面向金属增材制造的拓扑优化设计研究进展 
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Owing to the continuous developments in the Chinese aerospace industry, aviation structural parts need to ensure lightweight, high efficiency, long flight time, and high maneuverability characteristics. Therefore, it is a significant challenge in structural optimization design to further reduce the structural quality coefficient. Traditional lightweight designs are mostly based on the replacement of classical structures with equivalent parts, such as lean improvement and excavation of structural potential using the new processes and new materials, and have now approached the “ceiling”.
Topology optimization technology, as an important branch of structural optimization design, determines the optimal material distribution and best load-bearing path by defining material properties, load conditions, and constraints. It is an effective design method for obtaining lightweight structure design and high-performance innovative configurations, and has been widely used in aerospace, automobile manufacturing, and other fields.
However, topology configurations are usually complex. Limited by traditional manufacturing processes, designers often need to simplify the optimal topology configurations, which fails to fully reflect the structural advantages of topology optimization design. Additive manufacturing technology uses high-energy laser beams and adopts the superposition mode of “bottom to up” layer-by-layer material melting, which can realize rapid prototyping and solid-free manufacturing of complex topology configurations without molds. This method addresses the problem of “manufacturing determines design” in structure optimization, which greatly broadens the design space. However, additive metal manufacturing technology is not completely a “free manufacturing” technology, and it is limited by unique manufacturing constraints. Therefore, considering additive manufacturing constraints in topology optimization design, researching and developing topology optimization design for additive metal manufacturing has a broad application prospect.
This study reviews the progress of structural topology optimization design for metal additive manufacturing technology. First, it summarizes the common methods and characteristics of continuum structure topology optimization (Table 1) and compares the cantilever beam topology optimization results obtained with different methods (Fig.1). From the perspective of optimizing topology algorithms, it concludes the effective measures to improve structural continuity and manufacturability based on topology optimization methods of elements and boundary evolution (Figs.2-4). Then, it expounds on the principle, processing characteristics, and application range of the mainstream metal additive manufacturing technology (Fig.5). After that, it summarizes the topology optimization methods considering the geometric size (Fig.8), structural forming (Figs.9 and 10), and material property constraints (Fig.11) of metal additive manufacturing technology (Fig.7 and Table 2). Finally, it prospects the development directions of metal additive manufacturing and topology optimization technology.
In this study, the structural topology optimization of an advanced design technology is integrated with the metal additive manufacturing technology that is an advanced manufacturing technology. This study summarizes the methods, characteristics, and improvement measures of continuum structure topology optimization design. Moreover, it expounds on the principle, characteristics, and application of metal additive manufacturing technology. In addition, it summarizes and prospects the topology optimization methods considering the constraints of metal additive manufacturing, which will provide a reference for researchers to further study the topology optimization design for metal additive manufacturing technology.
Topology optimization design has shortcomings of numerous design variables, weak convergence, and low computational efficiency. It is often difficult for existing topology optimization algorithms to output the optimal structural performance solution that can be directly used in additive manufacturing. Therefore, combined with the parallel computing technology, it is crucial to carry out algorithm research with fewer design variables and better convergence, and output the optimal solution that can be directly used in additive manufacturing.The research on macroscopic topology optimization and microscopic lattice structure is becoming increasingly improved. By effectively integrating the two, fully leveraging the high-performance configurations of topology optimization design and broad design space provided by additive manufacturing technology, the pursuit of high-performance lightweight design has broad development prospects.The topology optimization methods considering the constraints of metal additive manufacturing adopt relatively ideal material models, which differ from the actual printing materials used in metal additive manufacturing technology. Therefore, establishing a precise topological model of material anisotropy under multiple process parameters, quantification of process parameters of metal additive manufacturing equipment, simulation of the metal additive manufacturing process, and prediction of warping deformation and cracking of parts can effectively reduce residual stress and deformation, and improve forming accuracy and surface quality.Topology optimization design for metal additive manufacturing technology is often based on the optimization of a single material. Effectively combining multi-material topology optimization and metal additive manufacturing, studying the topology optimization design and metal additive manufacturing technology of functional gradient materials, and realizing the integrated design of materials, structure, process, and performance, are breakthrough points in the pursuit of high-performance, multi-function, and lightweight.
1 引言
随着我国航空航天事业的持续发展，航空结构件需满足轻质高效、长航时、高机动性等要求，因此，进一步降低结构质量系数是结构优化设计领域面临的一项严峻挑战［1-3］。传统轻量化设计大多是基于经典结构的等效替换，例如通过新工艺、新材料等精益改善和挖掘结构潜能，现已趋近“天花板”［4］。
拓扑优化技术作为结构优化设计的重要分支［5］，通过定义材料属性、载荷工况与约束条件，寻求给定设计域内材料的最优分布形式，是结构轻量化设计、获得高性能创新构型的有效设计方法，现已被广泛应用到航空航天［6-9］、汽车制造［7-8］等领域中。例如，应用填充微观点阵结构的卫星支架多尺度拓扑优化设计，使卫星支架减重17%，动态响应减少25%［8］；考虑切口、保持传统钣金轮廓的涡轮发动机支架的拓扑优化设计，使发动机支架减重25%；考虑增材制造工艺、扩大设计空间的拓扑优化设计，使发动机支架减重66%，最大位移减少约50%［9］；由30多个单独部件组成的稳定器前翼梁支架，应用拓扑优化一体化设计，成功实现前翼梁支架减重30%，显著改善结构性能，提升加工效率［8］。
然而拓扑构型通常较为复杂，受制于传统制造工艺限制，设计人员往往需要简化最优拓扑构型，这导致拓扑优化的结构优势不能充分体现。增材制造技术使用高能束热源，采用“自下而上”材料逐层熔化沉积的叠加方式，无需模具，可实现复杂拓扑构型的快速“自由制造”，解决了结构优化存在的“制造决定设计”的问题［10-11］，极大地拓宽了设计空间。但金属增材制造技术并不是完全“自由制造”技术，仍存在特有的制造约束，如当拓扑构型最小尺寸小于设备精度时，则会出现打印失败现象［12］；受制于设备成形腔与结构热变形限制，增材制造大型构件时，需进行分块与连接处理［13］；增材制造零件有时会沿构建方向出现20%~40%的强度损失［14］；对于粉末床增材制造技术，在制造含有封闭孔洞的拓扑结构时会出现内部粉末与支撑难以去除［15］等问题。因此，在拓扑优化设计中考虑增材制造约束，发展面向金属增材制造的拓扑优化设计方法具有重要意义［2，15］。
本文首先介绍了连续体结构拓扑优化的常用方法与特点，对比了不同方法的拓扑优化结果，从算法优化的角度，总结了改善拓扑构型连续性的有效措施。随后，阐述了金属增材制造技术的原理、加工特点与适用范围，归纳了考虑金属增材制造几何尺寸约束、结构成形约束、材料性能约束的拓扑优化方法。最后，讨论了现有拓扑优化与金属增材制造领域的发展方向，为学者们深入研究面向金属增材制造的拓扑优化技术提供参考。
2 连续体结构拓扑优化常用方法与特点
根据优化算法迭代与更新的不同形式，连续体结构拓扑优化可分为：基于单元网格的拓扑优化方法，如均匀化法、变密度法、渐进结构法等；基于边界演化的拓扑优化方法，如水平集法、移动可变形组件法、特征驱动结构拓扑优化法等。
表 1. 连续体结构拓扑优化常用方法与特点
Table 1. Common methods and characteristics of continuum structure topology optimization
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2.1 基于单元网格的拓扑优化方法
均匀化方法［16］通过调整单胞结构的几何尺寸与空间方位函数，寻求结构最优拓扑形式，但是所采用的较为复杂的数学模型限制了其普遍应用。变密度法［17］通过定义每个单元的“伪密度”在0~1区间变动，建立了伪密度与弹性模量的关联函数，通过调整惩罚因子（p），减小中间密度，获得较为清晰的拓扑结构。该方法设计变量较少，计算效率较高，应用更为广泛［
图 1. 悬臂梁连续体结构拓扑优化。（a）基于HyperWorks的变密度法；（b）渐进结构法［19］；（c）水平集法［20］；（d）移动可变形组件法［21］；（e）特征驱动法［22］
Fig. 1. Continuum structure topology optimization about cantilever beam. (a) Variable density method using HyperWorks; (b) evolutionary structural optimization method[19]; (c) level set method[20]; (d) moving morphable component method[21]; (e) feature-driven method[22]
相较于基于边界演化的拓扑优化方法［
如
图 2. 基于单元网格的拓扑优化方法的数值不稳定现象与改善措施
Fig. 2. Numerical instability and improvement of topology optimization method based on elements
2.2 基于边界演化的拓扑优化方法
水平集法［40］使用零值水平集函数描绘结构边界，使用Hamilton-Jacobi方程更新水平集函数，结合形状导数与灵敏度分析技术，寻求最佳拓扑结构。移动可变形组件（MMC）法［41］与移动可变形孔洞（MMV）法［42］通过优化设计域中一系列组件轮廓/孔洞边界的尺寸、位置等显式几何信息，得到不同工况下的最优承力路径。相较于传统水平集方法，MMC/MMV法所采用的设计变量明显较少，计算效率较高，可与CAD/CAE软件无缝连接。特征驱动结构拓扑优化方法［43］结合隐式水平集函数描述的结构轮廓工程特征，通过基于梯度的优化方式控制特征结构的移动、缩放等，实现结构特征与拓扑优化的有效融合。
水平集法具有清晰的结构边界，无数值不稳定现象，但该类方法高度依赖初始参数值，存在不能自主开孔［44］（
图 3. 水平集法初始依赖性与不能自主开孔问题［44］。（a）2个初始孔洞；（b）9个初始孔洞；（c）40个初始孔洞
Fig. 3. Initial dependence and inability to open holes of level set method[44]. (a) 2 initial holes; (b) 9 initial holes; (c) 40 initial holes
图 4. MBB梁水平集拓扑优化。（a）拓扑导数［45］；（b）PS-Kriging插值［55］
Fig. 4. MBB beam level set topology optimization.(a) Topological gradient[45]; (b) PS-Kriging interpolation[55]
针对弱收敛问题，Luo等［54］提出基于紧支撑径向基函数，采用更稳定、更高效的积分形式，实现Hamilton-Jacobi方程在时间与空间上的解耦，改善传统水平集法求解困难等弱收敛问题。Guirguis等［55］基于Kriging［
尽管基于显式拓扑框架的移动可变形组件/孔洞法设计变量较少，计算效率较高，优化设计结果边界清晰，可与CAD/CAE软件实现无缝衔接，但该方法存在一定的初始依赖性［67-68］及结构低连续性［
现有拓扑优化方法往往仅考虑结构力学性能提升，而忽视了拓扑结构工程特征属性，常采取先性能后特征的设计模式，可能难以同时满足结构力学性能与工程特征的设计要求。特征驱动结构拓扑优化方法将结构工程特征贯穿到模型构建、有限元分析与拓扑优化整个流程中，设计变量规模较小，在求解大型工程问题时具有明显优势。然而，其优化结果对特征数目与布局有较强的依赖性。借助拓扑导数可以改善与消除结构初始依赖性，结合一阶符号距离函数与KS函数，可获得结构清晰的优化模型［72］。
3 金属增材制造技术原理与特点
增材制造技术是制造业的“革命性”飞跃，打破了传统制造技术的局限，解决了产品研发存在的“制造决定设计”问题。金属增材制造技术作为重要分支，已成为当前实施技术创新、提振本国制造水平的关键着力点［73-74］。如
图 5. 金属增材制造技术。（a）~（e）原理图［75］；（f）~（j）产品［73，76-78］
Fig. 5. Metal additive manufacturing technologies. (a)-(e) Schematics[75]; (f)-(j) products[73,76-78]
3.1 粉末床熔融增材制造技术
粉末床熔融技术通过对三维模型进行分层切片处理来提取每层轮廓信息，规划热源（激光、电子束）扫描路径与打印方向，逐层熔化预先铺放的金属粉末，实现自下而上的材料逐层叠加的零件快速制造。成形件精度较高、表面质量较好，结构复杂性基本不受限。但成形效率较低，成形尺寸受限，故主要应用于小批量、中小尺寸、结构较为复杂的零件加工与模具制造。
3.1.1 激光选区熔化技术
激光选区熔化（SLM）技术基于惰性气体的工作环境［
3.1.2 电子束选区熔化技术
电子束选区熔化（EBSM）技术使用电磁线圈精准且快速地驱动电子束逐层熔化金属粉末［
3.2 定向能量沉积增材制造技术
定向能量沉积技术选用金属粉末/丝材为原材料，依据三维模型进行分层切片与轮廓提取，规划沉积路径，使用高能束（激光、电子束、电弧）为热源，逐层熔化与沉积，实现零件快速制造。相比粉末床熔融技术，定向能量沉积技术具有成形效率更高、成形结构尺寸更大的技术优势［93］，但成形复杂度较低，成形精度较差，须结合后处理技术改善零件表面质量。
3.2.1 激光金属沉积技术
激光金属沉积（LMD）技术是在惰性气体的工作环境下，利用激光逐层熔化金属粉末，实现结构零件的“近净成形”［
3.2.2 电子束自由成形制造技术
电子束自由成形制造（EBF）技术是基于真空环境，运用高能量密度的电子束冲击并熔化金属丝材，依据预设轨迹移动与逐层累积，实现零件的快速加工与制造［
3.2.3 电弧增材制造技术
电弧增材制造（WAAM）技术利用熔化极惰性气体保护焊、非熔化钨极惰性气体保护焊及等离子弧焊等焊接方法产生的电弧作为热源［
4 面向金属增材制造的拓扑优化设计研究进展
金属增材制造技术虽有效解决了复杂拓扑结构可制造性差的问题，但仍存在某些制造约束，如当结构最小尺寸小于束斑直径时，零件实际打印轮廓会超出设计轮廓；激光选区熔化技术所能制造的零件几何尺寸受限；当悬垂角度选择不当时，会产生零件装配孔材料塌陷［
图 6. 增材制造打印失效［102］。（a）装配孔材料塌陷；（b）支撑结构断裂；（c）内部支撑无法去除
Fig. 6. Additive manufacturing printing failure[102]. (a) Assembly hole material collapse; (b) fracture of support structure; (c) internal support cannot be removed
图 7. 面向金属增材制造的拓扑优化设计
Fig. 7. Topology optimization design for metal additive manufacturing
表 2. 考虑金属增材制造约束的拓扑优化常用方法
Table 2. Topology optimization common methods considering constraints of metal additive manufacturing
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4.1 考虑结构几何约束的拓扑优化方法
拓扑构型往往有细小的杆状分支，若杆状最小尺寸小于高能束的束斑直径、结构最大几何尺寸大于设备成形腔尺寸，则存在无法制造的难题。因此合理设计结构构型及分块与连接方式，利用考虑增材制造成形件几何约束的拓扑优化方法，可有效降低加工难度，减少结构热变形。学者们主要从最大、最小尺寸约束两个方面出发。
4.1.1 考虑最小尺寸约束的拓扑优化设计
基于单元网格的最小尺寸优化方法主要有投影滤波函数、鲁棒公式、功能梯度函数等。Guest等［105］使用节点体积分数为设计变量，将其投影到由单元质心和最小允许半径确定的单元空间，提出线性投影函数和使用正则化Heaviside阶跃函数的非线性投影函数，实现最小尺寸控制。但线性投影函数在边界处存在衰落效应［
图 8. 考虑最小尺寸约束的拓扑优化。（a）节点设计变量与投影函数［105］；（b）鲁棒公式［106］；（c）空间梯度算子［108］；（d）骨架提取与最小特征优化［12］
Fig. 8. Topology optimization considering minimum size constraint. (a) Nodal design variable and projection functions[105]; (b) robust formulation[106]; (c) spatial gradient operators[108]; (d) skeleton extraction and minimum feature optimization[12]
针对基于边界演化的拓扑方法，学者们提出一系列最小尺寸约束函数。Chen等［111］在目标函数中引入二次能量泛函，将几何特征尺寸信息引入到水平集框架中，实现梁状柔性机构最小尺寸优化。随后Luo等［112］将二次能量泛函引入到无铰链柔性机构中，对原始目标函数进行增广处理，采用半隐式算法，避免了传统水平集方法存在的数值求解困难问题，实现了更为高效的结构最小特征尺寸控制与优化，但该方法未提供明确的几何信息，无法实现结构最小尺寸的精确控制。Liu等［12］提取拓扑优化结果骨架［
4.1.2 考虑最大尺寸约束的拓扑优化设计
Guest［116］以映射方法［105］为基础，构建基于局部区域体积比的最大尺寸约束，靳绍猛［19］基于双向渐进结构法定义最大尺寸约束，皆实现了结构最大尺寸控制。Zhang等［117］基于固体各向同性材料惩罚模型，提取拓扑结构骨架，实现了结构最大/最小尺寸控制。
Guo等［118］基于水平集框架提出显式几何尺寸控制方法，通过限制符号距离函数的最大/最小值，同时实现了结构最大/最小尺寸精确控制。Xia等［119］引入内切圆的概念，通过约束结构边界到骨架的距离，实现了结构最大/最小尺寸控制。Wang等［120］将结构边界与水平集轮廓曲线进行偏移，建立基于偏移距离的显式特征尺寸约束函数，实现了结构最大/最小尺寸精确控制，该方法不需要提取几何骨架，可有效提升计算效率。白伟等［121］通过构造映射新模型与全局约束函数，对违反最大尺寸约束的单元实行“挖孔”处理，结合鲁棒公式，实现结构最大/最小尺寸协同控制。Niu等［122］基于移动可变形组件方法，通过调整组件宽度的上下限，实现结构最大/最小尺寸控制。Liu［13］通过识别与分割结构骨架，提出基于分段长度的尺度控制与滤波方法，实现了结构动态极限尺寸控制。
总体来说，考虑结构几何尺寸约束的研究较为完善，几何尺寸约束现已集成到OptiStruct等商用软件中。但引入滤波器与非线性投影函数后，在结构边界处存在灰度单元，需结合适当的后处理技术加以完善。引入梯度约束、几何尺寸约束后，计算量较大，计算效率与稳定性有待提升。因此，结构清晰、稳定收敛及便于数值实现与精确控制特征尺寸的新方法有一定发展空间。此外，针对多喷头打印、打印方向精度等特定增材制造工艺的特征尺寸控制新方法仍有待完善。
4.2 考虑结构成形约束的拓扑优化方法
由于增材制造逐层加工的特征，大悬垂结构需调整悬垂角度与长度并添加支撑结构以防止材料坍塌；增材制造过程中存在较大的温度梯度，支撑结构可以将上层热量传递到基板，减小热应力与变形，提升结构精度与表面质量，但成形完成后支撑结构的添加与去除会增加材料成本与打印时间。此外，若结构内部存在封闭孔洞，粉末床增材制造技术易出现内部粉末与支撑无法去除等问题。因此，考虑结构成形约束的拓扑优化方法备受关注。
4.2.1 考虑悬垂角度与悬垂长度的拓扑优化设计
考虑悬垂角度与长度的结构自支撑优化设计可有效提升结构可制造性与经济性。学者们主要从悬垂角度与长度约束方面开展深入研究。Leary等［123］通过改变结构几何形状、角度等参数并采用适当的后处理技术，避免了大悬垂结构的材料坍塌现象。Brackett等［124］基于双向渐进结构法，设定悬垂角度与悬垂长度成正比，实现了自支撑优化。Langelaar［125］提出三维增材制造滤波器，严格将违反悬垂角约束的几何形状排除在设计空间之外，实现了三维结构的自支撑设计。Qian［126］提出基于投影周长约束和密度灰度约束的悬垂角度控制新方法，应用Heaviside投影积分函数，协同优化中间密度，引入基于侧区的投影周长约束，避免了边界不可打印现象。Gaynor等［127］在Heaviside投影中嵌入悬垂角度约束，实现了最小长度尺度与悬垂角度控制［
图 9. MBB梁自支撑优化。（a）悬垂投影约束［127］；（b）优化悬垂角度与打印方向［21］；（c）多边形特征孔［22］；（d）非线性虚拟温度场［128］
Fig. 9. MBB beam self-supporting optimization. (a) Overhang projection constraint[127]; (b) optimizing overhang angle and printing direction[21]; (c) polygon-featured holes[22]; (d) nonlinear virtual temperature method[128]
Wang等［129］提出一种水平集函数梯度积分域的悬垂约束形式，使用单域积分代替点约束，利用水平集函数符号距离性质，简化悬垂约束形状导数，避免惩罚参数值过大导致的弱收敛问题，获得相对平滑、自然的结构边界。Guo等［21］提出基于移动可变形组件/孔洞的显式拓扑优化框架，协同考虑悬垂角度与工作平面倾斜角度，结构性能牺牲小，可以以更明确、更自然的几何处理方式实现结构自支撑优化［
总而言之，采用基于单元网格投影与过滤的自支撑拓扑方法，优化后结构性能损失较大，存在边界振荡等不足，有待研究结构清晰、收敛性好的三维空间滤波器。采用基于边界演化的悬垂约束控制方法，结构性能损失较小，但存在V形不可打印区域，结构连续性有待完善。现有方法多是引入全局悬垂角度约束，针对支撑去除较为容易的外轮廓，可通过优化支撑质量与数量，减少结构性能损失。此外，考虑特定增材制造工艺，协同优化悬垂长度与倾角的拓扑优化具有一定发展前景。
4.2.2 考虑连通性约束的拓扑优化设计
考虑结构连通性约束的拓扑优化设计，主要从消除孔洞、构建孔洞与边界的连接隧道及实现孔洞自支撑入手。Liu等［102］提出一种虚拟温度场，将含有封闭孔洞的连通性约束转为最大温度梯度约束，实现熔化粉末流动与水溶性支撑去除［
图 10. 考虑连通性约束的拓扑优化。（a）虚拟温度场［102］；（b）最短连接隧道［131］；（c）边约束［132］；（d）应力最小化［133］
Fig. 10. Topology optimization considering connectivity constraints. (a) Virtual temperature method[102]; (b) shortest connection tunnels[131]; (c) side constraint[132]; (d) stress minimization[133]
Luo等［128］使用非线性虚拟温度场（N-VTM）识别封闭孔洞［
考虑结构连通性约束的拓扑优化设计可有效提升结构可制造性，但消除封闭孔洞往往以牺牲柔度为代价。构建孔洞与边界的最短连接隧道以实现封闭孔洞自支撑等方法可有效减少结构性能损失，具有重要参考价值。此外，现有方法往往是基于单一特征的优化，协同考虑结构强度、构建方向、尺寸约束与成形约束具有重要现实意义。
4.3 考虑材料性能约束的拓扑优化方法
相较于传统制造技术，金属增材制造技术制备的零件存在材料各向异性、残余热应力与变形、翘曲与开裂等缺陷与不足，因此，考虑材料性能约束的拓扑优化方法具有重要现实意义。
4.3.1 考虑材料各向异性的拓扑优化设计
增材制造逐层叠加的成形方式使显微组织与力学特性具有一定的方向性，呈现出材料各向异性，因此合理规划沉积路径与构建方向，设计考虑材料各向异性的拓扑结构，可有效提高结构的承载能力与使用寿命。
Liu等［135］通过提取等值水平集轮廓来计算沉积路径，消除了域积分项，简化了灵敏度分析结果，大部分沉积路径与主应力方向保持一致；针对固定几何形状，提出一种多步方法，实现了优化结果的快速光滑收敛。Liu等［136］采用多个水平集函数表示每层切片轮廓，提出基于轮廓偏移和骨架提取的沉积路径优化模式，实现了材料各向异性与自支撑约束的协同优化。Mirzendehdel等［14］提出一种基于广义失效准则的各向异性强度灵敏度分析方法，考虑了构建方向拉伸强度低于其他方向各向异性强度的设计准则，提升了构建方向的极限承载能力［
图 11. 考虑材料各向异性的拓扑优化。（a）强度各向异性［14］；（b）量化增材制造工艺参数［138］
Fig. 11. Topology optimization considering material anisotropy. (a) Strength anisotropy[14]; (b) quantified additive manufacturing process parameters[138]
现阶段考虑材料各向异性的拓扑模型过于简化。结合增材制造过程，建立多元工艺参数下的各向异性精准三维模型，构建特定增材制造工艺参数与材料性能的定量相关性，提升数值稳定性与计算效率等方面仍有待完善。
4.3.2 考虑残余应力与变形等制造缺陷的拓扑优化设计
金属增材制造技术在逐层快速加热与迅速冷却过程中，较大的温度梯度易引起结构内部残余热应力累积，导致结构翘曲变形与开裂等问题。因此，考虑残余应力与变形等制造缺陷的拓扑优化设计，可有效提升拓扑结构工艺性与可靠性。Chen等［140］基于固有应变法，通过热电偶实验修正热源参数与热边界条件，提取固有应变作为热膨胀系数，预测零件残余应力与变形。Zhang等［141］提出密度拓扑优化与固有应变法的并行计算模型，设计仅在重力与残余应力下的刚性更好的支撑结构，改善了结构残余变形，提升了计算效率。Takezawa等［142］提出基于逐层固有应变法的晶格结构分布优化方法，结合灵敏度分析技术，模拟增材制造逐层叠加的过程，输出有效弹性模量，控制残余变形。
该领域现有研究主要集中在固有应变法的分层变形研究方面，轮廓扫描对表面残余应力的影响尚未量化，过程仿真模型仅模拟各层内部区域扫描，忽略了激光扫描速度较高、功率较低的边界轮廓扫描。为提高零件残余应力与变形的预测精度，应考虑零件几何形状对固有应变值的热效应影响，不同高度层可能经历不同的热积累。此外，非均匀变形的固有应变精确模型对提升预测精度与效率具有一定现实意义。
5 结束语
拓扑优化设计可以依据材料属性、约束条件及载荷工况，在给定设计区域内寻求材料最佳分布形式与最优承力路径，实现高性能轻量化设计。金属增材制造技术基于高能束热源，采用快速熔化与逐层叠加的成形方式，可实现复杂拓扑构型的快速原型制造与实体自由制造。将拓扑优化设计与金属增材制造结合，归纳了基于单元网格与边界演化的拓扑优化方法在改善结构连续性与可制造性方面的有效措施，总结了考虑金属增材制造几何尺寸约束、成形约束及材料性能约束的拓扑优化方法，为学者们进一步研究面向金属增材制造的拓扑优化设计提供了参考。
拓扑优化设计存在设计变量巨大、计算效率较低、求解困难、弱收敛等不足，现有拓扑优化算法往往难以输出可直接应用于增材制造的结构性能最优解，学者们往往基于最优拓扑构型进行二次简化设计，损失了结构性能。因此，结合并行计算技术，开展设计变量较少、收敛性较好的算法研究以输出可直接应用于增材制造的最优拓扑结构具有重要现实意义。
宏观拓扑优化与微观点阵结构研究日趋完善，将宏观拓扑优化设计与微观点阵结构有效融合，建立多尺度结构之间的高度衔接性，充分利用拓扑优化的高性能构型及增材制造提供的广阔设计空间，追求高性能的轻量化设计具有广阔发展前景。
考虑金属增材制造约束的拓扑优化方法采用较为理想的材料模型，与金属增材制造技术实际打印过程存在一定的差异，因此，通过建立多元工艺参数下的材料各向异性精准拓扑模型，量化金属增材制造设备工艺参数，模拟金属增材制造加工过程及预测零件翘曲变形与开裂，可有效减少残余应力与变形，改善成形精度与表面质量。
面向金属增材制造的拓扑优化往往是基于单一材料的优化，将多材料、拓扑优化及金属增材制造有效结合，研究功能梯度材料的拓扑优化设计与金属增材制造技术，实现材料、结构、工艺、性能一体化设计，是追求高性能、多功能、轻量化的又一突破点。
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Article Outline
刘博宇, 王向明, 杨光, 邢本东. 面向金属增材制造的拓扑优化设计研究进展[J]. 中国激光, 2023, 50(12): 1202301. Boyu Liu, Xiangming Wang, Guang Yang, Bendong Xing. Research Progress on Topology Optimization Design for Metal Additive Manufacturing[J]. Chinese Journal of Lasers, 2023, 50(12): 1202301.
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