that always are of loose constraint (non active constraint) features in the conventional structural optimization method. A novel structural topological optimization method with stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed method is feasible and very effective. © 2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

Simultaneous optimization of force and placement of friction dampers under seismic loading

It is known that the use of passive energy-dissipation devices, such as friction dampers, reduces considerably the dynamic response of a structure subjected to earthquake ground motions. Nevertheless, the parameters of each damper and the best placement of these devices remain difficult to determine. Some articles on optimum design of tuned mass dampers and viscous dampers have been published; however, there is a lack of studies on optimization of friction dampers. The main contribution of this article is to propose a methodology to simultaneously optimize the location of friction dampers and their friction forces in structures subjected to seismic loading, to achieve a desired level of reduction in the response. For this purpose, the recently developed backtracking search optimization algorithm (BSA) is employed, which can deal with optimization problems involving mixed discrete and continuous variables. For illustration purposes, two different structures are presented. The first is a six-storey shear building and the second is a transmission line tower. In both cases, the forces and positions of friction dampers are the design variables, while the objective functions are to minimize the interstorey drift for the first case and to minimize the maximum displacement at the top of the tower for the second example. The results show that the proposed method was able to reduce the interstorey drift of the shear building by more than 65% and the maximum displacement at the top of the tower by approximately 55%, with only three friction dampers. The proposed methodology is quite general and it could be recommended as an effective tool for optimum design of friction dampers for structural response control. Thus, this article shows that friction dampers can be designed in a safe and economic way.     

离散变量结构优化的拟满应力设计方法

本文以满应力设计思想为基础,提出了适用于离散变量结构优化设计计算的拟满应力设计方法。该方法能直接计算具有应力约束和截面尺寸约柬的离散变量结构优化设计问题,也能处理同时具有稳定性约束和位移约束的多工况、多约束、多变量的离散变量结构优化设计问题。算例结果表明,拟满应力设计方法对于离散变量结构优化计算是非常有效的。