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.     

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.     

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。     

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.     

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.     

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     

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.     

CO-EVOLUTIONARY HYBRID DIFFERENTIAL EVOLUTION FOR MIXED-INTEGER OPTIMIZATION PROBLEMS

Evolutionary algorithms are promising candidates for obtaining the global optimum. Hybrid differential evolution is one or the evolutionary algorithms, which has been successfully applied to many real-world nonlinear programming problems. This paper proposes a co-evolutionary hybrid differential evolution to solve mixed-integer nonlinear programming (MINLP) problems. The key ingredients of the algorithm consist of an integer-valued variable evolution and a real-valued variable co-evolution, so that the algorithm can be used to solve MINLP problems or pure integer programming problems. Furthermore, the algorithm combines a local search heuristic (called acceleration) and a widespread search heuristic (called migration) to promote the search for a global optimum. Some numerical examples are tested to illustrate the performance of the proposed algorithm. Numerical examples show that the proposed algorithm converges to better solutions than the conventional MINLP optimization methods     

GENERATING OPTIMAL CONFIGURATIONS IN STRUCTURAL DESIGN USING SIMULATED ANNEALING

This paper presents a combinatorial optimization procedure based on the simulated annealing approach for generation of optimal configuration of structural members. The work is based on altering the finite element model of structure by removing or restoring elements to minimize the material use subject to constraints on maximum stress value and maintenance of connectivity between elements. Such an optimization problem is categorized as a large-scale, non-convex and non-linear problem. Thus, the problem can have multi-minima and it is important to find the global optimum solution as opposed to a local minimization. To improve the computational efficiency, the non-linear shape optimization problem has been linearized and to account for the difference between the non-linear and the linearized values a correction factor is implemented. To illustrate the approach, several design examples are presented and the effect of the parameter of the simulated annealing on the final configuration design is examined. © 1997 by John Wiley & Sons, Ltd.     

空间桁架结构动力学形状优化设计

在多阶固有频率约束条件下,采用“渐进结点移动法”,对空间桁架结构的形状进行优化设计,使结构重量达到最小。首先分析固有频率相对结点位置的一阶导数,确定结点移动的效率,即灵敏度数。根据灵敏度分析结果,优先移动效率较高的结点。然后,利用库恩-塔克优化条件检验所得结果,保证优化过程收敛于最小重量设计。最后,用二个典型数值算例验证本文算法的有效性和可靠性。     

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。     

基于虚载荷变量的无网格法形状优化的研究

将选择施加在"虚结构"控制点上的虚载荷作为形状优化的设计变量,并将它与无网格Galerkin法相结合来开展结构形状优化研究,采用罚函数法来施加边界条件,通过直接微分法建立了结构形状优化的离散型灵敏度分析算法,利用无网格法研究了节点坐标关于设计变量导数的计算。所提出的算法简单明了,它不仅解决了网格的畸变问题,而且简化了优化模型和迭代流程,并可使结构的受力特性得到进一步的改善。最后用2个工程实例验证了所建立的算法,并得到了形状优化结果。     

NEW DEVELOPMENTS IN STRUCTURAL OPTIMIZATION USING ADAPTIVE MESH REFINEMENT AND MULTIPOINT APPROXIMATIONS

Combining Shape Optimization (SO) with Adaptive Mesh Refinement (AMR) potentially offers a higher accuracy and higher computational efficiency, especially if the applied target error for AMR is reduced in the course of the optimization process. The disadvantage of that approach is that the rate of convergence of the corresponding optimization processes can be significantly lower as compared to processes which apply a fixed target error for AMR. In the present paper the so-called Multipoint Approximation Method (MAM) is used as a basis for SO in conjunction with AMR. Several techniques for improvement of the rates of convergence are presented and investigated. Firstly, alternative algorithms for determining the approximation functions using a weighted least squares method are investigated. The focus is on weights which depend on the discretization errors. Secondly, different strategies for moving and resizing the search sub-regions in the space of design variables are presented. The proposed methods are illustrated by means of several optimization problems in which the effect of AMR with changing discretization errors is modelled by artificially introduced numerical noise.     

NON-LINEAR STRUCTURAL RESPONSE USING ADAPTIVE DYNAMIC RELAXATION ON A MASSIVELY PARALLEL-PROCESSING SYSTEM

A parallel adaptive dynamic relaxation (ADR) algorithm has been developed for non-linear structural analysis. This algorithm has minimal memory requirements, is easily parallelizable and scalable to many processors, and is generally very reliable and efficient for highly non-linear problems. Performance evaluations on single-processor computers have shown that the ADR algorithm is reliable and highly vectorizable, and that it is competitive with direct solution methods for the highly non-linear problems considered. The present algorithm is implemented on the 512-processor Intel Touchstone DELTA system at Caltech, and it is designed to minimize the extent and frequency of interprocessor communication. The algorithm has been used to solve for the non-linear static response of two- and three-dimensional hyperelastic systems involving contact. Impressive relative speed-ups have been achieved and demonstrate the high scalability of the ADR algorithm. For the class of problems addressed, the ADR algorithm represents a very promising approach for parallel-vector processing.