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.     

基于可靠度的结构优化的序列近似规划算法

基于可靠度的优化的最直观解法是把可靠度和优化的各自算法搭配一起形成嵌套两层次迭代。为改善其收敛性提高计算效率,人们提出了功能测度法、半无限规划法、单层次算法等多种改进方法。本文对传统结构优化界的经典序列近似规划法改造并扩展应用于求解基于可靠度的结构优化问题,构造该问题的序列近似规划模型和求解过程;其核心思想是在每个近似规划子问题中采用近似可靠度指标对设计变量的线性近似,在优化迭代过程中同步更新设计变量和随机空间中的近似验算点坐标,以达到可靠度分析和优化迭代同步收敛的目标。为了算法的实施,还推导出近似可靠度指标的半解析灵敏度计算公式,编制了程序,最终实现与通用软件的连接。论文用算例证实算法的有效性。     

热-应力耦合结构灵敏度分析方法

研究稳态／瞬态热传导灵敏度分析、以及热与机械荷载同时作用的热结构应力灵敏度分析问题。考虑了温度场随设计变量的变化及其对应力的影响,提出温度场与结构热应力耦合问题的灵敏度计算方法。特别指出了热－应力耦合灵敏度分析中温度场导数的影响, 说明了在热－应力耦合结构灵敏度分析中必须考虑耦合灵敏度。在应用软件系统JIFEX中实现了所提出的方法,数值算例验证了灵敏度算法的精度。     

连续体结构拓扑优化的节点独立连续映射法

提出基于节点拓扑变量的独立连续映射连续体结构拓扑优化方法,将常规ICM中基于单元拓扑变量描述改进为基于节点拓扑变量描述,通过单元形函数插值保证了拓扑变量场的连续性.引入拓扑变量和设计变量倒数关系,根据复合函数链式求导法则,推导了各类结构响应量的敏度分析公式.以位移约束问题为例,遵循ICM法建模方式建立了连续体结构拓扑优化模型.基于数字图像处理过滤技术,提出两种改进的过滤方式以消除数值不稳定性现象.最后通过二维拓扑优化数值算例证明方法和模型的可行性和有效性.     

基于改进灵敏度过滤策略的SIMP方法

在传统变密度法SIMP(Solid Isotropic Microstructures with Penalization)中,Sigmund灵敏度过滤策略是解决拓扑优化结果数值不稳定问题的重要方法之一。但在SIMP方法中采用Sigmund灵敏度过滤策略时,有时优化结果会有较多的灰度单元。为了减少拓扑优化结果中灰度单元的数量,提出了一种基于改进灵敏度过滤策略的SIMP方法。在过滤后单元灵敏度的计算中,增加了一个与中心单元过滤前灵敏度有关的部分,该部分在过滤后灵敏度中所占的比重需要通过过滤权重来调节。通过对灵敏度过滤需要的分析,确定了过滤半径和中心单元密度值是影响过滤权重的因素。根据拓扑优化实例的优化结果给出了过滤半径影响过滤权重的情况,并基于过滤半径对过滤权重的影响情况构建了过滤权重的函数。在改进灵敏度过滤策略的基础上,结合双重SIMP算法给出了本文SIMP方法的流程。最后,以悬臂梁结构和简支梁结构为例,验证了本文方法的有效性。与Sigmund灵敏度过滤策略相比,改进的灵敏度过滤策略能有效地减少灰度单元。与双重SIMP方法相比,本文SIMP方法能有效地减少数值不稳定现象。     

变截面压杆稳定非线性微分方程边值问题的最优化算法研究

针对任意约束类型的变截面受压杆件的稳定临界载荷计算问题,结合非线性微分方程数值算法和最优化方法,以起点边界的初始条件、待求临界荷载和附加约束力为设计变量;以终点边值条件满足的函数关系与位型条件构建目标函数,提出变截面压杆临界载荷和稳定位型的优化求解算法。应用VB编制通用的优化计算程序,分析了典型算例;通过对比发现,本文以较少设计变量实现了临界载荷的高精度计算,为工程应用提供参考。     

弹塑性摩擦接触问题形状设计灵敏度分析

提出了一种计算二维有限变形弹塑性摩擦接触问题形状设计灵敏度的算法. 采用主动集策略和mortar方法处理接触边线上的约束条件. 在mortar接触边线的切线和法线方向上采用相同的名义罚函数,提出基于名义罚函数的移动摩擦锥算法来正则化接触约束条件,发展了一种新的二维多体有限变形摩擦接触算法. 在此基础上, 通过将离散形式的摩擦接触问题控制方程对形状设计变量微分,得到了该路径相关问题的直接微分法解析设计灵敏度计算格式, 其节点位移灵敏度方程在每个增量步不用迭代、直接求解. 与国际上现有的二维多体有限变形摩擦接触问题的解析设计灵敏度算法相比,本算法不需分解为法向和切向推导,表达式较简洁,便于编程实现. 数值算例验证了算法的精度和有效性.      

精确频率约束下框架结构的尺寸优化

本文研究受到频率约束的框架结构优化设计问题,采用基于Bernoulli-Euler梁的动力刚度法求解框架结构的自振频率,采用W-W算法精确求解自振频率及频率对设计变量的灵敏度.分析比较了将框架结构的杆件按普通静刚度梁单元和动力刚度梁单元分别建立计算模型时得到的频率分析结果.以杆件面积为设计变量,以结构重量为优化目标,对基频约束下的框架结构进行了尺寸优化,比较了节点集中质量对优化结果的影响.通过2个数值算例,验证了动力刚度阵法在框架结构频率性能分析与优化设计中的适用性,并具有较高的计算精度.     

岩体渗透结构类型的划分及其渗透特性研究

提出了一种计算二维有限变形弹塑性摩擦接触问题形状设计灵敏度的算法. 采用主动 集策略和mortar方法处理接触边线上的约束条件. 在mortar接触边线的切线和法线方向上 采用相同的名义罚函数,提出基于名义罚函数的移动摩擦锥算法来正则化接触约束条件,发 展了一种新的二维多体有限变形摩擦接触算法. 在此基础上, 通过将离散形式的摩擦接触问题 控制方程对形状设计变量微分,得到了该路径相关问题的直接微分法解析设计灵敏度 计算格式, 其节点位移灵敏度方程在每个增量步不用迭代、直接求解. 与国际上现有 的二维多体有限变形摩擦接触问题的解析设计灵敏度算法相比,本算法不需分 解为法向和切向推导,表达式较简洁,便于编程实现. 数值算例验证了算法的精度 和有效性.     

Accuracy Analysis of the Semi-Analytical Method for Shape Sensitivity Calculation∗

ABSTRACT

The semi-analytical method of design sensitivity analysis that is widely used for calculating derivatives of static response with respect to design variables for structures modeled by finite elements is studied in this paper. It is shown that the method can have serious accuracy problems for shape design variables in structures modeled by beam, plate, truss, frame, and solid elements. Errors are shown to be associated with an incompatibility of the sensitivity field with the structure. An error index is developed to test the accuracy of the semi-analytical method. It characterizes the difference in errors between a general finite difference method and the semi-analytical method. A method for improving the accuracy of the semi-analytical method (when possible) is provided. Examples are presented to demonstrate the use of the error index.     

Optimum design and sensitivity analysis of piezoelectric truss structures

This paper is concerned with the optimum design and sensitivity analysis methods of piezoelectric intelligent trusses on the structural stiffness and free-vibration frequency. On the basis of the finite element method and taking into account the mechanical-electric coupling effect under electric load and mechanical load, the computational formula of sensitivities of structural displacement and free-vibration frequency with respect to the size and shape design variables are proposed. A new kind of design variable of the electric voltage and the calculating method of displacement sensitivity with respect to the electric voltage variable is presented. The new method for structural deformation control by optimizing the voltages of piezoelectric active bars is given. The numerical methods of optimum design and sensitivity analysis for piezoelectric truss structures are implemented in JIFEX software. Numerical examples demonstrate the effectiveness of the methods and the program. Project supported by the Special Funds for National Key Basic Research of China (No. G1999032805) and the Foundation for University Key Teachers by the Ministry of Education of China.     

压电智能桁架的灵敏度分析与优化设计

在压电桁架结构的有限元分析方法基础上，考虑电荷载和机械荷载联合作用的机电耦合效应，给出了压电智能桁架的位移和自振频率对常规尺寸设计变量和形状设计变量的灵敏度计算公式，增加了电压这一类新的设计变量，给出了位移对电压的灵敏度计算方法。在此基础上实现了通过优化压电主动桁架的电压进行变形控制的方法，并在JIFEX软件中实现了压电桁架的灵敏度分析与优化设计。文中给出的数值算例验证了算法和程序的有效性。     

等几何壳体分析与形状优化

首先基于Reissner-Mindlin理论进行了三维壳体等几何分析,而后基于此对三维壳体进行形状优化,提出了形状优化中灵敏度的全解析计算方法,包括位移应变阵、雅克比阵和刚度阵等相对控制顶点位置的灵敏度解析计算公式;通过实例验证了壳体等几何分析和灵敏度全解析计算方法的有效性。与传统的基于网格的灵敏度半解析计算方法相比,基于NURBS的灵敏度全解析计算具有精确、计算效率高的特点,且可以避免优化迭代中的网格畸变。     

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.     

Sensitivity and randomness in homogenization of periodic fiber-reinforced composites via the response function method

The main issue this paper addresses is the derivation and implementation of a general homogenization method, including the simultaneous determination of sensitivity gradients and probabilistic moments of the effective elasticity tensor. This is possible with an application of the perturbation method based on Taylor expansion and with the effective modules method. The computational procedure is implemented using plane strain analysis carried out with the finite element method (program MCCEFF) and the symbolic computations system MAPLE. The sensitivity gradients and probabilistic moments are commonly determined on the basis of partial derivatives for the homogenized elasticity tensor, calculated using the response function method with respect to some composite parameters. They are subjected separately to a normalization procedure (in deterministic analysis) and the relevant algebraic combinations (for the stochastic case). This enriched homogenization procedure is tested on a periodic fiber-reinforced two component composite, where the material parameters are taken as design variables and then, the input random quantities. The results of computational analysis are compared against the results of the central finite difference approach in the case of sensitivity gradients determination as well as the direct Monte-Carlo simulation approach. This numerical methodology may be further applied not only in the context of the homogenization method, but also to extend various discrete computational techniques, such as Boundary/Finite element and finite difference together with various meshless methods.     

Sensitivity of a general class of shape functionals to topological changes

The topological derivative represents the first term of the asymptotic expansion of a given shape functional with respect to the small parameter which measures the size of singular domain perturbations. The topological derivative has been successfully applied in the treatment of problems such as topology optimization, inverse analysis and image processing. In this paper, the calculation of the topological derivative for a general class of shape functionals is presented. In particular, we evaluate the topological derivative of a modified energy shape functional associated to the steady-state heat conduction problem, considering the nucleation of a small circular inclusion as the topological perturbation. Several methods were proposed to calculate the topological derivative. In this paper, the so-called topological-shape sensitivity method is extended to deal with a modified adjoint method, leading to an alternative approach to calculate the topological derivative based on shape sensitivity analysis together with a modified Lagrangian method. Since we are dealing with a general class of shape functionals, which are not necessarily associated to the energy, we will show that this new approach simplifies the most delicate step of the topological derivative calculation, namely, the asymptotic analysis of the adjoint state.     

A computational analysis of the hydrodynamic instability of a liquid jet focused into a converging microchannel

The instability of a focused liquid jet is studied by semi-analytical methods and by methods of computational fluid dynamics. The semi-analytical approach relies on earlier work on the instability of an extending liquid thread and is based on the Stokes flow regime and small-amplitude perturbations. The evolution of different excitation modes is evaluated and compared. Through hydrodynamic focusing and the corresponding extensional flow an initially stable mode may become unstable and it depends on the position away from the inlet which mode is to be regarded as the most unstable one. When plotting a hypothetical jet decay length against the excitation wave number, a comparatively broad minimum is exhibited. The CFD simulations based on the volume-of-fluid method show that the jet may break up either in the conical focusing zone or in the attached capillary, depending on the flow velocity. When the deformation of the jet surface reaches a certain amplitude, the jet assumes a “beads-on-a-string” structure instead of a shape derived from a harmonic perturbation. A jet decay within the capillary produces elongated droplets with cusped ends. When comparing the results of the CFD and the semi-analytical model, it turns out that the CFD simulations predict more stable jets with a larger decay length. An analysis of the flow velocity field shows that the increased stability might be due to the interaction of the jet with the channel walls.     

强迫谐振动下连续体结构拓扑优化

应用结构拓扑优化ICM(独立连续映射)方法,对强迫谐振动下结构拓扑优化问题建立了以重量极小为目标,位移幅值为约束的优化模型.位移幅值采用一阶泰勒展式近似,由于拓扑优化中设计变量数目通常很多,对强迫谐振动位移幅值的敏度分析推导了伴随法公式,使得一次敏度分析可以计算出对所有设计变量的偏导数,克服了采用直接法敏度分析中一次只能计算出对一个设计变量的偏导数的不足.算例表明用伴随法分析敏度在结构拓扑优化中可以大幅提高计算效率,ICM方法采用独立于截面及形状参数的拓扑优化设计变量更清晰地反映了拓扑优化的本质.