Concurrent topology optimization for minimization of total mass considering load-carrying capabilities and thermal insulation simultaneously

The present work introduces a novel concurrent optimization formulation to meet the requirements of lightweight design and various constraints simultaneously.Nodal displacement of macrostructure and effective thermal conductivity of microstructure are regarded as the constraint functions, which means taking into account both the loadcarrying capabilities and the thermal insulation properties.The effective properties of porous material derived from numerical homogenization are used for macrostructural analysis. Meanwhile, displacement vectors of macrostructures from original and adjoint load cases are used for sensitivity analysis of the microstructure. Design variables in the form of reciprocal functions of relative densities are introduced and used for linearization of the constraint function. The objective function of total mass is approximately expressed by the second order Taylor series expansion. Then, the proposed concurrent optimization problem is solved using a sequential quadratic programming algorithm, by splitting into a series of sub-problems in the form of the quadratic program. Finally, several numerical examples are presented to validate the effectiveness of the proposed optimization method. The various effects including initial designs, prescribed limits of nodal displacement, and effective thermal conductivity on optimized designs are also investigated. An amount of optimized macrostructures and their corresponding microstructures are achieved.

A modified model for concurrent topology optimization of structures and materials

This paper presents a study on the concur- rent topology optimization of a structure and its material microstructure. A modified optimization model is proposed by introducing microstructure orientation angles as a new type of design variable. The new model is based on the assumptions that a structure is made of a material with the same microstructure, and the material may have a different orientation within the design domain of the structure. The homogenization theory is applied to link the material and structure scales. An additional post-processing technique is developed for modifying the obtained design to avoid local optima caused by the use of orientation angle variables. Numerical examples are presented to illustrate the viabil- ity and effectiveness of the proposed model. It is found that significant improvement in structural performance can be achieved by optimizing the orientation of microstructures in concurrent topology optimization of structures and materials.

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 independent 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 interpolation scheme has been applied to achieve the discrete optimization design of the fiber winding angle. An optimization model based on the minimum structural compliance and the specified fiber material volume constraint has been established. The sensitivity information about the two geometrical scales design variables are also deduced considering the characteristics 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 further explore the coupling effect between the macro-structure and micro-material of the composite to achieve an ultralight design of the composite frame structure. The novel two geometrical scales optimization model provides a new opportunity for the design of composite structure in aerospace and other industries.

Concurrent optimization of structural topology and infill properties with a CBF-based level set method

In this paper,a parametric level-set-based topology optimization framework is proposed to concurrently optimize the structural topology at the macroscale and the effective infill properties at the micro/meso scale.The concurrent optimization is achieved by a computational framework combining a new parametric level set approach with mathematical programming.Within the proposed framework,both the structural boundary evolution and the effective infill property optimization can be driven by mathematical programming,which is more advantageous compared with the conventional partial differential equatiodriven level set approach.Moreover,the proposed approach will be more efficient in handling nonlinear problems with multiple constraints.Instead of using radial basis functions(RBF),in this paper,we propose to construct a new type of cardinal basis functions(CBF)for the level set function parameterization.The proposed CBF parameterization ensures an explicit impose of the lower and upper bounds of the design variables.This overcomes the intrinsic disadvantage of the conventional RBF-based parametric level set method,where the lower and upper bounds of the design variables oftentimes have to be set by trial and error;A variational distance regularization method is utilized in this research to regularize the level set function to be a desired distanceregularized shape.With the distance information embedded in the level set model,the wrapping boundary layer and the interior infill region can be naturally defined.The isotropic infill achieved via the mesoscale topology optimization is conformally fit into the wrapping boundary layer using the shape-preserving conformal mapping method,which leads to a hierarchical physical structure with optimized overall topology and effective infill properties.The proposed method is expected to provide a timely solution to the increasing demand for multiscale and multifunctional structure design.

Self-consistent Clustering Analysis-Based Moving Morphable Component(SMMC)Method for Multiscale Topology Optimization

Current multiscale topology optimization restricts the solution space by enforcing the use of a few repetitive microstructures that are predetermined,and thus lack the ability for structural concerns like buckling strength,robustness,and multi-functionality.Therefore,in this paper,a new multiscale concurrent topology optimization design,referred to as the self-consistent analysis-based moving morphable component(SMMC)method,is proposed.Compared with the conventional moving morphable component method,the proposed method seeks to optimize both material and structure simultaneously by explicitly designing both macrostructure and representative volume element(RVE)-level microstructures.Numerical examples with transducer design requirements are provided to demonstrate the superiority of the SMMC method in comparison to traditional methods.The proposed method has broad impact in areas of integrated industrial manufacturing design:to solve for the optimized macro and microstructures under the objective function and constraints,to calculate the structural response efficiently using a reduced-order model:self-consistent analysis,and to link the SMMC method to manufacturing(industrial manufacturing or additive manufacturing)based on the design requirements and application areas.

时域动载荷作用下多微结构多尺度并行动力学拓扑优化

采用拓扑优化方法对含多种多孔材料的结构进行结构与材料微结构构型一体化设计,可以获得具有优良力学性能的结构设计。该文面向多晶胞双尺度结构的时域动刚度最优设计问题,考虑不同晶胞间的可连接性,并行设计微结构的构型及其宏观布局。首先,引入双Helmholtz平滑-分块投影方案,识别不同多孔材料的宏观结构域。其次,通过均匀化方法计算多孔材料的宏观等效力学性能,利用有序SIMP(soid isotropic material with penalization)方法优化不同微观结构的宏观布局。同时,为了保证不同晶胞间的可连接性,在不同多孔材料微结构的边界区域设置为相同拓扑描述的可设计连接域。然后,基于先离散-后微分的伴随敏度分析方法,实现了时空离散动力系统的一致性敏度计算。最后,以双尺度结构动柔度最小化为目标,以材料用量为约束条件,提出了时域动载荷作用下多微结构多尺度并行动力学拓扑优化方法。数值算例结果表明,提出的优化方法能够实现多晶胞结构的构型与宏观布局设计,充分提高了多孔结构的承载性能,同时保证不同晶胞之间的几何连续性,研究结果可为高承载多孔材料结构设计提供理论参考。

Multi-scale design and optimization for solid-lattice hybrid structures and their application to aerospace vehicle components

By integrating topology optimization and lattice-based optimization,a novel multi-scale design method is proposed to create solid-lattice hybrid structures and thus to improve the mechanical performance as well as reduce the structural weight.To achieve this purpose,a two-step procedure is developed to design and optimize the innovative structures.Initially,the classical topology optimization is utilized to find the optimal material layout and primary load carrying paths.Afterwards,the solid-lattice hybrid structures are reconstructed using the finite element mesh based modeling method.And lattice-based optimization is performed to obtain the optimal crosssection area of the lattice structures.Finally,two typical aerospace structures are optimized to demonstrate the effectiveness of the proposed optimization framework.The numerical results are quite encouraging since the solid-lattice hybrid structures obtained by the presented approach show remarkably improved performance when compared with traditional designs.

双尺度分级结构时域动刚度问题的并行拓扑优化

提出双尺度分级结构时域动刚度问题的并行拓扑优化方法,实现结构宏观拓扑构型和材料微观分布协同优化。利用三场密度方法实施宏观与微观结构的设计,基于能量的均匀化方法(energy-based homogenization method, EBHM)在微观结构上计算等效的宏观力学特性。针对比例阻尼系统,将HHT-α方法作为时间积分算法,求解多尺度结构的动力学有限元模型。融合先离散-后微分方法与伴随方法,在时间与空间离散的拓扑优化模型上实施敏度分析,避免了将时间作为连续变量而导致的灵敏度计算的一致性误差问题。以多尺度结构的动柔度最小化为目标,宏观与微观结构体积为约束,分别求解了正弦半波和余弦半波冲击载荷作用下多尺度结构的并行拓扑优化问题。最后,通过2种典型算例的数值结果验证了所提算法的有效性。

利用聚类分析的宏细观多尺度并行拓扑优化设计方法研究

多孔材料因为具有质量轻、比刚度和比强度大、隔振和隔热效果好等优点,越来越多的应用在航空航天和制造装备等领域。为充分发挥多孔材料的性能,本文提出同时考虑结构的宏观性能和细观微结构性能的多尺度并行拓扑优化设计方法,获得性能优良的多孔结构。论文采用聚类方法有效降低了计算成本,针对并行优化难收敛的问题提出改进模型,使迭代平稳收敛。最后以经典的悬臂梁、MBB梁和Michell结构为例进行优化设计,通过对宏细观多尺度并行优化结果的分析,验证了所提方法的有效性与正确性。

考虑尺寸效应的模块化结构两层级优化设计

出于构造、美观、生产工艺、质量控制、降低成本等考虑,在很多工业应用中,整个结构或装备可被规整的分成几个相同的子区域,各个子区域的结构形式完全相同,这样的子区域称为基本设计模块.整个结构或装备可通过有限个基本设计模块重复拼装而成.针对此类模块化结构,考虑结构与模块间的耦合作用,提出了结构、模块两层级并发优化设计的模型与求解方法,研究了基本设计模块的绝对尺寸对优化结果的影响.通过在结构和模块两个层次上分别引入独立的人工密度变量,借助拓扑优化技术和惩罚策略,给出了最优的设计模块构型以及模块在结构尺度上的最优分布,通过单工况和多工况数值算例验证了该方法的有效性.

基于均匀材料微结构模型的热弹性结构与材料并发优化

研究由宏观上均匀多孔材料制成的结构的优化设计问题,待设计的结构受到给定的外力与温度载荷作用,优化设计旨在给定结构允许使用的材料体积约束下,设计宏观结构的拓扑及多孔材料的微结构,使得结构柔度最小。本文提出了一种宏观结构与微观单胞构型并发优化设计的方法,在此方法中,引入宏观密度和微观密度两类设计变量,在微观层次上采用带惩罚的实心各向同性材料法SIMP(Solid Isotropic Material with Penalty),在宏观层次上采用带惩罚的多孔各向异性材料法PAMP(Porous Anisotropic Material with Penalty),借助均匀化方法建立两个层次间的联系,通过优化方法自动确定实体材料在结构与材料两个层次上的分配,得到优化设计;提供的数值算例检验了本文所提方法及计算模型,并讨论了温度变化、材料体积及计算参数对优化结果的影响。研究结果表明同时考虑热和机械载荷时,采用多孔材料可以降低结构柔顺性。

考虑均一微结构的结构／材料两级协同优化

以结构最小柔顺性为目标,提出了考虑均一微结构的结构/材料两级协同优化方法。出于制造考虑,假设了材料微结构在宏观上具有相同的构形。为实现拓扑优化,本方法在两个尺度上独立定义了单元密度作为设计变量,分别引入了SIMP(Solid Isotropic Material with Penalization)和PAMP(Porous Anisotropic Material with Penalization)方法对密度进行惩罚,并且采用了周长约束控制微结构拓扑的复杂度。借助均匀化方法建立了结构和材料之间的联系,从而将两个尺度上的设计纳入到一个优化模型中,实现了协同求解。数值算例验证了本方法的有效性和正确性,讨论了各参数的影响,优化结果体现了类桁架材料的优越性。

结构构型与结构间连接方式协同优化设计

通常的结构拓扑优化是在给定外部作用下的构型设计.然而对于复杂结构,各部件由连接构件相连并通过它传递外部作用,所以连接方式影响部件承受载荷的分布,即连接构件的布局对部件的最优构型有重要影响.研究结构部件构型与部件之间连接方式的协同优化问题,在连接区域引入一种特殊的材料模型,以材料的分布描述连接方式.将承力结构与连接构件域的材料密度同时作为设计变量,提出了一种基于拓扑优化思想的连接方式与结构拓扑优化协同设计的优化模型和相应的求解方法.给出的算例证明了这种设计方法与优化策略的有效性.

基于IPTO算法的连续体结构可靠性拓扑优化

研究结构几何尺寸、结构材料体积及作用载荷不确定条件下的连续体结构可靠性拓扑优化问题,提出基于改进比例拓扑优化(Improved Proportional Topology Optimization,IPTO)算法的可靠性拓扑优化方法。基于概率方法描述变量的不确定性,建立综合考虑材料体积约束和可靠性约束且使柔度最小的连续体结构可靠性拓扑优化数学模型;为提高计算效率,使用基于解耦思想的混合法将可靠性拓扑优化问题解耦成可靠性分析和当量确定性拓扑优化两个独立子问题。在可靠性分析阶段,遵循一阶可靠度法中可靠性指标的几何意义,获取满足可靠性约束且服从正态分布的随机变量,并通过逆变换对其进行修正;在当量确定性拓扑优化阶段,以修正后的随机变量作为设计参数,使用具有无需获取灵敏度信息功能的IPTO算法对结构执行拓扑优化设计。最后使用MBB梁和悬臂梁两个算例证实了所提方法的有效性。

基于热固耦合问题的多尺度拓扑优化设计

基于均质材料的拓扑优化逐渐难以适应现代化生产对工业产品高品质、轻量化的需求,同时很多工业设备经常面临着高温和高负载的工作环境。为了提高结构在温度载荷和机械载荷共同作用下的力学性能,本文提出了一种在稳态热源作用下的热固耦合连续体结构并行化拓扑优化方法。以结构刚度作为目标函数,材料的体分比为约束,利用能量均匀化方法预测微结构的等效属性,建立热固耦合结构并行化拓扑优化数学模型。为便于数值计算,利用直接法进行灵敏度分析,同时采用Heaviside非线性密度过滤技术抑制数值不稳定性,将OC准则算法用于求解优化问题。数值算例表明,本文方法能够有效地进行优化设计且能显著地提高结构的刚度性能和散热性能。

Reduced Order Machine Learning Finite Element Methods:Concept,Implementation,and Future Applications Dedicated to Professor Karl Stark Pister for his 95th birthday

This paper presents the concept of reduced order machine learning finite element(FE)method.In particular,we propose an example of such method,the proper generalized decomposition(PGD)reduced hierarchical deeplearning neural networks(HiDeNN),called HiDeNN-PGD.We described first the HiDeNN interface seamlessly with the current commercial and open source FE codes.The proposed reduced order method can reduce significantly the degrees of freedom for machine learning and physics based modeling and is able to deal with high dimensional problems.This method is found more accurate than conventional finite element methods with a small portion of degrees of freedom.Different potential applications of the method,including topology optimization,multi-scale and multi-physics material modeling,and additive manufacturing,will be discussed in the paper.

Concurrent Analysis and Design of Structure and Its Periodic Material

The specific good properties of cellular materials and composite materials, such as low density and high permeability, make the optimal design of such materials necessary and at- tractive. However, the given materials for the structures may not be optimal or suitable, since the boundary condition and applied loads vary in practical applications; hence the macro-structure and its material micro-structure should be considered simultaneously. Although abundant studies have been reported on the structural and material optimization at each level, very few of them considered the mutual coordination on both scales. In this paper, two FE models are built for the macro-structure and the micro-structure, respectively; and the effective elastic properties of the periodic micro-structure are blended into the analysis of macro-structure by the homogenization theory. Here, a topological optimum is obtained by gradually re-distributing the constituents within the micro-structure and updating the topological shape at the macro-structure until converges are achieved on both scales. The mutual coordination between the roles of micro-scale and macro-scale is considered. Some numerical examples are presented, which illustrate that the proposed optimization algorithm is effective and highly efficient for the micro-structure design and macro-structure optimization. For the composite design, one can see reasonable effects of the stiffness of base materials on the resultant topologies.

尺度关联的微结构构型与排布的材料/结构动力学设计

考虑微观尺度和宏观尺度的关联,将微结构胞元设计和多尺度均匀化设计结合,建立了由相同尺寸和内部构型的微结构胞元组成的复合材料结构的材料/结构一体化动力学优化设计方法,给出了相应的算例。方法中引入了微观和宏观两个尺度上的独立密度变量,采用材料属性的合理近似模型对密度进行惩罚,利用有限元超单元技术建立材料与结构的联系。基于算例获得了超单元尺寸与微观、宏观材料用量对结构拓扑构型的影响规律。将算例结果与已有方法的结果比较,表明本文方法具有合理有效性,可作为对轻质结构进行动力学设计的一种新方法。

考虑泊松效应的材料/结构一体化设计方法

为实现含有不同泊松比组分复合材料的优化设计,并考虑宏观结构及复杂的边界条件,提出了考虑泊松效应的材料/结构一体化设计方法,其显著特征在于不同组分材料中引入了泊松比插值,假设宏观结构由周期性排列的复合材料组成,复合材料含两种各向同性且泊松比不同的组分材料,以静态问题中柔顺度最小化或动态问题中特征值最大化为目标以及宏微观体积比为约束建立了拓扑优化模型。采用均匀化理论预测了复合材料等效性能,推导了目标函数对宏微观密度变量的敏度表达式。分别采用密度过滤和敏度过滤来消除宏微观拓扑优化中的不稳定性现象。采用优化准则法分别更新宏观、微观密度变量,考察了微观体积比和组分材料泊松比参数对优化结果的影响。三维数值算例结果表明所提出的一体化方法具有可行性和优越性。

稳态热传导下基于多相材料的一体化设计

提出一种基于多相多孔材料的一体化模型。模型以结构散热弱度最小为目标,以结构体积比与材料微结构质量为约束。引入独立的宏观设计变量和微观相设计变量,通过单元相密度建立两类变量间的联系;基于对等混合材料插值模型建立单元相密度与热传导系数间的惩罚关系;推导得到目标函数灵敏度表达式。求解偏微分方程实现单元散热弱度过滤,消除棋盘格及网格依赖性现象。通过二维数值算例讨论并分析了材料特性、热载荷、体积比约束以及质量约束对一体化优化结果的影响。数值实验结果表明,该建模方法在多相材料/结构一体化稳态热传导优化设计中具有可行性和有效性。