Chinese Journal of Aeronautics, (2021), 34(1): 91–110

Chinese Society of Aeronautics and Astronautics

& Beihang University
Chinese Journal of Aeronautics
cja@buaa.edu.cn
www.sciencedirect.com

A review of topology optimization for additive

manufacturing: Status and challenges
Jihong ZHU a,b,*, Han ZHOU a, Chuang WANG a, Lu ZHOU a, Shangqin YUAN c,
Weihong ZHANG a

a
State IJR Center of Aerospace Design and Additive Manufacturing, Northwestern Polytechnical University, Xi’an 710072, China
b
MIIT Lab of Metal Additive Manufacturing and Innovative Design, Northwestern Polytechnical University, Xi’an 710072, China
c
Institute of Intelligence Material and Structure, Unmanned System Technologies, Northwestern Polytechnical University, Xi’an
710072, China

Received 7 June 2020; revised 13 July 2020; accepted 9 August 2020

Available online 13 October 2020

KEYWORDS Abstract Topology optimization was developed as an advanced structural design methodology to
Additive manufacturing; generate innovative lightweight and high-performance configurations that are difficult to obtain with
Aerospace applications; conventional ideas. Additive manufacturing is an advanced manufacturing technique building as-
Hierarchical structure; designed structures via layer-by-layer joining material, providing an alternative pattern for complex
Manufacturing constraints; components. The integration of topology optimization and additive manufacturing can make the
Material feature; most of their advantages and potentials, and has wide application prospects in modern manufactur-
Topology optimization ing. This article reviews the main content and applications of the research on the integration of topol-
ogy optimization and additive manufacturing in recent years, including multi-scale or hierarchical
structural optimization design and topology optimization considering additive manufacturing con-
straints. Meanwhile, some challenges of structural design approaches for additive manufacturing are
discussed, such as the performance characterization and scale effects of additively manufactured lat-
tice structures, the anisotropy and fatigue performance of additively manufactured material, and
additively manufactured functionally graded material issues, etc. It is shown that in the research
of topology optimization for additive manufacturing, the integration of material, structure, process
and performance is important to pursue high-performance, multi-functional and lightweight produc-
tion. This article provides a reference for further related research and aerospace applications.
Ó 2020 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is
an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

* Corresponding author at: State IJR Center of Aerospace Design and Additive Manufacturing, Northwestern Polytechnical University, Xi’an
710072, China.
E-mail address: jh.zhu_fea@nwpu.edu.cn (J. ZHU).
Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

https://doi.org/10.1016/j.cja.2020.09.020
1000-9361 Ó 2020 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
92 J. ZHU et al.

1. Introduction

Topology optimization1 is an advanced structural design

method which can obtain the optimal structure configuration
via reasonable material distribution satisfying specified load
conditions, performance and constraints. Compared to sizing
and shape optimization, topology optimization is independent
of the initial configuration and has a broader design space.
Consequently, it has been developed as a mainstream struc-
tural design technique for high-performance, lightweight as
well as multifunctional structures and been widely used in
aerospace,2,3 automotive,4 architecture,5 etc. A typical example
of topology optimization solution is the leading edge droop Fig. 2 Topology optimization design of aircraft pylon.2
nose ribs for Airbus 380, as shown in Fig. 1, which achieved
structural weight saving design meeting all mechanical perfor-
mance requirements.6 Fig. 2 shows a typical topology opti- come this shortcoming, the bi-directional evolutionary
mization design of an aircraft pylon, which satisfied stiffness, structural optimization (BESO) method, an extension of
strength and weight requirements.2 Since Bendsøe and ESO with allowing to add as well as remove material to modify
Kikuchi7 proposed the seminal work, topology optimization structure, was proposed34–36 and this method was proved to be
has been developed rapidly. Apart from compliance-based reliable by Huang and Xie.37
structural topology optimization, periodic microstructure Different from the above methods, LSM adopts high-
topology optimization for prescribed performance (e.g. nega- dimensional level set functions to describe structural bound-
tive Poisson’s ratio, extreme thermal expansion, etc.),8–14 heat ary.38,39 The optimum configuration with smooth structural
conduction structural topology optimization,15–18 layout opti- boundary can be obtain via iteratively solving the Hamilton-
mization of complex multi-component system,19–21 concurrent Jacobi equation to update level set functions. Recently, Zhang
optimization of microstructure and macrostructure22–31 etc. et al.40,41 proposed a feature-driven optimization method
have also attracted attentions of numerous researches. (FDO) derived from LSM, where complex engineering struc-
Over last three decades, several topology optimization ture are explicitly decomposed into a set of simple geometric
methods have been proposed, among which the density- features (e.g. superellipse). Compared with density-based
based method, the evolutionary structural optimization method, not only the number of design variables can be greatly
(ESO), the level set method (LSM) are the most representa- reduced, but also zigzag boundary can be avoided in the opti-
tives. In density-based method, a 0–1 discrete optimization mum solutions. Guo et al.42,43 proposed moving morphable
problem is transformed into a continuous optimization prob- components (MMC) and moving morphable voids (MMV).
lem in order to relax the binary design form. Originally, the Literatures2,44,45 summarized the progress and applications
homogenization method was utilized to map specified of topology optimization in detail.
microstructure controlled by density variable to effective prop- Topologically optimized structures are generally character-
erties, but it is difficult to implement for mathematical compli- ized with complex geometric configuration. Therefore it is dif-
cation.7 Subsequently, Bendsøe and Kikuchi32 proposed an ficult to manufacture these innovate structures via
alternative approach named solid isotropic material with conventional processes (e.g. machining, cast), thus additional
penalization (SIMP). Compared to homogenization, element treatments are often necessary to improve manufacturability
elastic modulus penalized exponentially in terms of density which hardly realizes the full potential of topology optimiza-
variables. SIMP has soon become the most popular topology tion, as shown in Fig. 3.
optimization and been embedded in commercial software to
solve engineering problems for its concise form. 2. Direct applications of topology optimization and additive
The ESO method, first proposed by Xie and Steven,33 grad- manufacturing
ually remove inefficient material via heuristic strategies until
meeting prescribed material volume requirement, yet the effec- Additive manufacturing (AM), also known as 3D printing,
tive solutions can be hardly obtained in some cases. To over- manufactures parts via joining material layer-by-layer. AM
opens the possibility to manufacture complex structures espe-
cially for topology optimization structures. Without additional
tools, molds and complicated procedures, AM is flexible for
any complicated structures, which not only saves cost of pro-
duction but shortens the manufacturing cycle especially in
rapid prototyping and small batch production. In addition,
AM’s potential for complex structures also promotes the
design of integral structures that reduces the number of parts
and assembly processes. Over last several decades of studies
and applications, quite a lot of AM techniques have arisen
including stereolithography (SLA),46 selective laser sintering
(SLS),47 selective laser melting (SLM),48 fused deposition mod-
Fig. 1 Topology, sizing and shape optimized A380 droop nose
eling (FDM),49 etc. AM materials cover metal, polymer, com-
ribs.6
A review of topology optimization for additive manufacturing 93

Fig. 3 A topology optimization design cannot be fabricated as an integrity in NC machining, which have to be segmented then
assembled (https://whuborhub.wordpress.com/2014/10/16/thesis-summary/).

posite, biomaterial etc. AM parts ranges from micro-nano

components to large structures of meters. Due to powerful per-
sonalized manufacturing capabilities, AM has broken the
shackles of conventional manufacturing techniques and played
a significant role in advanced manufacturing industry, which
has broad application prospects in aerospace, mechatronics,
medicine and civil engineering.3,50–53
For topology optimized structures, AM allows engineers to
get rid of the limitations of conventional manufacturing tech-
niques and pay main attention to design lightweight and high-
performance structures. In turn, topology optimization is an
effective approach for additively manufactured products with Fig. 5 Antenna bracket for RUAG’s sentinel satellite (https://
lightweight and innovative configuration. The integration of www.eos.info/en/3d-printing-examples-applications/innovation-sto-
topology optimization and AM is an important way to achieve ries/ruag-aerospace-3d-printed-satellite-components).
matching of structural design and manufacturing.
As shown in Fig. 4, topology optimization and AM were
adopted to develop a cable mount on the front spar of the ver- and AM. The optimized component’s minimum rigidity
tical stabilizer. Compared with the conventionally produced requirements exceeded by more than 30% and weight reduced
component consisting of more than 30 individual parts, this to 940 g from 1.6 kg. Moreover, many competitions such as
integrated solution merely consisting of a single part not only ‘‘Backbone cup” sponsored by Northwestern Polytechnical
realized 30% weight reduction, but also significantly reduced University (NWPU), ‘‘Tiangong cup” sponsored by China
construction and installation times. The antenna bracket for Aerospace Science and Industry Corporation Limited
RUAG’s sentinel satellite (Fig. 5) is another representative (CASIC) and 3D printed structural optimization design com-
example of successful application of topology optimization petition sponsored by Commercial Aircraft Corporation of
China Ltd (COMAC), were held in order to inspire researchers
and engineers to design innovative structures and promote
application of topology optimization and AM.
Fig. 6 shows a typical optimization design and AM process
for an aerospace bracket developed by NWPU and CASIC.
Three key steps, i.e. topology optimization as well as recon-
struction, sizing optimization and fabricated part via SLM
are involved in this procedure. Generally, topology optimiza-
tion is used for structural conceptual design while sizing and
shape optimization are used for structural detailed design.
Besides enormous manufacturing potential, AM leads to
new structural design constraints and manufacturing defects,
such as accuracy, structural connectivity, additional support
Fig. 4 A cable mount on the front spar of the vertical stabilizer structure, surface roughness, material properties, etc. Design
for Airbus A350 XWB (https://www.eos.info/en/3d-printing- for additive manufacturing (DAM)54–57 requires to deeply
examples-applications/innovation-stories/airbus-a350-xwb-3d-printed- integrate product design and manufacturing via considering
cable-mount).
94 J. ZHU et al.

Fig. 6 Aerospace bracket designed by topology optimization and manufactured by AM.3

AM process constraints and real material properties during factured such as SLM quickly and accurately.63,64 Fig. 8
AM, and takes full advantage of AM to maximize product exhibits a satellite bracket filled with lattice developed by
performance. Northwestern Polytechnical University, of which the dynamic
In mainstream structural design and manufacturing pat- response is reduced by 25% and the weight is reduced by 17%
tern, topology optimization is generally regarded as a concep- compared with the original design.
tual design method. Manufacturing constraints, detailed
engineering features and subsequent sizing and shape opti- 3.1.1. Hierarchical lattice structural optimization
mization are considered and added in the reconstructed struc-
In recent years, the optimization design of multi-scale or hier-
ture that may be inconsistent with the topologically optimized
archical lattice structures has been a key research field in the
configuration. Consequently, this sequential design procedure
integration of topology optimization and AM and attracted
cannot completely exploit the potential of topology optimiza-
widespread attention from researchers, which include: (1) hier-
tion and AM versus more advanced design approaches need be
archical structural design derived from mapping combined
explored to pursue efficient and innovative structures.
with density-based method, (2) hierarchical structural design
based on large scale strut-sizing optimization, (3) concurrent
3. Hotspots in topology optimization for AM
optimization of macrostructure and microstructure based on
homogenization.
The reconstruction of topology optimized schemes, direct The mapping combined with density-based method is a
additive manufacturing for applications are the main means degenerate SIMP method, and the density distribution tends
to achieve rapid optimization design and manufacturing in towards continuousness and gradient when canceling exponen-
the industry. In order to further take full advantages of both tial penalization. Although the computational procedure is
topology optimization and AM, the current research on topol- easy to implement, the lattice configuration is generally simple
ogy optimization for AM mainly concentrates on two aspects: to some extent. Zhang et al.68 proposed a variable-density
one is to design high-performance as well as additive manufac- hexagonal cellular structural optimization design method.
turing multi-scale/multi-hierarchical structure, and the other is More specifically, homogenization method was first imple-
to integrate AM design constraints into topology optimization mented to obtain the equivalent material constitutive law to
to achieve product design and manufacturing integration. represent cellular structures. Thereafter, topology optimization
was conducted to obtain the optimum density distribution,
3.1. Optimization design of multi-scale/multi-hierarchical which was mapped into explicit cellular structure in the later
structure construction stage. Fig. 9(a) shows optimal density distribu-
tion and mapped lattice structure of a half simply supported
As a class of lightweight and multifunctional (specific beam. In the researches of Jing et al.,69 an irregular cellular
strength,58 heat,59 energy dissipation,60 vibration,61 etc.) struc- structural modeling technique based on tangent circles was
tures, lattice structures as well as porous microstructures have proposed, which automatically generate the main outline of
attracted more and more attention from researchers62–67. irregular cellular structure.
Fig. 7 shows several lattice structures with different configura- Likewise, all cellular structures could be determined
tions. Not only complicated procedures but long working according to topology optimized solution. Panesar et al.65 con-
hours are required to manufacture lattice structures via con- ducted SIMP topology optimization to obtain multiple lattice
ventional techniques (machining, welding, weaving, etc.).62 structures, including uniform lattice structure, graded lattice
As a type of self-supporting structures, complex lattice struc- structure (also called solid-lattice hybrid structure) and scaled
tures as well as hierarchical structures can be additive manu- lattice structure. The finite element analysis results show that
A review of topology optimization for additive manufacturing 95

Fig. 7 Typical lattice samples.

Fig. 8 A satellite bracket filled with lattice.

Fig. 9 Some topology optimization examples for hierarchical structures.

96 J. ZHU et al.

the mechanical performance of lattice solution is worse than et al.22,25 proposed a hierarchical structural optimization
solid solution, but graded lattice solution and scaled lattice method with parameterized lattice microstructure, in which
solution can improve significant mechanical performance com- two types of design variables were introduced into density-
pared with uniform lattice structure. Although the density based topology optimization to determine both macrostruc-
mapping-based method is easy to implement, it is hard to ture and pointwise lattice microstructure concurrently. More
obtain hierarchical structure with multiple lattice specifically, parameterized interpolation for lattice material
configurations. (PILM), i.e. a series of polynomial functions bridging lattice
For truss-based lattice structure, lattice unit cell only serves control variables and effective mechanical properties, were
as an auxiliary modeling tool, while each strut is utilized as constructed before optimization to save computational cost.
basic unit in numerical analysis procedure. Strut-sizing opti- An optimized three-point bending beam and additive manu-
mization is implemented to obtain graded hierarchical struc- factured samples were shown in Fig. 10. Both numerical simu-
tures. As presented in literature,70,71 a so-called size lation and mechanical experiment demonstrated that non-
matching and scaling (SMS) method focusing on the sizing uniform lattice structure is better than uniform lattice struc-
optimization of struts was utilized to design mesoscale lattice ture. In this method, parametrized lattice ensured both the
structure. In this method, lattice structural topology was gen- connectivity of adjacent microstructures and
erated via a set of pre-defined lattice configuration and struts’ manufacturability.
size was directly determined by stress distribution of solid-
body finite element analysis. It is worth noting that this 3.1.2. Hierarchical stiffened thin-wall structural optimization
method could directly generate a lattice structure based on a Hierarchical stiffened thin-wall consisting of major stiffeners
heuristic strategy with higher computing efficiency and less and minor stiffeners is a kind of widely used lightweight struc-
computing costing, but could hardly get the optimum tural configuration in aerospace structures, e.g. launch vehicles
configuration. and aircraft wings,77 as shown in Fig. 11 and Fig. 12. Since
Chen et al.72 combined meshing technique and pre-defined increased global dimensions and multiple local features, bulk-
lattice cell units based on element type, element nodes and ing analysis and structural optimization generally suffer from
their connect relationships to model a large-scale parametric heavy computational costs. The minor stiffeners are smeared
lattice structure. Moving iso-surface threshold method and siz- to improve computing efficiency and the major stiffeners are
ing optimization algorithm were implemented to modify struts’ retained to accurately predict the dominated partial overall
cross-sectional areas to obtain optimum graded lattice struc- bulking mode and overall bulking mode.78,79 Thus, Smeared
ture. Since pure lattice structures without functional solids Stiffener Method (SSM) and asymptotic homogenization
and surfaces are difficult to assemble for practical structures, method were adopted to convert minor stiffeners and corre-
Tang et al.64 proposed a lattice-skin structural optimization sponding skin into an equivalent unstiffened skin. Hao et al.79
design method, in which only functional volumes with lattice and Wang et al.78,80 proposed a hybrid optimization frame-
were optimized. Meanwhile, functional volumes with solids work of hierarchical stiffened shell to explore loading effi-
as well as functional surfaces were preserved, as shown in ciency. Notably, imperfection sensitivity should be taken into
Fig. 9(b). consideration during optimization due to the bulking load
Another representative approach for hierarchical structures deviations between imperfect shell structures and perfect shell
is multi-scale concurrent topology optimization of material structures.81 Hao et al.82 also developed a hybrid optimization
and structure, in which the homogenization method plays a of cylindrical stiffened shells with reinforced cutouts by curvi-
significant role in bridging microstructures and material prop- linear stiffeners. More specifically, the near field around the
erties, thus both microstructure and macrostructure can be cutout which would be filled with curvilinear stiffeners was first
optimized to obtain higher-performance scheme. Rodrigues determined as design domain via an adaptive method and a
et al.30 proposed a hierarchical algorithm to optimize material bilevel optimization strategy was adopted to design the curvi-
distribution and local microstructures. And this algorithm was linear stiffener layout, number, section profile and location.
extended to 3D structural optimization by Coelho et al.29 here-
after. However, two independent computational procedures 3.2. Topology optimization considering AM constraints
were executed to determine macrostructure and microstructure
respectively, which cannot realize concurrent multiscale struc-
Typical AM constraints includes minimum length constraints,
tural optimization. Liu et al.28 and Chen et al.23 proposed a
connectivity constraints and overhang constraints.83 As shown
concurrent topology optimization strategy for macrostructure
in Fig. 13, the downward face becomes rougher and eventually
and microstructure. Uniform microstructure hypothesis was
the part will fail with overhang angle decreasing. The critical
formulated to guarantee manufacturability as well as low com-
overhang angle and strut diameter of Ti-6Al-4V part fabri-
putational cost. Moreover, literatures73–76 reported the above
cated by SLM were measured via experiment in.84 How to inte-
hypothesis was utilized for thermomechanical design and
grate these AM constraints into topology optimization is a hot
dynamic response design.
issue in recent research.
To pursue higher-performance structures, concurrent
topology optimization of material and structure based on
FE2 nonlinear multiscale analysis work was proposed by Xia 3.2.1. Length scale constraints
and Breitkopf.24,27 As shown in Fig. 9(c), cellular material var- Limited by the manufacturing accuracy of 3D printing equip-
ied from point to point at the macro scale to adapt the macro- ment, topology optimized structure could hardly be additively
scopic structural physical response, which not only lead to manufactured directly. Consequently, imposing length scale
intensive computational cost but weaken manufacturability. constraints in topology optimization is crucial to avoid
To balance computational cost and design freedom, Wang unmanufacturable geometric feature such as small holes,
A review of topology optimization for additive manufacturing 97

Fig. 10 Three-point bending beam infilled with uniform and graded lattice.22

projection, which can constraint minimum length scale of both

solid and void. Zhou et al.88 proposed density gradient
approach to impose minimum length scale by identifying solid
and void phase. Yang et al.89 analyzed and compared three
gradient operators including the orthogonal central difference
scheme, Prewitt operator and Sobel operator. It is found the
Prewitt operation considering boundary modification could
calculate density gradient more accurately.
Moreover, minimum length scale was also found imposing
in LSM-based topology optimization.90–93 For example, in
MMC topology optimization framework, minimum length
scale of each component can be controlled by dimensional
variables. Special treatments are required to control the mini-
mum length scale on the intersection regions of compo-
nents.92,93 Liu90 proposed a piecewise length scale control
Fig. 11 NASA Space Launch System rocket and stiffened
method for LSM-based topology optimization. The topology
shell.80
design can be decomposed into several strip-like components
and each component’s length scale can be dynamically con-
strained according to real-time status. Xia and Shi91 character-
ized structural length scale as the size maximal inscribable ball
and introduced structural skeleton obtained via fast marching
method to eliminate invalid tiny geometric features. Based on
the above concepts, length scale control could be explicitly for-
mulated as distance constraints from boundary to skeleton.
Length scale control technique in topology optimization has
been relatively mature and been integrated into the commercial
software such as Altair OptiStruct. As shown in Fig. 14,
unmanufacturable branches disappears when imposing the
minimum length scale.

Fig. 12 Lattice infilled shell designed and fabricated by Shang- 3.2.2. Connectivity constraints
hai Institute of Astronautical System Engineering, Northwestern
During AM process, it’s impossible to remove unmelt powders
Polytechnical University and Bright Laser Technologies Co.
and supports in enclosed voids. Thus, eliminating enclosed
voids is quite crucial to ensure structural manufacturability.
Liu et al.94 and Li et al.95 proposed a so-called virtual temper-
thin-walls, tiny-hinges, etc. For density-based topology opti- ature method to convert connectivity constraints to an equiv-
mization, density filtering is an effective approach to constraint alent maximum temperature constraints. The void region is
minimum length scale.85–87 For instance, Heaviside step filter filled with high heat conductive material and continuously
introduced by Guest et al.85 and modified Heaviside filter heated, while the solid is filled thermal insulation material.
introduced by Sigmund86 ensure minimum length scale on Since enclosed voids lead to local high temperature, constrain-
the solid phase and void phase, respectively. Wang et al.87 ing the maximum temperature can force to eliminate enclosed
put forward a general filtering formulation called threshold voids. But extra finite element analysis procedure increases
98 J. ZHU et al.

Fig. 13 Manufacturability of cantilever structures with decreased overhang angles (https://www.additivemanufacturing.media/

blog/post/7-helpful-numbers-quantify-design-rules-for-am).

approach can generate a material distribution which is similar

to free topology optimization, as shown in Fig. 15.

3.2.3. Self-support structural design

In AM process, additional supports are required to avoid over-
hang structures collapsing, as shown in Fig. 16(a). Supports
should be removed generally through machining after manu-
facturing, as shown in Fig. 16(b). Additional supports reduce
material efficiency and their removal increases post-
processing time and difficulties.
Many investigations have been made to slim down addi-
Fig. 14 Topology optimization of C-clip in Altair OptiStruct. tional supports during additive manufacturing process. Mor-
gan et al.99 pursued the optimum building orientation to
minimum the volume of additive supports via single objective
computational cost. Zhou and Zhang96 proposed an effective
optimization technique. To further reduce the additional sup-
approach to eliminating enclosed voids in feature-driven topol-
ports, Hu et al.100 proposed an orientation-driven shape opti-
ogy optimization by bonding design variables related to center
mization approach to achieve self-supporting structure. Global
points of void features outside the design domain. Another
rigidity energy is formulated as objective to minimize the shape
representative approach to avoiding enclosed voids is selec-
variation. But this approach is only applicable for the case
tively creating tunnels that connect voids with structural
where the initial design is allowed to be adjusted.
boundary during topology optimization process.97 This

Fig. 15 Topology optimization of a platform.97

A review of topology optimization for additive manufacturing 99

For LSM-based topology optimization, overhang con-

straints could also be directly imposed via explicit geometric
constraints. Allaire et al.114 adopted anisotropic perimeter
functions to penalize overhang features. While these geometric
constraints could not avoid unprintable V-shape regions
within critical overhang angle. To overcome the limitation,
an alternative mechanical overhang constraint formulation
based on simplified layer-wise printing process was pro-
posed.114,115 The self-weight manufacturing compliance of
intermediate shape during the layer-by-layer assembly was
selected to estimate whether the final shape could be manufac-
tured smoothly. However, these mechanical constraints could
Fig. 16 Additively manufactured upright.98 not eliminate completely plat overhang features due to that
each layer was assumed to be deposited instantaneously. A
combination of geometric constraints and mechanical con-
Designing innovative and efficient support structure instead straints was developed to pursue self-support structures effec-
of conventional supports is another case for slimming down tively.114 Guo et al.116 proposed two effective approaches
supports. Mezzadri et al.101 formulated the generation of sup- based on MMC and MMV frameworks respectively. The treat-
ports as a topology optimization problem with length scale ment of geometric constraints includes two parts, i.e. the con-
control and manufacturability constraints. The optimized sup- straint on each void feature to obtain a printable shape, and
ports as shown in Fig. 17(a) employ less material than those the constraint on any different voids from intersecting during
generated in existing software. Zhou et al.102 proposed a optimization procedure. Zhang and Zhou117 proposed an anal-
tree-like support generation approach based on the L-system ogous approach based on polygon void features, which are
and octree as shown in Fig. 17(b). These supports are easier easier to impose geometric angle constraints. The difference
to remove and require less material as well as building time. is that polygon fusion algorithm is formulated to eliminate
Integrating overhang constraints into topology optimiza- unprintable V-shape region caused by intersecting polygons.
tion to pursue self-supported structure is one of current
research hotspots. Fig. 18 exhibits a comparison of non-self- 3.3. Remarks
supporting design and self-supporting design. Similar to length
scale constraints, overhang constraints can also be imposed via
density projection in density-based topology optimization.103 A summary collection of research about topology optimization
Only those should be projected to solid and satisfy overhang for AM is shown in Table 1. Note that there are still lots of
constraints are preserved via a combination of a local projec- related study not included in this section due to limited space.
tion. This merely requires a series of projection operations Topology optimization for AM reveals abroad engineering
without additional explicit geometric constraints. This method application prospects of pursuing high-performance and light-
was extended to design 3D self-supported structures by John- weight structures. It is highly convinced that more and more
son and Gaynor.104 Additionally, an improved overhang map- industrial structures will be designed and fabricated with the
ping procedure was adopted to exactly impose allowable help of the integration of topology optimization and AM in
overhang angle via adjusting the spacing of the design points. the future. However, practical industrial structures are gener-
And adjoint approach to sensitivity analysis was adopted to ally characterized with irregular shape and large geometric
speed up the calculation procedure, which could be easily dimensions, which leads to their numerical simulation and
applied to large scale structural optimization problem. Simi- optimization cost-prohibitive in term of time and computing
larly, an AM filtering procedure was proposed in Langelaar’s burden. Fortunately, the development of massively parallel
work,105,106 which acted as a simplified layer-wise printing sim- computing versus high-performance simulation118,119 makes
ulation to exclude unprintable parts from the optimum resolu- it applicable for large-scale engineering structures. Moreover,
tion. And this layer-wise filtering concept was also further most researches were subjected to ideal material model, e.g.
applied and studied in literatures.107–110 isotropic material, which hardly exists in practical structures,
Based on Heaviside projection-based aggregations, especially additively manufactured structures. In this regard,
Qian111 and Wang112 explored an effective overhang angle some challenges for further integration and application of
constraint approach, which transformed pointwise overhang topology optimization and AM are presented in the following
angle constraints into two integral inequalities, i.e. directional section.
gradient based global constraint to eliminate internal sup-
ports and gradient-based global constraint to avoid external 4. Challenges for future integration and application
supports. Moreover, Mass and Amir113 proposed an alterna-
tive two-step approach using a virtual skeleton. Overhang In recent years, AM integrated with topology optimization has
constraints are converted into a combinatorial optimization become a hotspot in the field of mechanical design and manu-
problem. Truss optimization is firstly implemented to obtain facturing. Numerous achievements have been made in engi-
a discrete skeleton that satisfies overhang constraints. Then neering applications and academic research. However, due to
the optimum discrete pattern is mapped to a continuum the feature of additively manufactured structures and their
structure and density-based topology optimization is imple- optimization design, more challenges are encountered. On
mented to relax stress concentration around intersection of the one hand, facing the requirements of multiscale and multi-
bars. functional structures, topology optimization and its numerical
100 J. ZHU et al.

Fig. 17 Innovative support structures.

Fig. 18 3D printed topologically optimized industrial frame.109

design schemes have many contents to be developed. On the the energy-based approaches to predict effective coefficient of
other hand, AM itself shall be improved by predicting and thermal expansion and thermal conductivities are also pro-
controlling its material mechanics behaviors, fabricating more posed in literatures.123,124
complicated functional structures etc. to realize the integration Homogenization is based on periodic hypothesis and scale
of material, structure, process and performance. separation hypothesis,8 while additive manufactured lattice
and hierarchical structures are always non-periodic and
4.1. Effective properties prediction of lattice structure scale-dependent. Thus, how to characterize the microstruc-
tures’ equivalent performance is an essential challenge. As
In large-scale hierarchical structural analysis and topology shown is Fig. 19, Dirichlet bulk modulus and Neumann bulk
optimization for additive manufacturing, homogenization the- modulus converge on the homogenized bulk modulus of peri-
ory plays a significant role in bridging microscale and macro- odic composite only when the scale factor converges to infin-
scale. Homogenization120 relies on an asymptotic expansion of ity.125 Many efforts have been done trying to solve this
the governing equations to realize structural analysis at sepa- challenge. Yan et al.126 used the extended multiscale finite ele-
rate scales. With the help of homogenization, not only material ment method to study the scale effect of lattice microstructure
effective properties but also physical field (e.g. stress) on the and minimize the weight of scale-related lattice structure.
microscopic scale can be simulated. Thus homogenization is Wu et al.127 developed an approximation of reduced substruc-
widely used in multi-scale structural analysis of composite ture with penalization model to optimize scale-related hierar-
material121,122 and structural design of periodic chical structure, which regards each lattice unit cell as a
microstructures.9,14 substructure.
To overcome the difficulties of numerical calculation and The beam element approach, which employs 3D beam ele-
sensitivity derivation, an equivalent energy-based homogeniza- ments for modeling the lattice structure,72,128,129 is an alterna-
tion was also proposed to predict the effective properties of tive approach without scale effect. Dong et al.128 proposed a
microstructure.13 The stress and strain tensors of the equiva- hybrid elements model to simulate elastic properties of solid-
lent homogenous medium are equal to the average stress and lattice hybrid structures. In his method, Timoshenko beam ele-
strain tensors of the periodic microstructures. The strain ments and 3D solid elements are applied to mesh lattice part
energy density in microstructure is equal to that in the equiv- and solid part respectively. Rigid Body Elements (RBEs) are
alent homogenous medium. Only four simple load cases are applied to create connection between beam elements and solid
required to calculate stiffness tensor for 2D orthotropic elements. Fig. 20 shows a topologically optimized solid-lattice
microstructure and nine are required for 3D cases. Moreover, structure. Although beam element model can save computa-
A review of topology optimization for additive manufacturing 101

Table 1 Typical research hotspots in topology optimization for AM.

Hotspots Proposed approach References Natures
Hierarchical Mapping combined with Zhang et al.68; Panesar Easy to implement based on standard topology optimization method
structures degenerate SIMP et al.65 while only porosity can be optimized
Strut-size matching and Nguyen et al.70; Chang Can design strut-based lattice without time-consuming optimization
scaling and Rosen71 and suitable for large-scale structures
Strut-size optimization Chen et al.72; Tang Conformed lattice can be optimized effectively while only strut-based
et al.64 lattice can be used
Hierarchical optimization Rodrigues et al.30; Structure and pointwise material can be optimized sequentially with
of material and structure Coelho et al.29 huge computational cost.
Concurrent optimization of Xia et al.24,27 Both scales can be designed simultaneously while intensive
material and structure computational cost is required
Concurrent optimization Liu et al.28, Chen Decrease computational cost to some degree while unified
with unified microstructure et al.23 microstructure cannot exploit material potential completely
Concurrent optimization Wang et al.22 Manufacturability, structural performance and computational cost are
based on PILM model all balanced
Stiffened Hybrid optimization based Hao et al.79; Wang High computational efficiency and vigorous engineering significance
thin-wall on homogenization et al.78,80
Length scale Density filtering and Wang et al.87 Computational stable and heavy computational burden caused by
constraints projection three separate finite element analysis
Gradient-based geometric Zhou et al.88; Yang Impose minimum length scale constraints in a computational cheap
constraints et al.89 manner without additional finite element analysis
MMC-based method Wang et al.92; Zhang Minimum length scale can be controlled by thickness of components,
et al.93 but additional geometric constraints are required on the intersection
Structural skeleton based Xia and Shi91; Liu90 Both minimum and maximum length scale can be effectively controlled
method
Connectivity Virtual temperature Liu et al.94; Li et al.95 Easily applied in topology optimization while additional thermal
constraints method analysis increases computational cost
Side constraint scheme Zhou and Zhang96 Only proper bound values of variables require to be imposed without
additional constraints
Selectively generating Xiong et al.97 Can obtain material distribution similar to conventional topologically
tunnels optimized solution
Self-support Density projection-based Gaynor and Guest103; Only projection operations rather than additional constraints are
structures method Johnson and Gaynor104 needed, but it is difficult in convergence
Density gradient based Qian111; Wang and Overhang constraints are formed with explicit geometric meaning, but
integral method Qian112 boundary V-shape cannot be avoided
Virtual skeleton method Mass and Amir113 Cheap in computational terms but cannot eliminate overhang features
completely
Combination of geometric Allaire et al.114 Geometric and mechanical constraints complement each other to avoid
and mechanical constraints overhang features but it leads to heavier computational burden
MMC/MMV-based Guo et al.116 Distinctive features make it easier to impose overhang constraints via
method explicit geometric treatment
Polygon features based Zhang and Zhou117 Add polygon fusion procedure to eliminate boundary V-shape
method

tional cost and avoid poor elements, it suffers from two devi- to some extent, they are generally empirical and their analysis
ations, i.e., overestimation of material volume at overlapping accuracy closely depends on the complexity of the vertices.
domains and underestimation of both stiffness and strength Meng et al.133 proposed a systematic inverse approach to
in the vicinity of the vertices.130 accurately modelling additively manufactured lattice core
According to Luxner et al.,130 material distribution around based on variable cross-section beam elements. To be specific,
vertices could be approximated by a sphere whose radius equal each idealized uniform beam was substituted by three beams
to strut radius and material stiffness within the spherical consisting two enhanced tapered beams close to intersecting
domain was magnified by 1000 times to compensate these devi- nodes and one uniform beam in-between. The proposed vari-
ations. Labeas and Sunaric131 and Smith et al.132 adopted able cross-section beam could be characterized by four geo-
another compensation approach to approximating the devia- metric parameters. These parameters can be obtained via
tions derived from material aggregation, i.e., reasonably solving a minimum approximation problem. Notably, since
increasing the cross-section at the end of each strut. Although the four geometric parameters can be identified by real exper-
abovementioned approximations can compensate deviations iment data, this approach can also take geometric accuracy
102 J. ZHU et al.

Besides metal material, additively manufactured polymer

material also exhibits the anisotropy of both microstructure
and mechanical properties. Yang et al.67 found that SLA man-
ufactured material is almost transversely isotropic in elastic
behavior and its mechanical properties, including elastic mod-
ulus, yield strength and fracture strain, in horizontal plane are
better than that along building direction, as shown in Fig. 22.
However, the idealized isotropic material models have been
mainly applied in topology optimization, ignoring the effects
of material anisotropy on structural design as well as perfor-
mance. To overcome this challenge, we can (1) optimize addi-
tive manufacturing process parameters, such as building
direction136,138 and nozzle scanning strategy135 together with
the structural configurations, (2) improve additive manufactur-
ing process, e.g. employ secondary processes consisting of
selective laser erosion and laser re-melting,139 (3) introduce
the real anisotropic material model into topology optimiza-
Fig. 19 Equivalent Dirichlet and Neumann bulk modulus of the tion67 to pursue higher-performance additively manufactured
stiff composite with increasing scale factor.125 structures.

4.3. Structural fatigue performance in additive manufacturing

Both long-life aircrafts and reusable space vehicles require

structures with high fatigue performance. But the fatigue life
and fatigue strength of additively manufactured material are
significantly lower than those of corresponding wrought mate-
rial,140 for instance, the fatigue performance of SLM manufac-
tured 17-4 PH stainless steel is weaker than its wrought form
according to Fig. 23(a),140 which not only limits additively
manufactured material’s application in aerospace field but also
brings new challenges for further research on topology opti-
mization for additive manufacturing.
Fig. 20 A topologically optimized solid-lattice hybrid structure Defects produced in additive manufacturing process includ-
(http://fea.ru/news/6697). ing un-melt regions due to lack of fusion as shown in Fig. 23(b)
and pores due to entrapped gas as shown in Fig. 23(c) are the
main cause of fatigue failure.138 These defects typically serve as
crack initiation sites under cyclic loading. The location, size
and structural defects derived from AM process into
and shape of void defects significantly affects the fatigue per-
consideration.
formance of additively manufactured material. The high stress
concentrations mainly occur around void defects of large size
4.2. Material anisotropy in additive manufacturing and irregular shape closing to surface, which are detrimental
to fatigue performance. Researches138,140 show that the loca-
The mechanical properties anisotropy in AM material is an tion of void defects is the leading contributor for fatigue failure
essential challenge for structural design and application. Plenty in most cases.
of experimental observations demonstrate that the as-built The main factors that affect additively manufactured parts’
material microstructures are diverse in horizontal plane per- fatigue performance also includes geometry, building direction
pendicular to the building direction and vertical plane parallel and surface roughness. The part geometry as well as the num-
to the building direction.134–137 The anisotropy of mechanical ber of parts manufactured per build operation directly affects
properties is mainly caused by the anisotropy of microstruc- inter-layer time interval, which is closely related to thermal his-
ture. As shown in Fig. 21(a), the material microstructures of tory experienced during manufacturing. The distinct thermal
the cold sprayed copper deposits exhibit an equiaxed shape history leads to various microstructural features including
in horizontal plane, while a lens-like shape is exhibited in ver- grain structure and defect distribution. In the study of direct
tical plane. As a result, the microhardness in horizontal plane laser deposited 316L stainless steel,141 longer inter-layer time
is lower than in vertical plane.135 According to in Fig. 21(b), interval and larger cooling rates along each layer lead to finer
during Ti-6Al-4 V additive manufacturing process, the prior- grain structure, higher yield and tensile strength as well as
b grains grow in a columnar way almost parallel to building lower elongation to failure. But longer cooling time diminishes
direction, which affects the fracture mechanisms and the crack the initial layer temperature and laser melting pool, which is
propagation in the parts.136 Similarly, the microstructure of easier to produce un-melted regions and weaken fatigue per-
17–4 PH stainless steel specimen manufactured via SLM is formance. The microstructural anisotropy of additively manu-
characterized by complex directional or columnar lath marten- factured material, including grain and defect directionalities,
site parallel to the building direction,137 as shown in Fig. 21(c). also leads to anisotropic fatigue performance. For instance,
A review of topology optimization for additive manufacturing 103

Fig. 21 Some additively manufactured material microstructures.

Via a combined action of high temperature and high pressure,

HIP can eliminate part of void defects and homogenize the
microstructure, which significantly improve fatigue perfor-
mance. In addition, surface treatment (e.g. machining, polish-
ing) can also improve the fatigue performance of additively
manufactured parts.143 However, affected by the defects distri-
bution and cutting thickness, machining may resurface the
inner defects occasionally, which leads to poorer fatigue per-
formance instead.140
Due to geometric difference, experimental samples and real
parts experience different thermal history and produce differ-
ent microstructures and defect distribution. Thus, the fatigue
performance of real parts cannot be accurately characterized
by that of experimental samples. Moreover, the post-
manufacturing treatments to improve fatigue performance also
Fig. 22 Load-displacement curves of tensile samples with
depends on material properties. For instance, HIP is less effec-
different building directions fabricated by SLA in tensile
tive to improve fatigue performance of 316L stainless steel due
experiments.67
to its high ductility.144
Moreover, efficient structural configuration can notably
reduce stress level subject to dynamic loads to ameliorate fati-
selective laser melted 17-4 PH stainless steel138 and Ti-6Al- gue performance. Generally, fatigue-related topology opti-
4V142 exhibit better fatigue performance perpendicular to the mization can be formulated as two optimization problems,
building direction than parallel. To improve the fatigue perfor- i.e. fatigue performance such as fatigue life is considered as
mance of as-built parts, hot isostatic pressing (HIP),142 the optimization objective functions and constraints respectively.
most effective post-manufacturing treatments to remedy Since the difficulty of sensitivity analysis and high non-
defects produced in additive manufacturing process is applied. linearity of damage with regard to design variables, taking fati-

Fig. 23 Fatigue data and microstructural defects of selective laser melted 17-4 PH stainless steel.
104 J. ZHU et al.

gue performance into topology optimization is still a crucial At present, the design of FGM mainly focuses on two-
challenge. Holmberg et al.145 proposed an efficient fatigue- phase materials. Material composition usually changes accord-
constrained structural optimization procedure including two ing to preset direction and mode, which hardly fully realizes
separate steps, i.e. fatigue analysis and topology optimization. FGM’s potential. Literature152 gives a detailed review of
Fatigue analysis was first performed to obtain critical fatigue FGM’s design method for basic engineering structures (e.g.
stress, which was subsequently set as an upper bound of fati- beam, plate, shell). The development of topology optimization
gue stress level during topology optimization. To pursue struc- further enriches FGM’s design method. Liu et al.151 utilized
tures with prescribed high-cycle fatigue life, Oest and Lund146 topology optimization to conceive tailored FGM with specific
directly treated fatigue damage accumulation subject to pro- stiffness-related and failure-related mechanical properties.
portional loads as constraints. Palmgren-Miner’s linear dam- SLA is adopted to manufacture these optimized functionally
age hypothesis, S-N curves and Sines fatigue criterion were graded structures as shown in Fig. 24(c). Concurrent optimiza-
utilized to save computational costs during optimization. For tion of material properties and structural topology is an effec-
non-proportional loads, Zhang et al.147 formulated loading tive approach for high-performance FGM. Xia and Wang153
history as a linear combination of a set of simple unit loads, integrated two design variables, i.e. the material distribution
which could efficiently compute pointwise stresses and sensitiv- and the structural boundary, into a compliance optimization.
ities via linear superposition. Zhao et al.148 constructed Lieu and Lee154 proposed a multi-material topology optimiza-
dynamic-fatigue-constrained topology optimization model tion approach in the framework of isogeometric analysis,
with dynamic random loads, where fatigue constraints could which utilizes bivariate B-spline basic functions to describe
be expressed as the peak value of the period fluctuating material distribution. Apart from single stiffness-related
dynamic stress. And a constraint-limit-variant method was design, Banh and Lee155 proposed a multi-material topology
proposed to solve this highly nonlinear optimization problem. optimization design dependent on crack patterns, which con-
Consequently, establishing an effective model relating currently optimizes both material distribution and structure
material, structure, process and fatigue performance remains to prevent the propagation of crack.
a challenging topic and is crucial for pursuing reliable engi- Multi-material AM is the most ideal manufacturing method
neering structures with high fatigue performance. for FGM,156,157 but there still exists plenty of problems to be
further investigated, including multi-material fusion mecha-
4.4. Design and manufacturing of functionally graded material nism, material properties and manufacturing process limita-
tions. Mixed metal powders repeatedly experience complex
Functionally graded material (FGM) is an advanced compos- thermal history including melting, molten pool flowing and
ite material composed of two or more materials, whose compo- crystallizing under the action of laser. And structural defects
sitions or properties vary gradually along one or more such as pores, un-melted regions and cracks are sensitive to
directions to adapt to extreme loading circumstance. FGM AM process parameters, which significantly affects FGM’s
has wide engineering application prospects and is also a new performance. Due to diversity thermodynamic properties of
direction for the integration of multi-phase material additive multi-materials, constant laser power easily causes material
manufacturing and topology optimization. with high melting points or low laser absorption rates to fail
Fig. 24(a) shows a typical FGM form with grain size grad- to melt sufficiently. Consequently, it is necessary to dynami-
ually changing. Compared to conventional composite mate- cally adjust the laser parameters to adapt to the gradient
rial, FGM can effectively relax or even avoid interfacial changes of the material distribution.158 In addition, thermal
stress due to steep shifts in material properties via changing histories (e.g. cooling rate) usually severely affects microstruc-
each material phase distribution gradually. As shown in tural feature such as grain morphology and defects, which in
Fig. 24(b), a C/SiC FGM designed by Kim et al. effectively turn affects FGM’s performance such as strength and hard-
relaxed the residual thermal stress between the carbon fiber- ness. The thermal history is not only related to process param-
reinforced carbon composites and the SiC coating layer, and eters (e.g. laser power, scanning speed, scanning path), but also
increased the oxidation resistance.149–152 closely related to geometric model and deposition height.159

Fig. 24 Illustrations of FGMs.

A review of topology optimization for additive manufacturing 105

Fig. 25 3D printed interdigitated Li-ion micro-battery.160

challenging topic, which holds considerable significance for

structural and functional integrated components in engineer-
ing applications.

5. Conclusions

The deep integration of Topology optimization and AM is an

efficacious approach to pursuing next-generation high-
performance, multi-functional and lightweight structures. This
article briefly recalls the structural optimization design meth-
ods for AM and corresponding applications in recent years.
Fig. 26 3D printed complex multi-layer PCBs (https://www. To cope the limitation of sequential structural design proce-
3dprintingmedia.network/electronics-additive-manufacturing/). dure, two primary topology optimization techniques for AM,
i.e. structural design based on innovative configurations (e.g.
For complex multi-material AM systems, how to accurately lattice, hierarchical structures and stiffened thin-wall) and
control thermal history is still a complicated and challenging topology optimization based on AM constraints (e.g. length
research topic. scale, connectivity and overhang constraints) are reviewed
Simultaneously design and 3D printing integral compo- and summarized. Not only powerful and flexible manufactur-
nents consisting of functional electronic device and substrate ing capabilities are fully exploited to accurately fabricate lat-
structure are challenging applications of multi-material func- tice and hierarchical structures with complex geometric
tional structural design and AM. Sun et al.160 fabricated 3D configuration, but also topologically optimized structures can
interdigitated Li-ion micro-battery on a sub-millimeter scale suffice AM design rules without additional structural adjust-
via accurately depositing cathode and anode inks with the ment in the reconstructing procedure.
assistance of fine deposition nozzles, as shown in Fig. 25. Structural optimization for AM is and will be still a hot and
The printed micro-battery was characterized by high-aspect attractive issue. And further researches need to be carried out
ratio electrodes in interdigitated architectures and exhibited to optimize and design practice industrial structures. This arti-
higher areal energy and power densities than its rechargeable cle also presents the current research challenges including char-
counterparts. Du et al.161 utilized a 3D solution printer to first acterizing accurately the performance of scale-related lattice
fabricate n-type flexible tungsten carbide/polylactic acid ther- structures with as little computational costs as possible, effect
moelectric composites with favorable stability in air atmo- of AM process on material anisotropy and fatigue perfor-
sphere, which has significant potential for flexible mance, and design and manufacturing of FGM and complex
thermoelectric devices. Yu et al.162 also proposed an AM pro- multi-material functional system. To cope these challenges,
cess for carbon-nanotubes-based microsupercapacitors. The the further study directions should include the follows.
printed multi-layer carbon nanotubes and the PVA-H3PO4
electrolyte solution were packaged as the final microsuperca- (1) How to systematically and accurately characterizes com-
pacitors, which exhibited an excellent areal capacitance and plex lattice structures especially surface-based lattice still
cycle stability. Benefit from AM’s rapid development, 3D requires further attention and study. And efficient mod-
printed circuit board (PCB) has already realized commercial- elling approach can promote hierarchical structural
ization. Fig. 26 shows complex multilayer PCBs fabricated optimization to pursue higher performance.
via AM system. Recently, HENSOLDT utilized dielectric (2) The prompt development of advance manufacturing
polymer ink and conductive ink developed by Nano Dimen- technologies, e.g. multi-axis AM system163 and additive
sion to achieved the world-wide first 10-layer PCB with high- and subtractive combined manufacturing technol-
performance electronic structures soldered to both sides. ogy,164,165 further improve the manufacturing capability
The integration of topology optimization and AM for com- for complex structures. The emerging manufacturing
plex multi-material functional system remains a promising and technologies relax conventional AM’s constraints such
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