On the optimal design of elastic beams

In this paper we investigate the optimal dynamics of simply supported nonlinearly elastic beams with rectangular cross-sections. We consider the elastic beam under the assumption of time-dependent intensive transverse loading. The state of the beam is described by a system of partial differential equations of the fourth order. We deal with the problem of choosing the optimal shape for the beam. The optimal shape is determined in such a way that the deflection of the nonlinearly elastic beam for any given time is minimal. The problem of choosing the optimal shape is formulated as an optimal control problem. To solve the obtained problem effectively, we use the optimality principle of Bellman (Bellman and Dreyfus 1962; Bryson and Ho 1975) and the penalty function method (Polyak 1987). We present a constructive algorithm for the optimal design of nonlinearly elastic beams. Some simple examples of the implementation of the proposed numerical algorithm are given.     

Generation of shape functions for straight beam elements

Straight beam finite elements with greater than two nodes are used for edge stiffening in plane stress analyses and elsewhere. It is often necessary to match the number of nodes on the edge stiffener to the number on a whole plane stress element side. Beam elements employ shape functions which are recognised to be level one Hermitian polynomials. An alternative to the commonly adopted method for determining these shape functions is given in this note, using a formula widely reported in mathematical texts which has hitherto not been applied to this task in the finite element literature. The procedure derives shape functions for beams entirely from the set of Lagrangian interpolating polynomials. Examples are given for the derivation of functions for a three and four-noded beam element.     

Dynamic problems of shape optimal design for shallow curved plates

The problem of optimal structural design of shallow thin-walled elements such as curved rectangular plates are formulated and solved for dynamic conditions. The distribution of the initial curvature of shallow plates in a nonstrained state is taken as the control function. Dynamic compliance is considered as the minimized performance functional. Optimality conditions are derived for the distributed parameter system considered and applied for the construction of the analytical solution. The rigorous analysis of extremum conditions and behavioural equations shows that the initial optimization problem is decomposed into several problems of classical structural analysis, which can be successfully solved analytically. Some optimal designs obtained for rectangular plates under stretching and bending, and a plate lying on an elastic foundation and subjected to lateral forces are presented. Received: November 27, 1998     

Optimal design of laminated composite beam structures

This work deals with design sensitivity analysis and optimal design of composite structures modelled as thin-walled beams. The structures are treated as a torsion-bending resistant beams. The analysis problem is discretized by a finite element technique. A two-node Hermitean beam element is used. The beam sections are made from an assembly of elements that correspond to flat layered laminated composite panels. Optimal design is performed with respect to the lamina orientations and thickness of the laminates. The structural weight is considered as the objective function. Constraints are imposed on stresses, displacements, critical load and natural frequencies. Two failure criteria are used to limit the structural strength: Tsai-Hill and maximum stress. The Tsai-Hill criterion is also adopted to predict the first-ply-failure loads. The design sensitivity analysis is analytically formulated and implemented. An adjoint variable method is used to derive the response sensitivities with respect to the design. A mathematical programming approach is used for the optimization process. Numerical examples are performed on three-dimensional structures.     

带集中质量复合材料层合屈曲梁参激振动的研究

基于欧拉梁理论,运用Reissner变分原理,导出了轴向周期激励下一端固定一端夹支,带集中质量的复合材料层合屈曲梁的非线性动力学控制方程.利用模态截断,对系统非线性偏微分控制方程进行Galerkin积分,并用四阶龙格-库塔法数值研究了主共振下梁随激励幅值变化的分岔图,讨论了集中质量大小和位置对系统一阶频率和倍周期分叉的影响,结果表明,外激励幅值及集中质量的大小和位置会对带集中质量的屈曲梁的动力学行为产生重要影响.     

Problems of structural optimization for post-buckling behaviour

A proposal of a new approach to the optimal design of structures under stability constraints is presented. It is shown that the standard problem of maximization of the instability load may be modified so as to obtain a specified post-critical behaviour of the designed structure. The modified optimal structure represents stable post-buckling behaviour either locally, that is, in the vicinity of the critical point, or for a specified range of generalized displacements. First, some rigid–elastic finite-degree-of-freedom models are optimized, giving an insight into the modified design problems. Then a classification of the new optimization problems is presented. Various forms of instability are taken into account and a broad selection of objective as well as constraint functions is proposed. Based on the presented classification and following the proposed optimization concept, detailed formulations of nonlinear problems of design for post-buckling behaviour are given.     

桁架形状优化的一种改进模拟退火算法研究

通过设计一种产生可行解的状态发生器,由该状态发生器产生的新状态均满足所有的约束条件,从而方便地处理约束条件,并提出一种求解桁架形状优化设计问题的改进的模拟退火算法。算例表明该方法能获得较高质量的解,具有现实的工程意义,同时指出改进的SA算法用于桁架形状优化问题得不到全局最优解。     

Integrated optimal topology design and shape optimization using neural networks

In this paper, neural network- and feature-based approaches are introduced to overcome current shortcomings in the automated integration of topology design and shape optimization. The topology optimization results are reconstructed in terms of features, which consist of attributes required for automation and integration in subsequent applications. Features are defined as cost-efficient simple shapes for manufacturing. A neural network-based image-processing technique is presented to match the arbitrarily shaped holes inside the structure with predefined features. The effectiveness of the proposed approach in integrating topology design and shape optimization is demonstrated with several experimental examples.     

On optimal design of supports in beam and frame structures

An algorithm of optimal design of supports including their number, position and stiffness is proposed. The number of supports constitute topological design parameters, their positions correspond to configuration parameters. Both, elastic and rigid supports are considered and the optimization is aimed to minimize the total structure cost. The topology bifurcation points correspond to generation of new supports. The topological sensitivity derivative is used in deriving the optimality conditionsThe optimization procedure provides number of supports, their position and stiffness of both supports and beam segments.Presented at WCSMO-2, Zakopane, Poland, May 26–30, 1997     

On the practical stability of optimal stochastic control systems with dead-time

The practical stability of optimal stochastic control systems for processes having dead-times is considered. Previously obtained necessary conditions for practical stability of such systems are generalized using Pontryagin's theorem on the roots of two-variable polynomials. The conditions are expressed in terms of the relations between the orders of the process, the process model and the disturbance model.     

关于产品外型概念设计方法的研究

本文主要介绍了计算机辅助设计中实体造型常用的方法。如基于树结构、基于关键点的遗传算法、同伦映射法等,并通过实验结果进一步加深了对上述各方法的解释。     

Integration of topology and shape optimization for design of structural components

This paper presents an integrated approach that supports the topology optimization and CAD-based shape optimization. The main contribution of the paper is using the geometric reconstruction technique that is mathematically sound and error bounded for creating solid models of the topologically optimized structures with smooth geometric boundary. This geometric reconstruction method extends the integration to 3-D applications. In addition, commercial Computer-Aided Design (CAD), finite element analysis (FEA), optimization, and application software tools are incorporated to support the integrated optimization process. The integration is carried out by first converting the geometry of the topologically optimized structure into smooth and parametric B-spline curves and surfaces. The B-spline curves and surfaces are then imported into a parametric CAD environment to build solid models of the structure. The control point movements of the B-spline curves or surfaces are defined as design variables for shape optimization, in which CAD-based design velocity field computations, design sensitivity analysis (DSA), and nonlinear programming are performed. Both 2-D plane stress and 3-D solid examples are presented to demonstrate the proposed approach. Received January 27, 2000 Communicated by J. Sobieski     

平面线弹性问题的等几何形状优化方法

介绍具有等几何分析功能的GeoPDEs平台的数据结构和分析流程,针对二维平面形状优化问题,以控制顶点为设计变量,在推导出等几何分析的灵敏度计算公式后,提出基于GeoPDEs平台的灵敏度分析的高效实现方法,并采用移动渐近线法(Method of Moving Asymptotes,MMA)算法进行等几何形状优化.形状优化实例表明该方法收敛速度快,优化结果较理想.     

一类双输入非线性系统的二维平面形状合同控制器设计

针对一般控制系统,利用微分几何中的曲线理论提出了形状变量、被控形状轨迹曲线和形状合同控制等概念.然后,针对一类具有双输入及二维形状变最的非线性系统,利用相对曲率概念和曲线论基本定理设计了三维平面形状合同控制器,这种形状合同控制器能够保证系统的被控形状轨迹曲线与参考轨迹曲线具有完全相同的形状.最后的仿真说明了所采用方法的有效性.     

Robust stability considerations in optimal design of dynamic systems under uncertainty

This paper presents a method for the incorporation of robust stability criteria in the design of dynamic systems under uncertainty. Process systems are modelled via dynamic mathematical models, variations include both uncertain parameters and time-varying disturbances, while control structure selection and controller design is considered as part of the design optimization problem. Stability criteria are included, based on the concept of the measure of a matrix, to maintain desired dynamic characteristics, in a multiperiod design formulation. A combined flexibility-stabiluty analysis step is also introduced to ensure feasible and stable operation of the dynamic system in the presence of parametric uncertainties and process disturbances. The potential of the proposed approach is illustrated with a ternary distillation column design and control problem (featuring a rigorous tray-by-tray model).     

On the design of three-ply nonlinearly elastic composite plates with optimal resistance to buckling

The optimal design of symmetric three-ply sandwich plates is studied in the context of finite deformation incompressible nonlinear elasticity. The overall shape of the plate is dictated, fixed amounts of two materials are at our disposal, and performance is defined solely in terms of resistance to buckling. The problem is characterized by three parameters: the volume ratio of the two materials, the stiffness ratio of the two (neo-Hookean) materials, and one of the aspect ratios of the plate. Two competing constructions are considered: one in which the stiffer material is used in the outer plies and the other in which the stiffer material is used for the central ply. It is found that if the material volume ratio and the material stiffness ratio are fixed, then there is a single aspect ratio dependent transition in the optimal design. The configuration with the stiffer material used for the central ply is the optimal design for plates that are sufficiently short in the direction of thrust, while the configuration with the stiffer material used for the outer plies is the optimal design for plates that are sufficiently long in the direction of thrust.     

How to design a shock absorber

We consider an elastic bar of length ℓ, subject to the longitudinal impact of a mass hitting one end of it, while the other end is fixed. If the cross-section of the bar is constant, it is known how to determine the influence of the inertia of the bar on the velocity of the impinging mass and on the maximal compression of the bar. If, however, the cross-section can vary, preserving the total volume of material, the question arises of distributing this material along the span of the bar so as to minimize the maximal compression. Received July 15, 2000     

Singular optimal control problems: On the order of a singular arc

The higher-order tests for optimality of singular arcs in optimal control problems, as well as the characterization of the continuity and smoothness properties of the optimal control at points, called junctions, joining nonsingular and singular arcs of the control, depend upon the orders of the arcs involved. To data, definitions of order have been rigorously given only for terminal unconstrained control problems or by means of a normality assumption in the order cases. We give here a new definition of the local order of a singular arc which unifies those already given in the literature. One advantage of the proposed definition is that, unlike the previous ones, it remains valid even when there are terminal constraints. One example will illustrate the fact that such a basic concept is essential to solve the very difficult synthesis problem as soon as the dimension of the state space is greater than two.     

On the design of a sensitivity-reducing optimal dead-beat controller

A method is proposed for the design of constant-gain feedback-control laws for linear multivariable discrete-time systems which reduce trajectory sensitivity to small system parameter variations and ensure the closed-loop eigenvalues at the origin. The given quadratic index of performance including a sensitivity term is minimized in an average sense. An efficient computational procedure based on direct cost optimization using gradient type algorithm is also reported. An example is worked out to illustrate the proposed technique.