EVOLUTION STRATEGIES IN ENGINEERING OPTIMIZATION

The available methods used for the solution of optimization problems in engineering are commonly classified as zero, first and second order methods according to the order of information which is required. In this paper a zero order method based on evolution strategies is described. Evolution strategies are stochastic search methods for solving optimization problems and have their philosophical basis in processes found in nature. The basic evolution strategy is presented first1 followed by generalizations for parallel computing and for the solution of discrete and mixed discrete-continuous problems. Some applications to test problems from the literature and to realistic problems of structural optimization are given.     

SHAPE OPTIMIZATION WITH ADAPTIVE MESH REFINEMENT: TARGET ERROR SELECTION STRATEGIES

The multi-point approximation method in conjunction with Adaptive Mesh Refinement (AMR) for shape optimization of thin-walled structures is studied. Application of AMR is done in such a manner that in the beginning of an optimization process large discretization errors are accepted, while finite element discretizations become more accurate as the optimization process progresses. In this paper several strategies for selecting the target discretization errors are investigated. Special attention is paid to both the overall computational effort and the convergence of the optimization processes.     

SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION FOR STRUCTURAL DURABILITY

In this paper, a design sensitivity analysis (DSA) method for fatigue life of 3-D solid structural components of mechanical systems with respect to shape design parameters is presented. The DSA method uses dynamic stress DSA obtained using an analytical approach to predict dynamic stress increment due to design changes; computes fatigue life of the component, including crack initiation and crack propagation, using the predicted dynamic stress; and uses the difference of the new life and the original life at the same critical point to approximate the sensitivity of fatigue life. A tracked vehicle roadarm is presented in this paper to demonstrate accuracy and efficiency of the DSA method. Also, this method is applied to support design optimization of the tracked vehicle roadarm considering crack initiation lives as design constraints. © 1997 by John Wiley & Sons, Ltd.     

基于并行混沌和复合形法的桁架结构形状优化

针对多工况下受应力、位移和局部稳定性约束的桁架形状优化问题,提出了基于并行混沌优化算法和复合形法的混合优化算法。该算法综合利用了并行混沌的全局搜索能力,复合形法的快速局部搜索能力和混沌细搜索。首先,利用并行混沌优化算法快速搜索到全局最优解附近,然后应用改进复合形法以并行混沌的优化解为初始复形进行搜索,提高了最优解的搜索速度,最后应用混沌细搜索策略提高最优解的精度。两个典型数值算例验证了该混合优化方法对桁架形状优化问题的有效性和稳定性。     

A SELF-SCALING IMPLICIT SQP METHOD FOR LARGE SCALE STRUCTURAL OPTIMIZATION

The basic idea of an implicit sequential quadratic programming (ISQP) method for constrained problems is to use the approximate Hessian of the Lagrangian without explicitly calculating and storing it. This overcomes one of the major drawbacks of the traditional SQP method where a large matrix needs to be calculated and stored. This concept of an implicit method is explained and an algorithm based on it is presented. The proposed method extends a similar algorithm for unconstrained problems where a two-loop recursion formula is used for the inverse Hessian matrix. The present paper develops a similar algorithm for not only the constrained problem but also the direct Hessian updates. Several scaling procedures for the Hessian are also presented and evaluated. The basic method and some of its variations are evaluated using a set of mathematical programming test problems, and a set of structural design test problems—small to larger scale. The ISQP method performs much better than a method that does not use any approximate Hessian matrix. Its performance is better than the full SQP method for larger scale problems. The test results also show that an appropriate scaling of the Hessian can improve both efficiency and reliability substantially.     

SHAPE OPTIMIZATION FOR FATIGUE PROBLEMS IN MECHANICAL ENGINEERING

High Cycle Fatigue (HCF) is a very important topic in mechanical engineering problems. Almost any machine part contains some changes in cross section or boreholes connected with stress concentration. It is the task of shape optimization to improve the strength of machine parts by changing the design at free boundaries. In the past, some work was done with the so-called classical stress concentration factors just to minimize the mentioned stress concentration, also for fatigue. This paper represents a more sophisticated view by including modern Continuum Damage Mechanics (CDM) to model material behavior in HCF. The central question is to design machinery parts so that maximum lifetime is obtained. The whole context for this is developed in this paper, including non-linear finite element algorithms. The paper concludes with numerical and experimental tests for the shape optimization of a notched tension bar in HCF.     

A TWO-LEVEL DECOMPOSITION METHOD FOR SHAPE OPTIMIZATION OF STRUCTURES

The paper proposes a two-level decomposition method for shape optimization of structures. The optimization problem is divided into two subproblems on the basis of the different effects on structural behaviour of different kinds of design variables. A minimum mass subproblem is solved to determine the sizing variables and a constraint evaluation function based on norm optimization is minimized to determine the shape variables. An efficient coupling technique is used between the subproblems to ensure very rapid and steady convergence. Numerical results are presented to demonstrate the effectiveness of the method. © 1997 by John Wiley & Sons, Ltd.     

OPTIMAL STRUCTURAL DESIGN BY ANT COLONY OPTIMIZATION

Ant colony optimization (ACO) is a relatively new heuristic combinatorial optimization algorithm in which the search process is a stochastic procedure that incorporates positive feedback of accumulated information. The positive feedback (;i.e., autocatalysis) facility is a feature of ACO which gives an emergent search procedure such that the (common) problem of algorithm termination at local optima may be avoided and search for a global optimum is possible.

The ACO algorithm is motivated by analogy with natural phenomena, in particular, the ability of a colony of ants to ‘optimize’ their collective endeavours. In this paper the biological background for ACO is explained and its computational implementation is presented in a structural design context. The particular implementation of ACO makes use of a tabu search (TS) local improvement phase to give a computationally enhanced algorithm (ACOTS).

In this paper ACOTS is applied to the optimal structural design, in terms of weight minimization, of a 25-bar space truss. The design variables are the cross-sectional areas of the bars, which take discrete values. Numerical investigation of the 25-bar space truss gave the best (i.e., lowest to-date) minimum weight value. This example provides evidence that ACOTS is a useful and technically viable optimization technique for discrete-variable optimal structural design.     

STRUCTURAL OPTIMIZATION USING A NEW LOCAL APPROXIMATION METHOD

A new method for solving structural optimization problems using a local function approximation algorithm is proposed. This new algorithm, called the Generalized Convex Approximation (GCA), uses the design sensitivity information from the current and previous design points to generate a sequence of convex, separable subproblems. The paper contains the derivation of the parameters associated with the approximation and the formulation of the approximated problem. Numerical results from standard test problems solved using this method are presented. It is observed that this algorithm generates local approximations which lead to faster convergence for structural optimization problems.     

结构可靠性优化求解的解耦融合策略

在工程结构的可靠性优化过程中,求解的效率和精度是优化方法的关键。该文提出一种针对解耦优化的融合策略。所提方法在优化迭代解耦所用的失效概率函数为前几次迭代设计点构建的局部失效概率函数的加权融合形式。在对原可靠性优化问题进行解耦后,结合序列近似优化方法进行迭代求解。相比于常规的仅使用当次局部建立的失效概率函数而言,所提融合策略最大限度利用了各次迭代中产生的信息用于优化解耦求解,能够提高失效概率函数的近似精度,从而间接达到减少迭代次数和计算量的目的。最后给出了屋架和十杆结构的可靠性优化算例,验证该文方法的正确性和可行性。     

叠层复合材料方板多孔形状优化

采用一种无梯度仿生技术——基于等限制Tsai-Hill值准则的固定网格渐进优化方法(FGESO),研究了叠层复合材料方板在拉剪荷载时不同孔数、不同叠层构造条件下的最优孔形问题.在孔的周围不断把限制Tsai-Hill值小于删除标准的材料删除,直到稳定状态达到,然后提高删除标准继续迭代,直到达到指定的开孔面积.与传统渐进优化方法(ESO)的不同之处在于利用节点而不是单元的限制Tsai-Hill值来确定需要删除的材料,因此得到了比ESO更光滑的结果.例子证明了方法的普适性和有效性.还研究了两孔方板最优孔形的优化历程,结果反映了相邻开孔相互影响的一些规律.     

拱坝体形优化中的三维解析敏度

本文针对基于有限元分析的拱坝体形优化，采用改进的半解析法进行三维敏度分析。推导了敏度分析有关公式，并研制了相应的程序。敏度验证和拱坝优化算例表明，本文方法精度高、数值稳定，对基于三维有限元分析的拱坝体形优化是有效的。     

SHAPE OPTIMIZATION IN SHELL THEORY: Design Sensitivity of the Continuous Problem

We consider a non-shallow shell made of an isotropic homogeneous material, working in linear elastic conditions, subjected to a given load. Our aim is to change the shape of the shell so that it resists better towards a given criterion. By shape, we mean essentially the midsurface of the shell. The thickness could be added without any difficulty. The important aspect that we study here is the midsurface.

This problem is worked by gradient type methods. We prove that if the criterion depends on the displacement field through a differentiable function, then it depends on the shape in a differentiable manner, because the displacement field is a differentiable function of the shape. Then we present an analytical formula giving the exact gradient of the criteria before any discretization. After that, we explain how to compute numerically an approximation to this exact gradient. Then we give numerical results.     

PREFORM DIE SHAPE DESIGN IN METAL FORMING USING AN OPTIMIZATION METHOD

An optimization algorithm for preform die shape design in metal-forming processes is developed in this paper. The preform die shapes are represented by cubic B-spline curves. The control points of the B-spline are used as the design variables. The optimization objective is to reduce the difference between the realized and desired final forging shapes. The sensitivities of the objective function with respect to the design variables are developed in detail. The numerical examples show that the optimization method and the sensitivity analysis developed in this paper are very useful and the design results are satisfactory. Importantly, the preform die shapes designed by this method are easily manufacturable and can be implemented in practical metal-forming operations. This optimization method and the sensitivity analysis can also be applied in the preform design of complex industrial metal-forming problems. © 1997 by John Wiley & Sons, Ltd.     

MOST EFFICIENT NLP TECHNIQUES IN OPTIMUM STRUCTURAL FRAME DESIGN

This paper identifies the two most efficient non-linear programming techniques for the solution of optimization problems of plane structural frames under multiple load systems. The problem is one of minimizing the mass or the frame subject to constraints on normal and shear stresses, the maximum transverse deflection and buckling load. In all, five different NLP techniques are tried, and it is found that the sequential linear and sequential convex programming techniques are the most efficient for the solution of the class of NLP problems under consideration     

位移、应力、尺寸约束下二维连续体的形状优化

以二级控制理论为切入点,通过构造设计变量与关键点坐标之间的关系和关键点坐标与节点坐标之间的关系,得到了设计变量和节点坐标之间的关系。根据这一关系建立了目标函数跟设计变量之间的关系,采用有限差分法得到应力和位移的约束函数,建立了满足应力、位移和尺寸约束的形状优化模型,并采用序列二次规划方法求解二维连续体的形状优化问题。以MSC/Nastran软件为结构分析求解器,借助PCL语言开发了二维连续体的形状优化程序。为了有效地避免约束近似造成的迭代振荡乃至发散,在程序实现中,采用了区间因子来调整设计变量的上下限。数值算例表明程序算法的可靠性、精确性、高效性。     

SHAPE OPTIMIZATION AND MINIMUM WEIGHT LIMIT DESIGN OF ARCHES

The paper is concerned with the optimization of arches, using classical beam finite elements, for the minimum elastic displacements and the minimum weight designs under ultimate loading conditions.

The concept of separate but dependent design spaces for node coordinates and member plastic capacities is introduced. The shape optimization problem, whereby a norm of the elastic displacement vector is minimized, is formulated in the space of node coordinate variables. Then the minimum weight limit design problem in the space of member plastic capacities is considered using the static theorem of limit analysis. An iterative procedure alternating these two approaches is presented in the paper. The nonlinear unconstrained optimization and linear programming techniques are used to solve the corresponding numerical problems. The proposed method is illustrated by numerical examples.     

周期性扩大框架的渐进结构优化方法研究

传统渐进结构优化法(ESO)删除的低效材料,仅是阶段性的低效材料,其中部分材料若不被删除,有可能成为后续优化中的非低效材料。误删材料可能使接下来的优化成为“将错就错”的优化。该文在ESO法基础上提出的“周期性扩大框架的渐进结构优化方法(PEFESO)”,周期性地恢复并检查保留材料周围被删除材料的效率,重新判定该部分材料是否应被恢复到结构中,有效地削弱了ESO法可能误删材料的不良影响。PEFESO法具备一定的全程寻优能力,计算结果优于传统的ESO法,易于在工程中推广使用。     

一种新的拱坝形状优化方法

虚拟层合单元可以应用于结构的形状优化.这一单元的主要思想在于引入了弹性模量等于零即无实际材料的虚拟层,基于这一特点,单元内层的厚度被取作设计变量,而决定外部边界的单元节点坐标在优化过程中却始终保持不变.于是,有限元网格只在优化初始生成一次,优化过程无需改变原始数据,由此避免了通常形状优化方法所需要的单元网格重新划分.拱坝形状优化过程中利用有限元法作为结构分析方法,而不是通常的拱梁分载法.1型拱坝的算例说明了这一方法的有效性和实用性.