Topology optimization of 3D structures with design-dependent loads

Topology optimization of continuum structures with design-dependent loads has long been a challenge. In this paper, the topology optimization of 3D structures subjected to design-dependent loads is investigated. A boundary search scheme is proposed for 3D problems, by means of which the load surface can be identified effectively and efficiently, and the difficulties arising in other approaches can be overcome. The load surfaces are made up of the boundaries of finite elements and the loads can be directly applied to corresponding element nodes, which leads to great convenience in the application of this method. Finally, the effectiveness and efficiency of the proposed method is validated by several numerical examples.     

Topology optimization of piezoelectric nanostructures

We present an extended finite element formulation for piezoelectric nanobeams and nanoplates that is coupled with topology optimization to study the energy harvesting potential of piezoelectric nanostructures. The finite element model for the nanoplates is based on the Kirchoff plate model, with a linear through the thickness distribution of electric potential. Based on the topology optimization, the largest enhancements in energy harvesting are found for closed circuit boundary conditions, though significant gains are also found for open circuit boundary conditions. Most interestingly, our results demonstrate the competition between surface elasticity, which reduces the energy conversion efficiency, and surface piezoelectricity, which enhances the energy conversion efficiency, in governing the energy harvesting potential of piezoelectric nanostructures.     

桁架结构非概率可靠性拓扑优化

考虑非概率可靠性的拓扑优化对于非确定参数和荷载条件下结构的概念设计具有重要意义,有关研究国内外少见报道.本文利用凸模型理论,考虑优化迭代过程的需要,提出改进的非概率可靠性指标的定义,并针对桁架结构拓扑优化设计问题建立了以杆件截面积为设计变量、结构重量极小化为目标、具有非概率可靠性指标约束的广义尺寸优化数学模型.本文指出,考虑桁架结构参数的不确定性的条件下所得到的最优杆件布局与确定性优化所得到的结果可能有显著不同.对文中提出的数学模型,采用数学规划算法求解,数值算例结果令人满意.本文工作表明了桁架结构非概率可靠性拓扑优化设计的可行性和所提出算法的有效性.     

狭长结构拓扑优化

通常的拓扑优化是在给定区域内,通过设计材料分布实现结构拓扑形式优化。对于设计区域的长和宽相近的平面问题,现行的方法可得到清晰的拓扑。但是,狭长结构的设计域具有大的长宽比。为了保证基结构包含足够多的拓扑形式,宽度方向要求有一定量的有限元分割,从而导致整体网格数和设计变量多、问题求解困难。本文提出了通过基本结构拼装的狭长结构拓扑优化方法,建立了以最小平均柔顺性密度为目标、同时设计材料分布和设计域几何尺度的基本结构的拓扑优化问题的数学提法和求解方法。利用所提出的问题提法和求解方法,设计了狭长悬臂梁的拓扑形式,讨论了危险截面的弯矩与剪力的相对值以及材料体积约束对拓扑形式的影响。数值结果表明,不同的弯矩与剪力的相对数值对应不同的拓扑形式,随着相对数值的增加,梁的拓扑形式由类桁架结构逐渐变成竖直立板加强的框架式结构。     

Topology optimization design of continuum structures under stress and displacement constraints

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应力约束下预应力平面实体钢结构拓扑优化设计

对预应力平面实体钢结构拓扑优化设计问题进行研究,建立了以索力值、单元尺寸和结构拓扑为设计变量,以应力为约束条件,以结构重量最小为目标函数的数学优化模型.在求解方法上,首先以满应力设计准则法建立索力值与结构重量之间的优化模型,通过该模型的求解确定索力值,并通过满应力设计选取单元尺寸,然后对单元体厚度按闻值进行分类,从而删除低厚度单元实现结构的拓扑优化.算例结果与预应力钢结构理论相吻合,表明本文所提出的方法是有效的,可为预应力钢结构体系创新提供理论方法.     

Concurrent multi-scale design optimization of composite frame structures using the Heaviside penalization of discrete material model

This paper deals with the concurrent multi-scale optimization design of frame structure composed of glass or carbon fiber reinforced polymer laminates. In the composite frame structure, the fiber winding angle at the micro-material scale and the geometrical parameter of components of the frame in the macro-structural scale are introduced as the inde-pendent variables on the two geometrical scales. Considering manufacturing requirements, discrete fiber winding angles are specified for the micro design variable. The improved Heaviside penalization discrete material optimization inter-polation scheme has been applied to achieve the discrete optimization design of the fiber winding angle. An optimiza-tion model based on the minimum structural compliance and the specified fiber material volume constraint has been estab-lished. The sensitivity information about the two geometrical scales design variables are also deduced considering the char-acteristics of discrete fiber winding angles. The optimization results of the fiber winding angle or the macro structural topology on the two single geometrical scales, together with the concurrent two-scale optimization, is separately studied and compared in the paper. Numerical examples in the paper show that the concurrent multi-scale optimization can fur-ther explore the coupling effect between the macro-structure and micro-material of the composite to achieve an ultra-light design of the composite frame structure. The novel two geometrical scales optimization model provides a new oppor-tunity for the design of composite structure in aerospace and other industries.     

基于移动可变形孔洞方法的超弹性结构拓扑优化

现有隐式拓扑优化方法在进行超弹性结构拓扑优化设计时,具有设计变量多、中间设计有限元分析存在严重的收敛性和设计结果无法直接导入CAD/CAE系统等问题。为解决这些问题,提出了一种基于移动可变形孔洞的显式拓扑优化方法来进行承受大变形的超弹性结构设计,材料本构采用常用的Mooney-Rivlin模型。首先,介绍了移动可变形孔洞方法的基本思想和可变形孔洞的显式描述方法;其次,构造了基于移动可变形孔洞方法的超弹性结构拓扑优化的数学列式,给出了相应的灵敏度结果;最后,通过数值算例验证了本方法的有效性。数值结果表明,该方法可以通过较少的设计变量和非常稳健的优化过程,给出边界由B样条曲线描述且可与CAD/CAE软件无缝连接的超弹性结构设计。     

多约束作用下连续体结构的拓扑优化

基于ICM(独立、连续、映射)方法建立了以结构重量最小为目标,以屈曲临界力、位移及应力三种约束同时作用的连续体拓扑优化模型:采用独立的连续拓扑变量,借助泰勒展式、过滤函数将目标函数作二阶近似展开。借助瑞利商、泰勒展式、过滤函数将屈曲约束化为近似显函数,借助于过滤函数,将位移约束用莫尔定理显式化;将应力这种局部性约束采用全局化策略进行处理,即借助第四强度理论、过滤函数将应力局部性约束转化为应变能约束,大大减少了灵敏度分析的计算量;将优化模型转化为对偶规划,减少了设计变量的数目,并利用序列二次规划求解,缩小了模型的求解规模。数值算例表明,ICM方法在解决屈曲、位移及应力三种约束共同作用的连续体拓扑优化问题上有优势。     

Plate/shell structure topology optimization of orthotropic material for buckling problem based on independent continuous topological variables

The purpose of the present work is to study the buckling problem with plate/shell topology optimiza-tion of orthotropic material.A model of buckling topology optimization is established based on the independent,con-tinuous, and mapping method, which considers structural mass as objective and buckling critical loads as constraints. Firstly, composite exponential function (CEF) and power function(PF)as filter functions are introduced to recognize the element mass,the element stiffness matrix,and the ele-ment geometric stiffness matrix.The filter functions of the orthotropic material stiffness are deduced. Then these fil-ter functions are put into buckling topology optimization of a differential equation to analyze the design sensitiv-ity.Furthermore,the buckling constraints are approximately expressed as explicit functions with respect to the design vari-ables based on the first-order Taylor expansion.The objective function is standardized based on the second-order Taylor expansion. Therefore,the optimization model is translated into a quadratic program.Finally,the dual sequence quadratic programming(DSQP)algorithm and the global convergence method of moving asymptotes algorithm with two different filter functions(CEF and PF)are applied to solve the opti-mal model.Three numerical results show that DSQP&CEF has the best performance in the view of structural mass and discretion.     

采用自动分组遗传算法的频率约束下桁架拓扑优化

研究了带有频率约束的桁架结构拓扑优化问题.首先比较了同为铰结情况下采用两种不同单元-杆单元和梁单元模拟对频率计算结果的影响,发现杆模型会丢失杆件弯曲振动模态,得到的最大自振频率设计往往含有很细截面的杆件,而实际上具有很低的频率.为克服这一困难,并考虑到实际工程结构的结点具有一定的刚性,建议此类优化问题可以采用刚架模型进...     

Topology optimization of truss structures with systematic reliability constraints under multiple loading cases

In this paper, a mathematical model for topology optimization of truss structures with constraints of displacement and system reliability under multiple loading cases is constructed. In order to avoid the difficulty of computing the structure's system reliability, a solving approach is presented in which the failure probability of system is divided into the sum of all bars' failure probability by means of reliability distribution. In addition, by drawing into the reliability safety factor and the fundamental relationship in structural mechanics, all probability constraints of displacement and stress are equivalently displayed as conventional form and linear function of the design variables. The optimization problem with multiple constraints is treated by the compact constraint tactics and is solved by the improved simplex method. The examples show that the approach proposed in this paper is feasible and efficient. The project supported by the National Natural Science Foundation of China.     

基于IGA-SIMP法的连续体结构应力约束拓扑优化

建立了一种IGA-SIMP框架下的连续体结构应力约束拓扑优化方法。基于常用的SIMP模型,将非均匀有理B样条（NURBS）函数用于几何建模、结构分析和设计参数化,实现了结构分析和优化设计的集成统一。利用高阶连续的NURBS基函数,等几何分析（IGA）提高了结构应力及其灵敏度的计算精度,增加了拓扑优化结果的可信性。为处理大量局部应力约束,提出了基于稳定转换法修正的P-norm应力约束策略,以克服拓扑优化中的迭代振荡和收敛困难。通过几个典型平面应力问题的拓扑优化算例表明了本文方法的有效性和精确性。应力约束下的体积最小化设计以及体积和应力约束下的柔顺度最小化设计的算例表明,基于稳定转换法修正的约束策略可以抑制应力约束体积最小化设计中的迭代振荡现象,获得稳定收敛的优化解;比较而言,体积和应力约束下的柔顺度最小化设计的迭代过程更加稳健,适合采用精确修正的应力约束策略。     

薄壁结构多层级并发加筋拓扑优化研究

新一代航天装备的主承力薄壁舱段在追求极致轻量化的同时,还具有更高的刚度和抗屈曲等设计指标.传统结构形式和设计方法难以满足轻质高承载的设计要求.为此,本文提出了一种薄壁结构多层级并发加筋拓扑优化方法,通过构建主层级稀疏加筋和次层级密集点阵增强结构整体和局部力学性能,扩展结构设计空间,有效提升材料利用率.其中,主层级加筋布...     

预应力索-桁架结构形状优化设计

就预应力索-桁架结构形状优化设计问题,考虑了施加预应力阶段、预应力与荷载共同作用阶段的性态约束条件,建立了设计变量包括截面尺寸、索力值、杆件及索节点坐标的形状优化数学模型;在求解方法上将设计变量分为两个子空间:第一子空间为索力值和截面尺寸优化设计空间,第二子空间为形状优化设计空间; 第一子空间给出新的求解方法以减轻结构重量,第二子空间用节点渐进法优化结构形状和布索位置以增加结构刚度.算例表明,该方法能使结构重量逐步减轻,结构刚度不减,形状逐步达到最优.     

基于离散变量的多材料结构拓扑优化设计

多材料结构拓扑优化相比于单材料结构优化具有更大的设计空间,展现出巨大的设计潜力和应用前景。本文基于离散设计变量研究多材料结构拓扑优化,通过建立有效的设计优化方法,获得轻质高效的结构设计。首先,根据离散变量的特点,建立多材料结构拓扑优化模型,由离散变量下约束函数和目标函数的灵敏度信息构造序列近似整数规划子问题,采用运动极限策略限制设计变量的改变量以保证整数规划子问题的近似精度。为了解决离散变量的组合复杂性,采用具有多项式复杂度的正则松弛算法来高效求解(和经典的连续变量算法效率相当)。本文利用多材料结构最小柔顺性优化问题以及最大传热效率优化问题研究离散变量方法的有效性。数值算例表明,离散变量优化方法可以获得稳定迭代收敛的多材料拓扑设计,优化构型中的材料界面清晰分明,有效避免了出现模糊区域。     

用拓扑优化设计多体分比蜂窝状结构

提出用拓扑优化及增材制造法设计同一材料制作的蜂窝状结构,并把设计可归结为在不同的设计域内使用不同的体分比的拓扑优化问题。首先推导了多体分比约束条件下的蜂窝状结构拓扑优化问题;然后提出了基于多阈值的等值面法及与商用有限元分析软件对接的算法。给出了有三个体积约束的最小柔顺性问题的数值算例。计算结果表明,对于具有三个体积约束的拓扑优化设计问题,存在一个最佳的体积约束组合;使用等量的材料,与两个体积约束的最佳拓扑设计相比,用最佳体积约束组合所得到的优化结构的柔顺性还要小10%。     

面向增材制造的双蒙皮夹层结构加筋拓扑优化方法

双蒙皮夹层结构是航空航天装备中的特殊承力结构,其典型代表为发动机尾喷管中的同步环构件。近年来,增材制造技术为该类薄壁结构的创新型设计提供了有利条件。但增材制造有其特殊的工艺要求,基于传统拓扑优化得到的设计结果往往存在大量的悬空区域,无法直接应用于增材制造工艺。因此,需要在优化设计阶段统筹考虑结构的力学性能和自支撑工艺约束。针对上述问题,本文提出了一种面向增材制造的双蒙皮夹层薄壁结构加筋拓扑优化方法,可在一次优化中同时得到优化的加筋布局和非均匀点阵分布,从而解决悬空结构的支撑问题,确保优化结果的工艺可达性。为了平衡计算成本和分析精度,本文采用渐进均匀化方法来求解不同类型单胞等效弹性性能,以适应不同复杂单胞构型。基于上述方法,本文给出了某发动机同步环结构的拓扑优化算例,结果表明,本文优化设计方法可以实现双蒙皮夹层结构中夹层加筋和点阵的共同优化,为航空航天装备中发动机同步环结构轻量化设计提供了思路。     

Influence of softening on fracture propagation in large-scale shell structures

In this study, fracture propagation in large shell elements is modelled with the softening law. This law is given in a general form, enabling investigations of different softening behaviours to be conducted. The final fracture is simulated by removing elements. The softening parameters are derived using the energy-based representative volume element (RVE) approach. Tracing crack propagation through the RVE defines the physically justified softening parameters for the current model. The softening model is implemented into ABAQUS using VUMAT subroutines for the shell elements. A large-scale tearing experiment is simulated with the current model and RVE-based softening parameters. In addition, the softening laws from the literature have been used. The fracture propagation is assessed in terms of plastic energy dissipation in the RVE and the whole structure, load–displacement, and crack growth. The RVE-based model is shown to have better performance compared with other models from the literature.     

On `best' shell models – From classical shells, degenerated and multi-layered concepts to 3D

Summary Problems of solid mechanics are most generally formulated within 3D continuum mechanics. However, engineering models favor reduced dimensions, in order to portray mechanical properties by surface or curvilinear approximations. Such attempts for dimensional reduction constitute interactions between theoretical formulations and numerical techniques. A classical reduced model for thin bodies is represented by shell theory, an approximation in terms of resultants and first-order moments. If the shell theory, with its inherent errors, is considered as qualitatively insufficient for a particular problem, a further improvement is given by solid shell models, which are gained by direct linear interpolation of the 3D kinematic relations. They improve considerably the analytic capabilities for shells, especially when their congenital locking effects are handled by variational `convergence tricks'. The next step towards 3D quality are layered shells or solid shell elements. The present paper compares these three approximation stages from the point of view of multi-director (integral) transformations of classical continuum mechanics. It offers physical convergence requirements for each of the treated models. Partial support to the present study by the German Science Foundation (DFG) within the Special Research Center (SFB) 398 is gratefully acknowledged.