OPTIMAL DESIGN FOR NON-STEADY-STATE METAL FORMING PROCESSES—I. SHAPE OPTIMIZATION METHOD

We suggest a shape optimization method for a non-linear and non-steady-state metal forming problem. It consists in optimizing the initial shape of the part as well as the shape of the preform tool during a two-step forging operation, for which the shape of the second operation is known. Shapes are described using spline functions and optimal parameter values of the splines are searched in order to produce, at the end of the forging sequence, a part with a prescribed geometric accuracy, optimal metallurgical properties and for a minimal production cost. The finite element method, including numerous remeshing operations, is used for the simulation of the process. We suggest using a least-squares-type algorithm for the unconstrained optimization method (based on external penalty) for which we describe the calculation of the derivatives of the objective function. We show that it can reduce to calculations which are equivalent to the derivative calculations of steady-state processes and to evolution equations. Therefore, the computational cost of such an optimization is quite reasonable, even for complex forging processes. Lastly, in order to reduce the errors due to the numerous remeshings during the simulation, we introduce error estimation and adaptive remeshing methods with respect to the calculation of derivatives.     

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.     

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。     

改进的响应面方法在二维连续体形状优化中的应用

在二维连续体结构形状优化中,采用响应面方法近似求解约束的敏度来建立模型:根据结构分析的响应值,可以拟合出复杂响应对设计变量的显函数。为克服各试验点的拟合值与结构真实响应值之间的误差,发展了一种在中心试验点精确拟合的方法,提高了模型的精度,收到了良好的效果。     

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.     

SIMULTANEOUS OPTIMIZATION OF SHAPE AND FLOW PARAMETERS FOR COMBINED FREE AND FORCED CONVECTION IN VERTICAL CHANNELS

Simultaneous optimization of shape and flow parameters is performed for a combined free and forced convection flow through vertical rectangular channels with moving walls. The laminar flow is assumed to be fully developed in the axial direction. The wall velocity, the axial pressure gradient and the channel height in the transverse plane are taken as the optimization parameters. The sensitivity expressions of both the objective function and the flow rate constraint of optimization are obtained in terms of the relevant physical variables, as well as adjoint variables which satisfy additional p.d.e.'s. All equations are discretized using the finite element method. Numerical results are provided for the present constrained optimization problem for various values of the problem parameters which include the moving wall segment size and the Rayleigh number. The results indicate that with increased Rayleigh number the optimal values of the wall velocity and the axial pressure gradient are increased, while the optimal value of the channel height is decreased. General sensitivity expressions are also presented in the appendix which might be utilized for arbitrary boundary variations along with arbitrary optimization objectives in other investigations. © 1997 by John Wiley & Sons, Ltd.     

基于有限元灵敏度分析的预锻模具形状优化设计

在控制锻件几何形状的前提下 ,采用有限元灵敏度分析方法 ,对预锻模具形状进行优化设计 .针对下模速度为零时 ,速度灵敏度边界条件为零 ,其形状在优化迭代过程中得不到优化的情况 ,对速度灵敏度边界条件提出改进措施 ,使上下模具形状同时能够得到优化 .最后给出了优化设计实例 ,验证该方法的可靠性 .     

THE ISSUE OF GENERALITY IN DESIGN OPTIMIZATION SYSTEMS

This paper attempts to convey some of the author's experiences in constructing and using systems for design optimization with as much generality as necessary to solve practical engineering design problems. A number of examples demonstrate that real-life design often calls for facilities that were difficult to foresee when the optimization system was devised. The conclusion contains two important points: a) in today's use of design optimization systems, the definition of the problem must often be modified to suit the capabilities of the system at hand, and b) the construction of design optimization systems with the necessary generality and flexibility to function in an engineering design environment is a problem that merits independent research beyond the traditional, fundamental investigations into mechanical properties of optimum structures and solution methods. The paper points out a number of important areas for further research.     

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.     

有限元优化技术在散热器结构设计中的应用

利用有限元优化技术系统地研究散热器结构参数对散热量以及散热器芯体重量的影响规律。并以散热器重量最小为优化目标，在满足散热量要求的前提下，应用对1301015—A1型管带式散热器的结构参数进行了优化设计，并给出了优化设计前后的性能对比分析。     

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.     

A SHAPE AND SIZE OPTIMIZATION ALGORITHM FOR MACHINE TOOL ELEMENT DESIGN

A shape and size optimization algorithm is developed to incorporate the shape and size variables in a recently developed machine tool simulation and design methodology known as Integrated Machining Process Design Simulator (IMPDS). This shape optimization algorithm, using coordinates of master nodes, parameters of geometric equations, coordinate linking and symmetry option approaches presents a general and flexible way of controlling and optimizing the structural shape of machine tool elements. An application of the proposed shape and size optimization algorithm indicates that the shape and size parameters have significant effects on the structural characteristics and the dynamic behavior of machine tool elements.     

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。     

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.     

EXTENSION OF THE ZIENKIEWICZ–ZHU ERROR ESTIMATOR TO SHAPE SENSITIVITY ANALYSIS

The application of the Zienkiewicz–Zhu estimator was extended to the estimation of the discretization error arising from shape sensitivity analysis using the finite element method. The sensitivity error was quantified from the sensitivity of the energy norm by using an estimator specially developed for this purpose. Sensitivity analyses were carried out using the discrete analytical approach, which introduced no additional errors other than the discretization error. In this work, direct nodal averaging was used for linear triangular elements and the SPR technique for quadratic elements in order to obtain the smoothed stress and sensitivities fields. Two examples with an exact solution are used to analyse the effectivity of the proposed estimator and its convergence with the h-adaptive refinement. © 1997 by John Wiley & Sons, Ltd.     

APPLICATION OF AI TOOLS FOR RE-ANALYSIS WITHIN STRUCTURAL OPTIMIZATION

Structural optimization can be viewed as a logical and rational base for modern CAD/CAE applications. However, problems are encountered if major finite element systems are to be optimized for which numerous optimization variables have been specified. Particularly, the implicit nature of the governing finite element equations can result in prohibitive computational effort.

Confining to linear discrete systems, a Taylor series based approximation concept is presented which may be expanded to further finite element problems (eigenvalue problems, etc.). Specific emphasis is placed on the abstract derivation of a rulebase for symbolic differentiation of the finite element matrices with respect to named optimization variables. By means of such a rulebase, in association with an expert system shell, the explicit approximation of finite element relationships can effectively be computerized.     

RECENT DEVELOPMENTS IN SHAPE SENSITIVITY ANALYSIS: THE PHYSICAL APPROACH

Because of increasing CAD requirements in industry, shape optimum design by finite element is an important field of research and development. The global solution of this problem involves two main aspects: the first one is the availability of powerful and robust optimizers and the second one is the necessity of producing the best sensitivity analysis for the approximate problem concerned by optimization. Nowadays, several good softwares or computer programs are available in order to solve a well stated approximate optimization problem. For sensitivity analysis the method using big design elements is quite successful but it appears that more general and powerful methods could be implemented. It is expected that the physical approach developed in this paper could be one of these better methods of the future.     

FINITE ELEMENT SOLUTION OF A MODEL FREE SURFACE PROBLEM BY THE OPTIMAL SHAPE DESIGN APPROACH

Optimal shape design approach is applied to numerical computation of a model potential free boundary value problem. The problem is discretized using the finite element method. To test the approach the problem is formulated in both velocity potential and stream function formulation and four different finite element discretizations are used. Associated minimization problem is solved using the quasi-Newton method. Gradient of the cost function is computed by solving the algebraic adjoint equation. Gravity and surface tension forces are included in the model. Viability of the method is showed by solving problems with important effects of gravity and surface tension forces. © 1997 by John Wiley & Sons, Ltd.     

OPTIMUM SHAPE DESIGN AND POSITIONING OF FEATURES USING THE BOUNDARY INTEGRAL EQUATION METHOD

A design optimization procedure is developed using the boundary integral equation (BIE) method for linear elastostatic two-dimensional domains. Optimal shape design problems are treated where design variables are geometric parameters such as the positions and sizing dimensions of entire features on a component or structure. A fully analytical approach is adopted for the design sensitivity analysis where the BIE is implicitly differentiated. The ability to evaluate response sensitivity derivatives with respect to design variables such as feature positions is achieved through the definition of appropriate design velocity fields for these variables. How the advantages of the BIE method are amplified when extended to sensitivity analysis for this category of shape design problems is also highlighted. A mathematical programming approach with the penalty function method is used for solving the overall optimization problem. The procedure is applied to three example problems to demonstrate the optimum positioning of holes and optimization of radial dimensions of circular arcs on structures.     

鲁棒优化设计方法在结构动力学中的应用

在传统的静力学鲁棒优化设计基础上,考虑时间t参数,通过优化系统目标函数和约束条件的鲁棒性,将鲁棒优化设计方法运用在动力学问题中。通过一个主系统的质量和刚度均有微小波动的二自由度模型减振器设计算例,与传统的优化设计方法相比,显示了鲁棒优化设计的优越性,能使结构具有更稳定的性能。