Accuracy of Finite Element Based Shape Design Sensitivity Analysis*

ABSTRACT Accuracy of finite element based shape design sensitivity analysis, for use in shape design optimization, is analyzed and tested. A shape design sensitivity formulation that uses stress averaging to treat stress constraint func-tionals is presented. A smooth boundary parameterization is introduced to avoid the "Babuska paradox" and to improve accuracy of design sensitivity analysis. Accuracy of design sensitivity analysis with displacement and hybrid stress finite element formulations is compared. Good accuracy is obtained using higher order displacement finite elements, such as linear stress triangular and 8-noded isoparametric elements.     

On optimum design of structures and materials

A survey of problems of optimum design of structures and materials is presented with the main emphasis on fundamental aspects and on current methods and capabilities for topology and shape optimization.The methods are selected from conditions of versatility and suitability for integration into an engineering design optimization system which realizes the design process as an iterative solution procedure of a multicriterion optimization problem based on the concept of integration of finite element analysis, sensitivity analysis, and optimization by mathematical programming.A picture of current possibilities and the present status of the field is given through a number of examples.General invited lecture presented at the 12th Italian Congress on Theoretical and Applied Mechanics AIMETA '95, Naples, 3–6 October 1995.     

Three-Dimensional Shape Optimization of Arch Dams with Prescribed Shape Functions*, †

Development and application of shape optimization of double-curved arch dams are presented. A mathematical formulation of arch dam design is described, where design variables, objective function, and constraint functions are defined. The geometrical form of the arch dam and part of its load-carrying foundation are described by three-dimensional hyperelements. Design variables are identified as geometrical parameters of these hyperelements. For static analysis of the dam-foundation system, the finite element method is employed with eight node solid elements and an incompatible displacement function. Elements are automatically generated within the hyperelements in each design step. Four different load cases are considered to account for important design specifications. A complete static sensitivity analysis is performed and used in each design iteration. Partial derivatives of element stresses with respect to all design variables are computed within the finite element context. The optimization problem is solved by sequential linear programming. It is demonstrated that practical three-dimensional shape optimization is feasible and economical.     

预锻模具形状优化设计与有限元灵敏度分析

采用刚－粘塑性有限元灵敏度分析方法研究锻造过程的预锻模具形状优化设计问题。表示预锻模具形状的三次Ｂ样条曲线的控制系数用作设计变量。以实际终锻件与理想终锻件的形状差异为目标函数,给出了与设计变量有关的目标函数,节点坐标和节点速度等方面的灵敏度及其数学关系,对于典型的轴对称锻造过程,优化设计的预锻模具形状可获得理想的的终锻件形状,为实现净成形锻造提供了一种有效方法。     

基于物理的变形曲线在结构形状优化中的应用

用各向异性模型定义NURBS曲线的变形能,基于所提出的变形能模型,用有限元法对NURBS曲线表达的设计边界进行模态计算,然后,将设计边界用模态向量的线性组合参数化表示。将这种基于边界特征向量的几何形状表示方法应用于优化参数定义．提出了一种适用于结构形状优化的自适应几何精化方法,它有效地将户型有限元分析、优化方法和设计边界的形状表达集成在一起。     

Structural–acoustic sensitivity analysis of radiated sound power using a finite element/ discontinuous fast multipole boundary element scheme

The complete interaction between the structural domain and the acoustic domain needs to be considered in many engineering problems, especially for the acoustic analysis concerning thin structures immersed in water. This study employs the finite element method to model the structural parts and the fast multipole boundary element method to model the exterior acoustic domain. Discontinuous higher‐order boundary elements are developed for the acoustic domain to achieve higher accuracy in the coupling analysis. Structural–acoustic design sensitivity analysis can provide insights into the effects of design variables on radiated acoustic performance and thus is important to the structural–acoustic design and optimization processes. This study is the first to formulate equations for sound power sensitivity on structural surfaces based on an adjoint operator approach and equations for sound power sensitivity on arbitrary closed surfaces around the radiator based on the direct differentiation approach. The design variables include fluid density, structural density, Poisson's ratio, Young's modulus, and structural shape/size. A numerical example is presented to demonstrate the accuracy and validity of the proposed algorithm. Different types of coupled continuous and discontinuous boundary elements with finite elements are used for the numerical solution, and the performances of the different types of finite element/continuous and discontinuous boundary element coupling are presented and compared in detail. Copyright © 2016 John Wiley & Sons, Ltd.     

考虑材料设计变量的热-固耦合结构的优化设计

从材料-结构协同设计的角度研究了热-固耦合结构的优化设计问题,将决定结构材料性质的细观参数与结构宏观几何参数作为设计变量,利用均匀化方法推导了细观设计变量灵敏度显式计算式,并结合耦合场有限元方程构造了耦合场设计变量灵敏度计算式;提出了材料-结构协同设计的三种优化设计模型.利用结构响应最小优化模型对算例进行了计算,比较了宏观设计变量优化和材料-结构协同设计优化的效果.计算结果显示,材料-结构协同优化设计可以取得较单一宏观设计变量更好的优化效果.     

CFD‐based optimization of aerofoils using radial basis functions for domain element parameterization and mesh deformation

A novel domain element shape parameterization method is presented for computational fluid dynamics‐based shape optimization. The method is to achieve two aims: (1) provide a generic ‘wrap‐around’ optimization tool that is independent of both flow solver and grid generation package and (2) provide a method that allows high‐fidelity aerodynamic optimization of two‐ and three‐dimensional bodies with a low number of design variables. The parameterization technique uses radial basis functions to transfer domain element movements into deformations of the design surface and corresponding aerodynamic mesh, thus allowing total independence from the grid generation package (structured or unstructured). Independence from the flow solver (either inviscid, viscous, aeroelastic) is achieved by obtaining sensitivity information for an advanced gradient‐based optimizer (feasible sequential quadratic programming) by finite‐differences. Results are presented for two‐dimensional aerofoil inverse design and drag optimization problems. Inverse design results demonstrate that a large proportion of the design space is feasible with a relatively low number of design variables using the domain element parameterization. Heavily constrained (in lift, volume, and moment) two‐dimensional aerofoil drag optimization has shown that significant improvements over existing designs can be achieved using this method, through the use of various objective functions. Copyright © 2008 John Wiley & Sons, Ltd.     

相变传热问题的灵敏度分析与优化设计方法

研究了相变传热问题的优化设计及其灵敏度分析方法. 在有限元-时间差分和等效热容 法求解相变温度场的基础上,提出了相变温度场对设计变量一阶灵敏度的计算方法,给出直 接法和伴随法两种计算格式并分析了它们的特点,建立了相变温度场优化的模型和算法,在有限元分析与优化设计软件JIFEX中实现了该方法. 数值算例表明了灵敏度计算的精度和优 化方法的有效性.     

Shape optimization and its extension to topological design based on isogeometric analysis

In most of structural optimization approaches, finite element method (FEM) has been employed for structural response analysis and sensitivity calculation. However, the approaches generally suffer certain drawbacks. In shape optimization, cumbersome parameterization of design domain is required and time consuming remeshing task is also necessary. In topology optimization, design results are generally restricted on the initial design space and additional post-processing is required for communication with CAD systems. These drawbacks are due to the use of different mathematical languages in design or geometric modeling and numerical analysis: spline basis functions are used in design and geometric modeling whereas Lagrangian and Hermitian polynomials in analysis. Isogeometric analysis is a very attractive and promising alternative to overcome the limitations resulting from the use of the conventional FEM in structural optimization. In isogeometric analysis, the same spline information such as control points and spline basis functions which represent geometries in CAD systems are also used in numerical analysis. Such unification of the mathematical languages in CAD, analysis and design optimization can resolve the issues mentioned above. In this work, structural shape optimization using isogeometric analysis is studied on 2D and shell problems. The proposed framework is extended to topology optimization using trimming techniques. New inner fronts are introduced by trimming spline curves in topology optimization. Trimmed surface analysis which was recently proposed to analyze arbitrary complex topology problems is employed for topology optimization. Some benchmarking problems in shape and topology optimization are treated using the proposed approach.     

非确定性结构静动态特性稳健优化设计

本文研究了考虑参数随机性的结构静动态特性稳健性优化设计问题的数学模型和数值求解。在考虑结构设计变量和其研究了考虑参数随机性的结构静动态特性稳健性优化设计问题的数学模型和数值求 解. 在考虑结构设计变量和其他参数随机分布的前二阶矩的条件下,采用基于二阶摄动法的 随机有限元方法对结构响应的平均值和方差进行近似求解. 在摄动法有限元分析的框架下, 提出以一般函数形式表达的结构性能的平均值和标准差及其灵敏度的计算格式. 将结构 稳健性优化设计问题构造为双目标优化问题,优化目标包含结构性能函数的期望值和标准 差,约束函数的变异也给予考虑. 优化问题采用基于函数梯度的算法进行求解. 文中给出的数值算例显示了方法的有效性.     

双向固定网格渐进结构优化方法

近年来发展的渐进结构优化方法是一种有前途的结构拓扑和形状优化方法.本文在渐进结构优化方法的框架内建立了统一敏感度的概念,并基于固定网格有限元技术,发展了一种新的增加材料技术,提出了双向固定网格渐进结构优化方法.将该方法应用于复合材料壳结构开孔形状优化,以孔周等Tsai-Hill强度值作为优化目标,可以得到合理的最优解,证明了双向固定网格渐进结构优化方法的适用性.不同的初始点能得到几乎相同的最优解,展示了本文方法良好的全局最优性.     

基于ANN-GA协同寻优的动态拉伸试样尺寸优化方法

材料动态拉伸力学性能测试中,动态拉伸试样的几何尺寸对测试结果的准确性与有效性有着较大影响。为对动态拉伸试样的结构进行优化设计,以使其在动态拉伸过程中更好地满足一维应力与变形均匀等基本假设。首先,建立了量化的试样测量准确度指标,即应力平衡达到时间、变形均匀度、非轴向应力相对水平、过渡段相对变形。然后,对试样结构参数进行正交试验设计,通过数值模拟的方法得到了关于试样尺寸与测量准确度指标的正交试验数据库,并对正交试验数据库进行多目标正交试验矩阵分析,得到了试样结构参数对各测量准确度指标影响的主次顺序和规律。最后,以正交试验数据库为训练集,采用人工神经网络（artificial neural network, ANN）协同遗传算法（genetic algorithm, GA）的全局寻优方法对试样的结构尺寸进行优化设计,得到了试样的最优结构尺寸,并对最优尺寸的有效性进行了验证。结果表明,优化后的试样结构在材料动态拉伸力学性能测试精度上的表现明显得以提升。因此,采用ANN-GA协同优化的方法对动态拉伸试样的结构进行优化具有可行性和有效性。     

结构优化半解析灵敏度及误差修正改进算法

提出结构半解析灵敏度分析及其针对刚体位移的误差修正方法的改进算法, 构建灵敏度分析与误差修正项可分离形式. 该方法实现简便, 数值精度不受摄动步长与单元数目的影响. 首先从总体角度推得静力问题的误差修正半解析灵敏度分析方法, 提出了位移误差修正灵敏度列式, 并给出算法实施途径; 然后将此思路推广于自振频率、屈曲临界载荷问题, 提出了相应的计算步骤. 随后, 给出梁单元与壳单元误差修正项的具体推导方法, 并分别使用两种单元构建有限元模型进行算例测试. 结果表明, 该方法适用于多种分析类型, 数值精度不受单元数目与摄动步长的影响. 由于灵敏度分析与误差修正项可以分开计算, 该方法支持将误差修正项直接叠加于灵敏度求解结果进行误差修正, 使已有灵敏度分析程序得到充分利用. 尤其对于复杂工程结构的优化设计, 特别是形状优化设计以及尺寸、形状混合优化设计, 相比于原误差修正方法, 实现更为简便, 效率有所提升, 能为半解析灵敏度分析方法及其程序实现提供新的思路.      

结构优化半解析灵敏度分析的改进算法

提出了结构半解析灵敏度分析的改进算法,该算法实现简便,对于设计变量摄动步长具有极佳的数值稳定特性。首先,从总体角度推导静力问题半解析法灵敏度分析新算法,提出了位移与应力灵敏度列式,并给出了算法实施途径;然后,将此思路推广于自振频率、屈曲临界荷载和瞬态响应等多种问题,提出了相应的计算步骤。以梁单元与壳单元等典型结构为例,开展了多个算例测试。测试表明,改进算法计算精度和效率均有提升,特别是设计变量步长有更大的数值稳定区域,为复杂工程结构形状优化的灵敏度分析提供了新途径。     

Complex variable function method fob hole shape optimization in an elastic plane

In this paper, a complex variable function method for solving the hole shape optimization problem in an elastic plane is presented. In this method, the stresses in hole problems are analysed by taking advantage of the efficiency of the complex variable function method. To optimize the hole shape, the coeffecients in conformal mapping functions are taken as design variables, and the sensitivity analysis and gradient methods are used to reduce the largest circumferential stress in absolute value and at the same time to make the second largest circumferential stress in absolute value not to exceed the largest one (in fact, these two stresses are the stationary values of the circumferential stresses). The coefficients in conformal mapping function are revised by iteration step by step until the largest circumferential stress in absolute value is reduced to the second largest stress. This method guarantees the continuity, differentiability and accuracy of the stress solution along the boundary, and it is evident that this method is better than either the difference method or the finite element method.     

Sampling-Based RBDO of Ship Hull Structures Considering Thermo-Elasto-Plastic Residual Deformation

We present a shape optimization method using a sampling-based RBDO method linked with a commercial finite element analysis (FEA) code ANSYS, which is applicable to residual deformation problems of the ship hull structure in welding process. The programming language ANSYS Parametric Design Language (APDL) and shell elements are used for the thermo-elasto-plastic analysis. The shape of the ship hull structure is modeled using the bicubic Ferguson patch and coordinate components of vertices, tangential vectors of boundary curves are selected as design variables. The sensitivity of probabilistic constraint is calculated from the probabilistic sensitivity analysis using the score function and Monte Carlo Simulation (MCS) on the surrogate model constructed by using the Dynamic Kriging (DKG) method. The sequential quadratic programming (SQP) algorithm is used for the optimization. In two numerical examples, the suggested optimization method is applied to practical residual deformation problems in welding ship hull structures, which proves the sampling-based RBDO can be successfully utilized for obtaining a reliable optimum design in highly nonlinear multi-physics problem of thermo-elasto-plasticity.     

基于虚荷载变量的形状优化和灵敏度分析

基于选择施加在结构“控制点”上的虚荷载作为优化设计变量,针对一种新的承受约束的形状优化数值方法进行了研究。借助于节点位移与虚荷载之间的线性关系,提出了一种新的计算灵敏度系数的解析方法。利用节点移动速度域概念构造了优化新形状产生的计算公式,以结构中节点的最大应力最小化作为优化目标,通过控制网格结点的最大位移量,较好地解决了单元网格在形状优化中的扭曲问题。对三个不同的实例成功地完成了形状优化。     

Topological shape optimization of power flow problems at high frequencies using level set approach

Using a level set method we develop a topological shape optimization method applied to power flow problems in steady state. Necessary design gradients are computed using an efficient adjoint sensitivity analysis method. The boundaries are implicitly represented by the level set function obtainable from the “Hamilton–Jacobi type” equation with the “Up-wind scheme.” The implicit function is embedded into a fixed initial domain to obtain the finite element response and sensitivity. The developed method defines a Lagrangian function for the constrained optimization. It minimizes a generalized compliance, satisfying the constraint of allowable volume through the variations of implicit boundary. During the optimization, the boundary velocity to integrate the Hamilton–Jacobi equation is obtained from the optimality condition for the Lagrangian function. Compared with the established topology optimization method, the developed one has no numerical instability such as checkerboard problems and easy representation of topological shape variations.