# Topology Optimization for Large-Scale Additive Manufacturing: Generating
designs tailored to the deposition nozzle size

E. Fernándeza, C. Ayasb, M. Langelaarb and P. Duysinxa Eduardo Fernández,
orcid: 0000-0003-3735-1886, Email<EMAIL_ADDRESS>Ayas, orcid:
0000-0002-9472-7424, Email<EMAIL_ADDRESS>Langelaar, orcid:
0000-0003-2106-2246, Email<EMAIL_ADDRESS>Duysinx, orcid:
0000-0001-7870-3628, Email<EMAIL_ADDRESS>aDepartment of Aerospace and
Mechanical Engineering, University of Liege, Allée de la Découverte 13A, B52,
4000, Liege, Belgium; bFaculty of Mechanical, Maritime and Materials
Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The
Netherlands.

###### Abstract

Additive Manufacturing (AM) processes intended for large scale components
deposit large volumes of material to shorten process duration. This reduces
the resolution of the AM process, which is typically defined by the size of
the deposition nozzle. If the resolution limitation is not considered when
designing for Large-Scale Additive Manufacturing (LSAM), difficulties can
arise in the manufacturing process, which may require the adaptation of the
deposition parameters. This work incorporates the nozzle size constraint into
Topology Optimization (TO) in order to generate optimized designs suitable to
the process resolution. This article proposes and compares two methods, which
are based on existing TO techniques that enable control of minimum and maximum
member size, and of minimum cavity size. The first method requires the minimum
and maximum member size to be equal to the deposition nozzle size, thus design
features of uniform width are obtained in the optimized design. The second
method defines the size of the solid members sufficiently small for the
resulting structure to resemble a structural skeleton, which can be
interpreted as the deposition path. Through filtering and projection
techniques, the thin structures are thickened according to the chosen nozzle
size. Thus, a topology tailored to the size of the deposition nozzle is
obtained along with a deposition proposal. The methods are demonstrated and
assessed using 2D and 3D benchmark problems.

###### keywords:

Nozzle Size; WAAM; Design for AM; Topology Optimization; Maximum Size

††articletype: ARTICLE TEMPLATE

##  1 Introduction

Topology Optimization (TO) and Additive Manufacturing (AM) are recognized as
promising technologies for the realization of high performance structural
components due to their ability to create designs with unprecedented
sophistication. The potential of these technologies has caught intense
interest across a wide range of industries (Zhu, Zhang, and Xia 2016; Culmone,
Smit, and Breedveld 2019; Tino et al. 2020), which promotes their accelerated
development. While AM techniques are becoming more versatile, stable and
precise (Li et al. 2019; Bajaj et al. 2020), TO is adapting to the limitations
of the AM processes (Liu et al. 2018; Meng et al. 2019).

Topology Optimization is well known for its ability to generate highly
efficient designs. However, TO designs often come with high geometrical
complexity which hinders manufacturing for even the most advanced AM
processes. This drawback has established a research field seeking to
facilitate the manufacturability of TO design (Liu et al. 2018). To date, a
number of TO methods have been proposed for specific AM processes, such as
powder-based processes, stereolithography or Fused Deposition Modeling (FDM).
For instance, there are methods to prevent closed cavities in optimized
designs in order to ensure the removal of support structures or unfused powder
(Li et al. 2016; Langelaar 2019; Gaynor and Johnson 2020). To reduce or avoid
the use of support structures, the maximum permissible overhang angle of parts
can be controlled during TO (Langelaar 2016; Gaynor and Guest 2016; Zhang and
Cheng 2020; van de Ven et al. 2020) along with the printing direction
(Langelaar 2018; Wang and Qian 2020). In addition, minimum member size and
minimum cavity size can be imposed in TO to ensure their printability (Wang,
Lazarov, and Sigmund 2011; Pellens et al. 2019). More complex methods take
into account thermal residual stresses in TO (Allaire and Jakabčin 2018), or
incorporate continuity constraints to facilitate the printability of multiple
materials (Yu et al. 2020). Experimental studies (Fu et al. 2019) and
industrial applications (Zhu, Zhang, and Xia 2016) have demonstrated that such
AM-specific TO methods enhance the manufacturability of the optimized design
when it comes to FDM, stereolithography or powder-based AM processes.

There is an emerging interest in the industry for Large-Scale Additive
Manufacturing (LSAM) processes, as they are able to produce components in
short time spans (Lim et al. 2012), either to manufacture large-scale parts
(Greer et al. 2019), or to speed up mass production (Bishop and Leigh 2020).
Among others, processes such as Direct Metal Deposition (DMD), Wire-Arc
Additive Manufacturing (WAAM), 3D Concrete Printing (3DCP), and large-scale
Fused Deposition Modelling (FDM) stand out. These LSAM processes achieve high
production rates by depositing relatively large volumes of material per unit
time. Though they operate using the layer-by-layer principle, they exhibit
distinctive limitations compared to low production rate processes, such as
powder bed fusion. For example, the large amount of deposited material
complicates the deposition path planning (Jiang and Ma 2020), since crossover
of deposition paths leads to local agglomerations of material that accumulate
geometrical defects from layer to layer (Mehnen et al. 2014), as shown in Fig.
1(a). Similarly, design features with sharp corners and small radii of
curvature may also cause local agglomeration of material (Geng et al. 2017;
Comminal et al. 2019), as shown in Fig. 1(b), while deposition paths with
T-joints can induce defects during stops and starts, such as excess or lack of
deposited material (Venturini et al. 2016), as shown in Fig. 1(c). These
difficulties have been included in path planning algorithms (Liu et al. 2020).
However, when the design to be manufactured has features with geometrical
details smaller than the nominal resolution of the LSAM process, the path
planning becomes extremely difficult, since the deposition process parameters
must be included in the planning algorithm. For example, the nozzle size can
be reduced, but if this is not possible, then the feed rate and travel speed
can be modified, or the deposition beads can be overlapped. Changing the
deposition parameters can reduce the process stability, induce geometrical or
metallurgical defects and demand extensive testing, thus increasing production
costs (Rodrigues et al. 2019). For these reasons, a discrete number of paths
must be considered when designing for LSAM, as shown in Fig. 1(d). Thus,
designs with resolution and geometrical features compatible with the nominal
production are obtained.

(a) Crossover weld paths (Mehnen et al. 2014).

(b) Sharp corner.

(c) T-type connection (Venturini et al. 2016).

(d) Discrete welding beads (Jackson 2016).

Figure 1: Some geometric features to be considered when designing for a Large-
Scale Additive Manufacturing (LSAM) process. Figure 1b is courtesy of
Institute Soudure (www.isgroupe.com). Figure 1d is courtesy of AML3D
(www.aml3d.com).

Ways to fully ensure the above mentioned compatibility between design and
process are not yet available. Only recently, few studies have appeared that
address this gap. Carstensen (2020) considers the size of the deposition
nozzle in the TO problem using the formulation that embeds discrete particles
in a continuous domain (Guest 2015). The author proposes a formulation that
binds the discrete particles to produce a continuous material bead. Numerous
examples in 2D are reported, which at first sight meet the imposed resolution
of the desired AM process. However, the method binds the particles in a
orthogonal pattern, which makes it difficult to generate inclined structural
members, therefore, the method is not able to fully exploit the form freedom
provided by the AM process. The work of Vantyghem et al. (2020) takes a
pragmatic approach to deal with the low resolution issue imposed by 3D
Concrete Printing. The authors post-process 2D topology-optimized solutions to
propose a large-scale three-dimensional design. In the post-processing step,
the authors enforce the design details to match the printing resolution of the
LSAM process. However, in this manner the optimality of the final design is
likely lost.

This article adopts a strategy different from that proposed by Carstensen
(2020), although with the same intent. Consequently, it is aimed to generate
topology-optimized designs complying with the low resolution of Large-Scale AM
processes, which is typically defined by the size of the deposition nozzle.
For this purpose, we propose two methods based on recent developments in
minimum and maximum size control in topology optimization (Fernández et al.
2020). The first method forces the minimum and maximum size of all solid
members to be equal to the size of the deposition nozzle. This approach proves
to be simple yet effective in producing designs suitable for the desired
resolution. However, the maximum size restriction prevents solid members
greater than the prescribed nozzle size, which can significantly reduce the
achievable design performance. The second method restricts the maximum size of
the solid members sufficiently to render a structural skeleton, which can also
be interpreted as the deposition path. Through filtering and projection
techniques (Wang, Lazarov, and Sigmund 2011), the skeleton is dilated
according to the nozzle size and thus the actual design is defined. Therefore,
a design adapted to the desired resolution is obtained and a low performance
penalization is introduced. The proposed strategies are explained, assessed
and compared using 2D and 3D numerical examples. Results exhibit designs that
conform to the size of the deposition nozzle, allowing optimized topologies to
be filled by a discrete number of material beads. Thus, the design requirement
shown in Fig. 1(d) is met. Furthermore, for a specific set of printing
parameters, deposition paths free of crossovers, T-type connections and sharp
corners can be obtained.

Although this paper addresses a purely geometric limitation that has to do
with the size of the deposition nozzle, the obtained results suggest that the
nozzle size restrictions proposed for topology optimization improve the
printability of optimized components. Other AM constraints that may apply such
as critical overhang angles or thermal-induced restrictions are outside the
scope of this work. Potential combinations and extensions with other AM
constraints are discussed in the Perspectives section towards the end of this
manuscript.

The remainder of this article is organized as follows. Section 2 introduces
the formulation of the TO problem that imposes the minimum and maximum size of
solid members. Section 3 presents and compares the two methods proposed in
this article using a compliance minimization formulation. In addition to 2D
examples in the preceding section, Section 4 compares the two methods on a 3D
test case. Section 5 validates the methods on benchmarks including stress
constraints and displacement functions. Finally, Section 6 provides the
conclusions and future perspectives of this work.

##  2 Base topology optimization formulation

The topology optimization problem is formulated using the density method
(Bendsøe and Kikuchi 1988). Therefore, the design domains are discretized into
$N$ finite elements and their densities are described by $N$ design variables.
The value of these design variables range from 0 to 1, where the value 0
represents the void phase and the value 1 represents the solid phase (Bendsøe
1989). To avoid the presence of intermediate densities, we use the Solid
Isotropic Material with Penalization (SIMP) interpolation law (Bendsøe 1989).
Note that, for simplicity and because the focus of this work is on the nozzle
size restriction, anisotropic material behavior is not considered.

It is well-known that SIMP exhibits numerical difficulties, such as the mesh-
dependency and the presence of checkerboard patterns (Sigmund and Petersson
1998). In addition, the formulation does not allow to impose precisely a
minimum size control in the solid and void phases (Wang, Lazarov, and Sigmund
2011), which is an essential requirement for the methods proposed in this
work. For this reason, we adopt the robust formulation proposed by Sigmund
(2009). This formulation considers manufacturing errors that may result in a
thinner (eroded) or thicker (dilated) design with respect to the reference one
intended for manufacturing (intermediate). These designs are constructed using
filtering and projections techniques, which are described below.

The filtering operation utilizes a weighted average of the design variables
within a circular (or spherical) domain of radius $r_{\mathrm{fil}}$ (Bruns
and Tortorelli 2001; Bourdin 2001). The numerical procedure is defined as
follows:

$\tilde{\rho}_{i}=\frac{\displaystyle\sum_{j=1}^{N}\rho_{j}\mathrm{v}_{j}w(\mathbf{x}_{i},\mathbf{x}_{j})}{\displaystyle\sum_{j=1}^{N}\mathrm{v}_{j}w(\mathbf{x}_{i},\mathbf{x}_{j})}\;,$
(1)

where ${\rho}_{i}$ and $\tilde{\rho}_{i}$ are the design variable and the
filtered variable associated with the finite element $i$, respectively. The
volume of element $i$ is denoted as $\mathrm{v}_{i}$ and the weight of the
design variable $\rho_{j}$ in the definition of $\tilde{\rho}_{i}$ is denoted
by $w(\mathbf{x}_{i},\mathbf{x}_{j})$. The weighting function depends on the
distance between the centroids of elements $i$ and $j$, which are denoted as
$\mathbf{x}_{i}$ and $\mathbf{x}_{j}$, respectively. Here, the weighting
function is defined by the following decreasing linear function:

$w(\mathrm{\mathbf{x}}_{i},\mathrm{\mathbf{x}}_{j})=\mathrm{max}\left(0\;,\;1-\frac{\|\mathrm{\mathbf{x}}_{i}-\mathrm{\mathbf{x}}_{j}\|}{r_{\mathrm{fil}}}\right)\;.$
(2)

The projection operation transforms the value of all the filtered variables
that are greater than a given threshold $\mu$ close to 1, and close to 0
otherwise (Guest, Prévost, and Belytschko 2004). This is done using the
following smoothed Heaviside function:

$\bar{\rho}_{i}=\frac{\mathrm{tanh}(\beta\mu)+\mathrm{tanh}(\beta\>(\tilde{\rho}_{i}-\mu))}{\mathrm{tanh}(\beta\mu)+\mathrm{tanh}(\beta\>(1-\mu))}\;,$
(3)

where $\bar{\rho}_{i}$ is the projected density and $\beta$ is the parameter
that controls the steepness of the smoothed Heaviside function (Wang, Lazarov,
and Sigmund 2011). The parameter $\mu$ controls the projection threshold and
can be used to impose a uniform manufacturing error. As can be seen in Fig. 2,
the eroded, intermediate and dilated designs are denoted by
$\bm{\bar{\rho}}^{\mathrm{ero}}$, $\bm{\bar{\rho}}^{\mathrm{int}}$ and
$\bm{\bar{\rho}}^{\mathrm{dil}}$, and they are created selecting the values of
thresholds $\mu^{\mathrm{ero}}$, $\mu^{\mathrm{int}}$ and
$\mu^{\mathrm{dil}}$, respectively. Fig. 2 depicts the eroded and dilated
designs colored green to emphasize the fact that the intermediate design is
the actual design intended for manufacturing.

Figure 2: A 2D cantilever beam solution (see boundary conditions in Fig. 6)
illustrating the three-field scheme. (a) Filtering, and (b) projection
operations to obtain the eroded, intermediate and dilated designs.

###  2.1 Minimum size formulation

As mentioned above, the eroded and dilated fields can be seen as the outcome
product of a manufacturing error. As illustrated in Fig. 3, the manufacturing
error can be caused by the misplacement of a milling tool with respect to the
reference line, where $t_{\mathrm{ero}}$ and $t_{\mathrm{dil}}$ represent the
placement errors.

Sigmund (2009) proposed a robust formulation with respect to manufacturing
errors, i.e. the component must provide high performance even if the optimized
design is eventually eroded or dilated during the manufacturing process. For
example, considering the compliance minimization problem subject to a volume
restriction, the robust formulation reads as follows:

$\displaystyle\begin{split}{\min_{\bm{\rho}}}&\quad\max\left(c(\bm{\bar{\rho}}^{\mathrm{ero}})\>,\>c(\bm{\bar{\rho}}^{\mathrm{int}})\>,\>c(\bm{\bar{\rho}}^{\mathrm{dil}})\right)\\\
\mathrm{s.t.:}&\quad\mathbf{v}^{\intercal}\bm{\bar{\rho}}^{\mathrm{ero}}\leq
V^{\mathrm{ero}}\\\
&\quad\mathbf{v}^{\intercal}\bm{\bar{\rho}}^{\mathrm{int}}\leq
V^{\mathrm{int}}\\\
&\quad\mathbf{v}^{\intercal}\bm{\bar{\rho}}^{\mathrm{dil}}\leq
V^{\mathrm{dil}}\\\ &\quad 0\leq{\rho_{i}}\leq 1\;,\end{split}$ (4)

where $c$ is the compliance, $\mathbf{v}$ is the array containing the
elemental volumes, and $V^{\mathrm{ero}}$, $V^{\mathrm{int}}$ and
$V^{\mathrm{dil}}$ are the maximum volume of material allowed in the eroded,
intermediate and dilated designs, respectively.

The compliance of each design field is computed as
$c=\bm{f}^{\intercal}\bm{u}$, where $\bm{f}$ is an array containing the
external forces and $\bm{u}$ is array containing the nodal displacement of the
design field. A linear-elastic response is assumed in the finite element model
that yields $\bm{u}$, and the finite element analysis is performed prior to
solving the iteration of the optimization problem (approach known as Nested
Analysis and Design (Arora and Wang 2005)).

Figure 3: Misplacement of a milling tool. Illustration of the manufacturing
error considered in the robust formulation.

An interesting outcome of the robust formulation is that the design intended
for manufacturing features a minimum size in both the solid and void phases
(Wang, Lazarov, and Sigmund 2011). The minimum size of the solid is defined by
the radius $r_{\mathrm{min.Solid}}$ of the largest circle (or sphere), that
can be inscribed in the thinnest solid member. Conversely, the minimum size of
the void is defined by the radius $r_{\mathrm{min.Void}}$ of the largest
circle (or sphere), that can be inscribed in the smallest cavity of the
topology. Thus, $r_{\mathrm{min.Solid}}$ may represent the radius of a
deposition nozzle in an AM process, while $r_{\mathrm{min.Void}}$ may
represent the radius of a cylindrical (or spherical) milling tool.

It is well known that in the robust formulation, the desired minimum length
scale is implicitly imposed through the parameters that define the filtering
and projection processes (Wang, Lazarov, and Sigmund 2011). Thus,
$r_{\mathrm{min.Solid}}$ and $r_{\mathrm{min.Void}}$ must be defined through
$r_{\mathrm{fil}}$, $\mu^{\mathrm{ero}}$, $\mu^{\mathrm{int}}$, and
$\mu^{\mathrm{dil}}$. The relation between the desired minimum length scales
and the parameters that impose it can be obtained numerically (Wang, Lazarov,
and Sigmund 2011) or analytically (Qian and Sigmund 2013). To facilitate the
description of the proposed methodology, the minimum length scale is fixed at
$r_{\mathrm{min.Solid}}=r_{\mathrm{min.Void}}$, which is imposed through:

$\displaystyle\begin{split}\mu^{\mathrm{ero}}&=0.75\;\;,\\\
\mu^{\mathrm{int}}&=0.50\;\;,\\\ \mu^{\mathrm{dil}}&=0.25\;\;,\\\
r_{\mathrm{fil}}&=2.0\>r_{\mathrm{min.Solid}}\;.\\\ \end{split}$ (5)

It is important to note that other combinations of length scales and threshold
parameters do not change the findings of this work, as long as
$0<\mu^{\mathrm{dil}}<\mu^{\mathrm{int}}<\mu^{\mathrm{ero}}<1$.

The eroded design is the most compliant. Therefore, in the compliance
minimization problem of Eq. (4), the intermediate and dilated designs can be
excluded from the objective function (Amir and Lazarov 2018). In addition, the
volume restriction can be applied to one field only and implicitly restrict
the volume of the other two (Sigmund 2009). Thus, the robust compliance
minimization problem reduces to the following formulation:

$\displaystyle\begin{split}{\min_{\bm{\rho}}}&\quad
c(\bm{\bar{\rho}}^{\mathrm{ero}})\\\
\mathrm{s.t.:}&\quad\mathbf{v}^{\intercal}\bm{\bar{\rho}}^{\mathrm{dil}}\leq
V^{\mathrm{dil}}(V^{\mathrm{int}})\\\ &\quad 0\leq{\rho_{i}}\leq
1\;.\end{split}$ (6)

In the simplified robust topology optimization problem of Eq. (6), the volume
constraint is applied to the dilated design. As the intermediate design is the
one intended for manufacturing, the desired volume constraint
$V^{\mathrm{int}}$ is used to scale the upper bound $V^{\mathrm{dil}}$ that
defines the volume constraint (Amir and Lazarov 2018). There are two main
reasons for this implicit approach to impose the volume constraint. The first
one is the most known in the literature (Amir and Lazarov 2018; Fernández et
al. 2020; Sigmund 2009; da Silva, Beck, and Sigmund 2019; Wang, Lazarov, and
Sigmund 2011) and it is due to the fact that applying the restriction in the
dilated design reduces numerical instabilities and promotes convergence to
better optimum. The second reason, which is rarely mentioned in the literature
(Trillet, Duysinx, and Fernández 2021), is that the implicit volume
restriction allows to involve all 3 design fields in the optimization problem,
which is an essential requirement to ensure the minimum size control on the
solid and void phases (Trillet, Duysinx, and Fernández 2021).

The minimum compliance optimization problem in Eq. (6) is denoted as the
Reference problem hereafter.

###  2.2 Minimum and Maximum size formulation

The topology optimization methods accounting for the size of the deposition
nozzle require in addition control over the maximum size of the solid members.
To this end, the approach proposed by Fernández et al. (2020) is used. In
(Fernández et al. 2020), the maximum size of a solid feature is imposed using
the following local volume restriction (Guest 2009):

$g_{i}(\bm{\bar{\rho}},\varepsilon,q,\Omega_{i})=\varepsilon-\frac{\displaystyle\sum_{j\in\Omega_{i}}v_{j}\>(1-\bar{\rho}_{j})^{q}}{\displaystyle\sum_{j\in\Omega_{i}}v_{j}}\leq
0\quad,$ (7)

where $g_{i}$ is the volume restriction applied within a region $\Omega_{i}$.
The expression $(1-\bar{\rho}_{j})^{q}$ is a measure of the amount of void
within the element $j$, where $q$ is a parameter penalizing intermediate
densities to avoid their appearance during the optimization process, similar
to the SIMP penalty parameter. Therefore, the volume restriction in Eq. (7)
computes the fraction of void within the local region $\Omega_{i}$ and
enforces it to be equal or greater than a given fraction $\varepsilon$. The
local region $\Omega_{i}$ is defined as an annulus (or spherical shell), with
an outer radius $r_{\mathrm{max}}$ and inner radius $r_{\mathrm{min.Solid}}$,
as shown in Fig. 4. Similar to (Fernández et al. 2020; Guest 2009), the void
fraction within the local region is set as $\varepsilon=0.05$ and the penalty
parameter is set as $q=2$.

Figure 4: The annular region $\Omega_{i}$ where the local volume constraint
$g_{i}$ is applied.

Fernández et al. (2020) showed that to obtain an intermediate design
$\bm{\rho}^{\mathrm{int}}$ satisfying the desired length scale, i.e. the
minimum member size ($r_{\mathrm{min.Solid}}^{\mathrm{int}}$), the minimum
cavity size ($r_{\mathrm{min.Void}}^{\mathrm{int}}$) and the maximum member
size ($r_{\mathrm{max}}^{\mathrm{int}}$), it is sufficient to apply the local
volume restriction $g_{i}$ at least in the dilated design. Thus, the
compliance minimization problem with minimum and maximum length scale control
is defined as:

$\displaystyle\begin{split}{\min_{\bm{\rho}}}&\quad
c(\bm{\bar{\rho}}^{\mathrm{ero}})\\\
\mathrm{s.t.:}&\quad\mathbf{v}^{\intercal}\bm{\bar{\rho}}^{\mathrm{dil}}\leq
V^{\mathrm{dil}}(V^{\mathrm{int}})\\\
&\quad\mathrm{G_{ms}}(\bm{\bar{\rho}}^{\mathrm{dil}})\leq 0\\\ &\quad
0\leq{\rho_{i}}\leq 1\;,\end{split}$ (8)

The optimization problem in Eq. (8) is named the Size-Constrained problem
hereafter. The $\mathrm{G_{ms}}$ constraint represents the global maximum size
restriction that aggregates the local ones ($g_{i}$). The aggregation is
performed with a p-mean function, as follows:

$\displaystyle\mathrm{G_{ms}}=\left(\frac{1}{N}\sum_{i=1}^{N}(g_{i}+1-\varepsilon)^{p}\right)^{1/p}-1+\varepsilon\leq
0\;.$ (9)

where $p$ is the aggregation exponent and $\varepsilon$ a shift parameter to
control aggregation accuracy. Similar to (Fernández et al. 2019, 2020), $p$ is
set to 100 and $\varepsilon$ to 0.05 in all the examples reported in this
work.

To apply the local volume constraint $g_{i}$ in the dilated design, the local
region $\Omega_{i}^{\mathrm{dil}}$ must be scaled according to the dilation
distance $t_{\mathrm{dil}}$, as illustrated in Fig. 5. Thus, the annular
region $\Omega_{i}^{\mathrm{dil}}$ is defined by the following outer and inner
radii:

$\displaystyle\begin{split}r_{\mathrm{max}}^{\mathrm{dil}}&=r_{\mathrm{max}}^{\mathrm{int}}+t_{\mathrm{dil}}\;\;,\\\
r_{\mathrm{min.Solid}}^{\mathrm{dil}}&=r_{\mathrm{min.Solid}}^{\mathrm{int}}+t_{\mathrm{dil}}\;.\\\
\end{split}$ (10)

Given the filter and projection parameters defining the minimum length scale
in Eq. (5), the dilation distance is (Fernández et al. 2020; Trillet, Duysinx,
and Fernández 2021):

$t_{\mathrm{dil}}=0.60\>r_{\mathrm{min.Solid}}^{\mathrm{int}}\;.$ (11)

Figure 5: The local $\Omega_{i}$ in the eroded, intermediate and dilated
designs.

Since the methods presented above are available in the literature, sensitivity
analysis is omitted for brevity. Interested readers are referred to (Bendsoe
and Sigmund 2004) for the sensitivity analysis of the objective function and
of the volume restriction, to (Wang, Lazarov, and Sigmund 2011) for the
sensitivity analysis of the eroded, intermediate and dilated designs, and to
(Fernández et al. 2020) for the sensitivity analysis of the maximum size
restriction.

###  2.3 Implementation details

Given the highly non-linearity of the Heaviside projection and of the
aggregation function that builds the maximum size constraint, it is useful to
implement a continuation scheme. As done in (Fernández et al. 2020), the
Heaviside parameter $\beta$ is initialized at 1.5 and is increased by 1.5
times up to a maximum of 38. The SIMP penalty parameter is initialized at 1.0
and is increased up to 3.0 in increments of 0.25. The parameter increments are
performed every 50 iterations. The stopping criterion of the complete TO
process is met either when 450 iterations are reached, or when the maximum
change in the design variables between two consecutive iterations is smaller
than 0.001.

The reference and the maximum-size-constrained problems, i.e. Eqs. (6) and
(8), are implemented in free access codes. Here we use the 88-line code
(Andreassen et al. 2011) and the TopOpt code (Aage, Andreassen, and Lazarov
2015). The first one is written in MATLAB aimed at solving 2D problems whereas
the latter is a C++ code intended to solve large scale 3D problems. These
codes use the density method, the SIMP interpolation scheme and the density
filter. Therefore, the Heaviside projection that builds the eroded,
intermediate and dilated designs has been added, along with the maximum size
restriction and the robust topology optimization formulation. The optimization
problems are solved using the Method of Moving Asymptotes (MMA) (Svanberg
1987).

The reader interested on implementation details is referred to (Fernández et
al. 2020), since the TopOpt code with the corresponding modifications to solve
the reference and the maximum size constrained problems are provided.

Figure 6: 2D cantilever beam design domain.

##  3 TO for LSAM

Having introduced the framework to control minimum and maximum size within a
topology optimization framework, we proceed with the discussion of the two
methods to ensure designs compatible with the intended LSAM process. First, a
test problem is introduced by which the methods are explained. Also solutions
from the reference optimization problem are discussed to clarify the need for
specific design restrictions, after which in subsection 3.1 a method based on
a nozzle size constraint is presented. Subsection 3.2 proceeds with the second
method, which is based on dilating a maximum size-constrained design.

The proposed strategies are explained using a 2D cantilever beam for
compliance minimization. For the sake of clarity, arbitrary yet realistic
dimensions are chosen in this test case. The design domain is defined by a
rectangle of 300 mm $\times$ 150 mm, as shown in Fig. 6. This domain is
discretized into 300 $\times$ 150 quadrilateral finite elements, resulting in
square elements of 1 mm length. The optimized design is intended for an AM
process equipped with a deposition nozzle of 5 mm diameter. Henceforth, the
size of the deposition nozzle is defined by its radius, which is denoted as
$r_{\mathrm{nozzle}}$.

To provide an insight into the printability of the reported designs, the
deposition paths obtained with the PrusaSlicer (v.2.2.0) software are
provided. PrusaSlicer (Pr$\mathring{\text{u}}$ša 2020) is a freely available
software that generates G-codes from STL files. The software is designed for
desktop 3D printers, where the nozzle diameter is approximately 0.4 mm, which
is rather small compared to a typical component intended for LSAM machines.
Although the nozzle size can be modified to a bigger size, the slicing
algorithms of PrusaSlicer (v.2.2.0) do not take into account the low-
resolution issue caused by large deposition nozzle sizes. Nonetheless, this
software is employed herein as a tool to provide the reader with an idea of
the printability of the component when large nozzle sizes are used. In this
work, the printability of the optimized designs appears next to the symbol *B
on the corresponding figures and it represents the percentage of the design
that can be filled using a nozzle of a given size. Thus, the printability
index (*B) is be defined as:

$\mathrm{Printability}:=\frac{\displaystyle\sum_{i=1}^{n_{\mathrm{Layers}}}A^{\mathrm{filled}}_{i}}{\displaystyle\sum_{i=1}^{n_{\mathrm{Layers}}}A_{i}}\times
100\;\;[\%]\quad,$ (12)

where $n_{\mathrm{Layers}}$ is the number of layers resulting from the slicing
procedure, which takes the value 1 in 2D designs. The surface area of the
layer desired to be filled by the deposition process is denoted by $A_{i}$,
and the actual surface area filled is denoted by $A^{\mathrm{filled}}_{i}$,
where $i$ is the index of the $i^{\mathrm{th}}$ layer.

PrusaSlicer offers different infill patterns, from which the perimeter one is
chosen. Thus, the slicing algorithm tracks the design perimeter sequentially,
which is a common practice in LSAM processes, especially those equipped with
large deposition nozzles. However, the software requires at least two
deposition paths per solid member, one for each side of the member. Therefore,
to ensure that each solid member found in the optimized design contains at
least two deposited beads, the minimum size of the solid phase must be twice
the size of the nozzle, i.e.
$r_{\mathrm{min.Solid}}^{\mathrm{int}}=2r_{\mathrm{nozzle}}=5$ mm.

Given that the deposition paths follow the contour of the design, it is
convenient to control the minimum size of the void phase in order to impose a
radius of curvature at the reentrant corners of the design and avoid the
difficulties associated to the sharp corners in the deposition path.
Arbitrarily, the minimum radius of curvature is set as
$r_{\mathrm{min.Void}}^{\mathrm{int}}=5$ mm, which is imposed with the filter
and projection parameters listed in Eq. (5).

Three reference problems (Eq. (6)) are solved for the cantilever beam depicted
in Fig. 6 with different volume restrictions, i.e. $V^{\mathrm{int}}=30\%$,
$V^{\mathrm{int}}=40\%$ and $V^{\mathrm{int}}=50\%$. It is recall that no
maximum size constraint is used. The optimized designs and the perimeter
deposition paths are shown in the first and second rows of Table 1,
respectively. The two circles next to each optimized solution indicate the
desired length scale. The blue circle represents the minimum size of the void
phase while the black circle represents the minimum size of the solid phase.
Both have been set to 5 mm. Regarding the deposition paths, the deposited
material is shown in light blue and the regions that cannot be filled are
shown in yellow. The printability of the optimized design is reported next to
the deposition paths. To complement the analysis and discussion, deposition
paths obtained with a rectilinear filling pattern, and combination of
perimeter and rectilinear are included in the third and fourth rows of Table
1, respectively.

Table 1: Optimized reference cantilever beams for compliance minimization with minimum size control, for the problem defined in Fig. 6. The circles next to each solution represent the minimum size of the solid (black circle) and void (blue circle) phases. The second row shows the deposition path provided by the PrusaSlicer software using a perimeter pattern, the third row using a rectilinear pattern, and the fourth row using a combination of rectilinear and perimeter pattern. Next to each deposition path, the printability (*B) is reported, as defined in Eq. (12). |  $V^{\mathrm{int}}=30\%$ |  $V^{\mathrm{int}}=40\%$ |  $V^{\mathrm{int}}=50\%$
---|---|---|---
Design |  |  |
|  c($\bm{\bar{\rho}}^{\mathrm{int}}$)=124.4 |  c($\bm{\bar{\rho}}^{\mathrm{int}}$)=88.8 |  c($\bm{\bar{\rho}}^{\mathrm{int}}$)=73.2
perimeter |  |  |
rectilinear |  |  |
combination |  |  |

Considering the perimeter infill pattern in Table 1, the reference designs
allow fabrication by at least two adjacent beads per solid member, which is
attributed to the minimum size control imposed on the topology. However,
perimeter paths present large unfilled areas that do not match the nozzle
size. Better printability can be achieved using other deposition paths or
combinations of infill patterns, as shown in the third and fourth rows of
Table 1. Even though, it is not possible to guarantee filling all or most of
the design during the deposition process, because it is highly likely that
numerous design features will not meet the resolution of the deposition
nozzle, as this condition is not imposed on the geometry of the reference
optimized designs. For example, it can be seen that a rectilinear pattern
covers more area than the perimeter pattern, especially in designs with large
volume ($V^{\mathrm{int}}$ equal to 40$\%$ and 50$\%$) as shown in the third
row of Table 1, but it does not cover more than 95$\%$ of the surface due to
the small gaps that are present at the outer edge of the design as a result of
the numerous path turns. Undoubtedly, the optimal deposition path must
consider not only the ability to cover the largest surface area but also other
aspects such as the anisotropy of the deposited material, the step-over
distance, the number of deposition passes and path continuity, surface finish,
and post-machining work among others (Ding et al. 2015; Liu et al. 2018).
These aspects are not considered in this work and the proposed methods for
topology optimization are limited to producing optimized designs tailored to
the process resolution in order to facilitate the path planning and guarantee
higher printability.

It is also worth noting that better printability can be achieved not only by
optimizing the deposition path, but also using a smaller deposition nozzle
size in the remaining unfilled areas, by overlapping the beads, or by
modifying the feed rate and travel speed. Nevertheless, each of these measures
adds a degree of complexity, cost and risk to the process, which is not
desired. Instead, the aim of this work is to propose optimized designs
suitable with the nozzle size and thus reduce or avoid the need to modify the
deposition parameters. For this purpose, the maximum size constraint is used
in two different forms, which leads to the two methods proposed in this
article.

###  3.1 Method 1: Nozzle size constraint

The first method conforms the optimized design to the size of the deposition
nozzle by tailoring the minimum and maximum size of the solid phase according
to the size of the deposition nozzle. Given that in this study it is decided
to place two deposited beads per solid member, the minimum and maximum length
scales are chosen as follows:

$\begin{split}r_{\mathrm{min.Solid}}^{\mathrm{int}}&=2r_{\mathrm{nozzle}}=5\text{mm}\;\;,\\\
r_{\mathrm{max}}^{\mathrm{int}}&=2r_{\mathrm{nozzle}}+\Delta
r=6\text{mm}\;,\end{split}$ (13)

where $\Delta r$ is a small distance to avoid contradictions between the
minimum size imposed by the Heaviside projection and the maximum size imposed
by the maximum size constraint. In this work, $\Delta r$ is equal to the size
of a finite element, which was found to work well. The specific choices in Eq.
(13) ensure the width of all structural features ranging from 5 mm to 6 mm,
i.e. twice the bead width, which facilitates manufacturability by a LSAM
process equipped with a large deposition nozzle fixed in size.

The resulting maximum-size-constrained designs and the corresponding
deposition paths are shown in Table 2. The desired maximum size of the solid
phase is indicated by a black circle next to each solution. Unlike the
reference designs, those constrained in maximum feature size possess
structural members of uniform width, which leads to fewer unfilled gaps in the
designs. The size-constrained members admit two parallel paths, notably
without gaps between the deposited beads. In addition, as designs are filled
with two parallel beads that follow the surface of the design, no T-joints or
path crossovers are obtained. Furthermore, as the minimum size of the void
phase is set equal to the radius of the deposition nozzle, no sharp corners
are obtained in the deposition path. Note also that the printability of all
designs with nozzle size constraint (95.1-98.1$\%$) is superior to that of the
reference designs (85.9-91.5$\%$). Therefore, designs constrained in maximum
size (Table 2) are demonstrated to be much easier to manufacture by LSAM than
the reference designs (Table 1).

Table 2: Optimized cantilever beams and the deposition paths obtained with PrusaSlicer. Design optimized for compliance minimization with simultaneous minimum and maximum size control, for the problem defined in Fig. 6. The difference in compliance with respect to the corresponding reference design (Table 1) is indicated in parentheses. Next to each design, its printability (*B) is reported, as defined in Eq. (12). The arrows point to areas where the length scale is not fulfilled. $V^{\mathrm{int}}=30\%$ |  $V^{\mathrm{int}}=40\%$ |  $V^{\mathrm{int}}=50\%$
---|---|---
|  |
|  |
c($\bm{\bar{\rho}}^{\mathrm{int}}$)=136.6 (+9.8 $\%$) |  c($\bm{\bar{\rho}}^{\mathrm{int}}$)=119.1 (+34.1 $\%$) |  c($\bm{\bar{\rho}}^{\mathrm{int}}$)=94.4 (+29.0 $\%$)
Figure 7: Maximum size-constrained cantilever beam solved for
$V^{\mathrm{int}}=40\%$ and
$r_{\mathrm{min.Solid}}^{\mathrm{int}}=4r_{\mathrm{nozzle}}=10\text{mm}$. The
design domain is defined in Fig. 6. The symbol *B represents the printability
index defined in Eq. (12).

Despite the benefits introduced by the maximum size restriction, some
drawbacks can be noted. For instance, small unfilled zones can be observed at
the intersection of two or more solid members. There, the perimeter deposition
path exposes triangular unfilled regions, which represent about $2\%$ of the
design obtained with $V^{\mathrm{int}}=30\%$ in Table 2. Due to these unfilled
zones, it is expected that performance of the manufactured component will be
different from that of the optimized design, although the performance
reduction is not quantified in this study. Regarding this geometrical drawback
at the intersection of two or more solid members, we point out that the
printability of such zones depends on the intended AM process and the
deposition path. For example, if the nozzle size is chosen to be equal to the
width of the solid members, the intersection between solid members would
represent a T-joint or a crossover of paths. In such cases, it may be
advantageous to have an intersection larger than the prescribed size since, in
practice, it is difficult to achieve a joint with high dimensional accuracy
(Venturini et al. 2016).

The size-constrained design in Table 2 for a volume restriction of
$V^{\mathrm{int}}=50\%$ exhibits zones that do not meet the desired minimum
size, such as those indicated by the red arrows in the deposition path in
Table 2. In addition, given that the global maximum size constraint
$\mathrm{G_{ms}}$ is defined by a p-mean aggregation function, it is possible
to find parts of the topology that do not meet the imposed maximum size, as
the one highlighted by the blue arrow in Table 2. These undesirable features
typical of a local optimum can be reduced by adjustment of parameters. For
instance, by increasing the $p$ exponent of the aggregation function and
increasing the value of the penalty parameters ($\beta$ and SIMP) more slowly,
i.e. by adapting the continuation method to involve a larger number of
iterations (Fernández et al. 2020).

It is worth noting that the choice of placing two material beads is to make
use of the perimeter deposition path of PrusaSlicer, which is used for
illustrative purposes. That is, the proposed method can be used to place any
number of beads along the structural members by choosing the maximum and
minimum size of the solid phase according to the desired number of beads. For
illustrative purposes, a cantilever beam is optimized taking into account 4
material beads along the structural members. Since an even number of beads is
assumed, the perimeter pattern can be used to illustrate the deposition path,
which is shown in Fig. 7. It is noticeable that a large part of the design is
filled with 4 material beads arranged in a parallel fashion, and the remaining
gaps at the intersection of solid members are filled with triangular patterns,
which may present difficulties when printing due to the sharp corners of the
trajectory, as outlined in Fig. 7. Therefore, positioning two material beads
per solid member may be a preferable strategy when using maximum size
constraints.

Certainly, one of the most significant drawbacks of the maximum size
restriction is the performance reduction. It can be seen that the size-
constrained design obtained with $V^{\mathrm{int}}=40\%$ has a $34\%$ higher
compliance compared to the reference design. Requiring all structural members
to have the same width of, in this case, two beads, is a rather strong design
restriction. Allowing for design features to have varying integer multiples of
adjacent beads in various places throughout the structure would also result in
a manufacturable design, while offering more design freedom. To conform the
optimized design to the size of the deposition nozzle and to avoid a
significant reduction on the structural performance, the following subsection
presents a second strategy, which is also based on maximum size constraints.

###  3.2 Method 2: Skeleton-based deposition paths

In the method discussed above, the chosen minimum and the maximum length
scales dictate structural members to have uniform width, which facilitates the
generation of a feasible deposition path. However, the structural members
remain separated from each other, which negatively affects the structural
performance. This subsection shows that the separation between members can be
eliminated through a dilation projection while retaining the
manufacturability. The proposed method is based on the numerical study
presented in (Fernández et al. 2020).

Fernández et al. (2020) revealed the behaviour of the maximum size constraint
in the context of the robust formulation by restricting the size of the
eroded, intermediate and dilated designs separately. The authors concluded
that in order to impose the desired minimum and maximum length scales, it is
necessary to restrict at least the dilated design. Part of the numerical study
developed in (Fernández et al. 2020) is reproduced in Table 3 in order to
highlight the property in interest for our work. To this end, the maximum-
size-constrained problem in Eq. (8) is solved but with variations on the field
where the maximum size constraint is applied. The first and second rows of the
table show the result obtained when the maximum size restriction is applied to
the intermediate and to the dilated design, respectively. The local region
$\Omega_{i}$ is shown next to the field in which the maximum size constraint
is applied, while the desired minimum length scale is shown by the blue and
black circles next to the intermediate design. As concluded in (Fernández et
al. 2020), the minimum and maximum length scales are met only when
$\mathrm{G_{ms}}(\bm{\bar{\rho}}^{\mathrm{dil}})$ is included in the topology
optimization problem. However, in this work we emphasize another important
observation.

Table 3: Optimized designs obtained with maximum size constraints applied on different design fields. Circles next to the intermediate design illustrate the desired minimum size in the void phase (blue circle), in the solid phase (black circle), and the maximum size in the solid phase (black ring). Constraint |  $\bm{\bar{\rho}}^{\mathrm{ero}}$ |  $\bm{\bar{\rho}}^{\mathrm{int}}$ |  $\bm{\bar{\rho}}^{\mathrm{dil}}$
---|---|---|---
$\mathrm{G_{ms}}(\bm{\bar{\rho}}^{\mathrm{int}})$ |  |  |
$\mathrm{G_{ms}}(\bm{\bar{\rho}}^{\mathrm{dil}})$ |  |  |

The first row of Table 3 suggests that if the maximum size constraint is
applied only in the intermediate design, the dilated design does not feature a
maximum member width, since the dilation projection closes the cavities
present in the intermediate design, as illustrated in Fig. 8. Under this
formulation, the eroded and intermediate designs can be interpreted as the
structural skeleton or the deposition lines of an AM process, and the dilated
design would be the component obtained after deposition. For suitably chosen
length scales, the bonding of adjacent beads can be represented in this way.
This observation allows us to formulate the following topology optimization
problem:

$\displaystyle\begin{split}{\min_{\rho}}&\quad\alpha\>c(\bm{\bar{\rho}}^{\text{\tiny{AM}}})+(1-\alpha)\>c(\bm{\bar{\rho}}^{\mathrm{ero}})\\\
\mathrm{s.t.:}&\quad\mathbf{v}^{\intercal}\bm{\bar{\rho}}^{\text{\tiny{AM}}}\leq
V^{\text{\tiny{AM}}}\\\
&\quad\mathrm{G_{ms}}(\bm{\bar{\rho}}^{\mathrm{int}})\leq 0\\\ &\quad
0\leq{\rho_{i}}\leq 1\;,\end{split}$ (14)

where $\bm{\bar{\rho}}^{\text{\tiny{AM}}}$ represents a dilated projection and
the design to be manufactured. Henceforth, this optimization problem (14) is
denoted as the AM-constrained problem.

Figure 8: Illustration of the bonding of bars resulting from the dilation
projection. Labels A, B and C represent the zones highlighted in Table 3. The
zone C emphasizes the fact that the design to be manufactured is the dilated
one, for which it is now defined as $\bm{\bar{\rho}}^{\text{\tiny{AM}}}$.

The AM-constrained problem contains the sum of two compliances in the
objective function. The compliance of the eroded design serves to impose the
minimum size of the solid members (Wang, Lazarov, and Sigmund 2011), while the
compliance of the dilated design is to ensure a good performance of the
component intended for manufacturing. The compliances are weighted by
$\alpha$, where $0\leq\alpha\leq 1$. To define $\alpha$, the following must be
considered. If the eroded compliance is prioritized ($\alpha\approx 0$), the
minimum size of the solid phase is guaranteed, but as insignificant emphasis
is placed on the performance of the printed structure, the bars of the AM
design could still be separated by a very thin layer of gray elements (as seen
in zone labeled as C in Table 3), which could impair structural performance.
In addition, if a large volume of material is allowed in the design domain,
floating structures can arise in the dilated design (Fernández et al. 2020).
On the other hand, if the dilated compliance is prioritized ($\alpha\approx
1$), the bonding of bars is promoted but the minimum size control is not
ensured. If the compliances are weighted equally ($\alpha=0.5$), a topology
that does not meet the desired minimum size and/or that does not completely
merge the bars could be obtained. To avoid these issues, the weighting
parameter $\alpha$ is defined with a continuation scheme. At the beginning of
the optimization problem, the minimum size is prioritized, and as the
optimization progresses, the structural performance is prioritized. Thus,
$\alpha$ can be defined by the following function:

$\alpha=\frac{\mathrm{Iter}-1}{\mathrm{Iter_{max}}-1}\;,$ (15)

where $\mathrm{Iter_{max}}$ is the maximum number of iterations defined for
the continuation scheme and $\mathrm{Iter}$ is the current iteration number.
Although this continuation scheme causes the objective to change in every
iteration, it was found to have no detrimental effect on convergence, and has
been used successfully in all the cases considered in this study.

To select appropriate length scales and obtain designs suitable for Large-
Scale AM processes, i.e. members composed of an integer number of parallel
beads, the following must be considered. Given that the design intended for
manufacturing is $\bm{\bar{\rho}}^{\text{\tiny{AM}}}$, the desired minimum
size in the solid phase is $r_{\mathrm{min.Solid}}^{\text{\tiny{AM}}}$, as
illustrated in Fig. 8. Therefore, the set of projection and filtering
parameters ($\mu^{\text{ero}}$, $\mu^{\mathrm{int}}$, $\mu^{\text{\tiny{AM}}}$
and $r_{\mathrm{fil}}$) must be found for a length scale defined in the
dilated design. The information available in the literature to obtain the
filtering and projection parameters are meant for a length scale defined in
the intermediate design (Wang, Lazarov, and Sigmund 2011; Fernández et al.
2020), but it can be used here knowing that the length scale of the
intermediate and dilated designs are related by the dilation distance, as
follows:

$r_{\mathrm{min.Solid}}^{\text{\tiny{AM}}}=r_{\mathrm{min.Solid}}^{\text{\tiny{int}}}+t_{\mathrm{dil}}$
(16)

Taking into account that the following set of thresholds is adopted:

$\displaystyle[\mu^{\text{\tiny{AM}}},\;\mu^{\mathrm{int}},\;\mu^{\mathrm{ero}}]$
$\displaystyle=[0.25\;,\;0.50\;,\;0.75]\;,$ (17)

it only remains to find the relationships between
$r_{\mathrm{min.Solid}}^{\text{\tiny{int}}}$, $t_{\mathrm{dil}}$ and
$r_{\mathrm{fil}}$, and to replace these relationships in Eq. (16) to obtain
the filter radius leading to $r_{\mathrm{min.Solid}}^{\text{\tiny{AM}}}$. To
assist the reader in understanding the procedure for obtaining the required
relationships, we include in Fig. 9 graphs relating the projection parameters
and the minimum sizes of the intermediate design. For the generation of these
graphs, the MATLAB code provided in (Trillet, Duysinx, and Fernández 2021) has
been run as NumericalSolution(10,38). The input parameter ”10” denotes a
representative size of the filter radius (given in finite elements) simply
needed to reduce rounding errors resulting from the finite element
discretization. ”38” represents the maximum value of the smoothed Heaviside
projection parameter. The procedure to obtain the filter radius for a given
nozzle size is detailed hereafter.

The minimum size of the solid phase in the intermediate design
($r_{\mathrm{min.Solid}}^{\text{\tiny{int}}}$) can be expressed as a function
of the filter radius using Fig. 9(a), which yields to:

$\displaystyle r_{\mathrm{min.Solid}}^{\text{\tiny{int}}}$
$\displaystyle=0.5r_{\mathrm{fil}}$ (18)

The dilation distance can be expressed as a function of the filter radius
using Fig.9(b), resulting in:

$\displaystyle t_{\mathrm{dil}}$
$\displaystyle=0.60r_{\mathrm{min.Solid}}^{\text{int}}$ (19)

Then, using Eqs. (19) and (18) in Eq. (16), the filter radius as a function of
the desired minimum length scale is obtained:

$\displaystyle r_{\mathrm{fil}}$
$\displaystyle=1.25\>r_{\mathrm{min.Solid}}^{\text{\tiny{AM}}}$ (20)

The procedure explained in Eqs. (18)-(20) can be used to obtain the filter
radius ($r_{\mathrm{fil}}$) as a function of the desired length scale
($r_{\mathrm{min.Solid}}^{\text{\tiny{AM}}}$) for any other set of thresholds
($\mu^{\text{\tiny{AM}}}$, $\mu^{\mathrm{int}}$ and $\mu^{\mathrm{ero}}$) that
may be required for a specific reason.

(a) Graph to obtain $r_{\mathrm{min.Solid}}^{\mathrm{int}}(r_{\mathrm{fil}})$

(b) Graph to obtain $t_{\mathrm{dil}}(r_{\mathrm{min.Solid}}^{\mathrm{int}})$

Figure 9: Graphs relating the projection thresholds, the filter radius, the
minimum size of the solid phase in the intermediate design and the dilation
distance. The highlighted points are used in Eqs. (18) and (19). The graphs
have been constructed with the MATLAB script provided in (Trillet, Duysinx,
and Fernández 2021), running the code as NumericalSolution(10,38).

It is emphasized that the relationships between parameters in Eqs. (16)-(20),
are obtained from the graphs of Fig. 9, which are built from a numerical
method that assumes a 1D design domain (Wang, Lazarov, and Sigmund 2011) and a
filter radius of the order of 10 finite elements. For this reason, it is
expected that the relationships provided by Fig. 9 will not be accurately met
in practice. A major source of error comes from rounding the number of finite
elements, especially when small distances (close to one or two finite
elements) are involved (Trillet, Duysinx, and Fernández 2021). This rounding
error explains the improved behavior of the method when using a slightly
smaller dilation projection threshold, which in our case is 0.20 instead of
0.25. Taking into account this observation, the set of parameters that are
used hereafter in the AM-constrained problem are:

$\displaystyle\begin{split}r_{\mathrm{fil}}&=1.11\>r_{\mathrm{min.Solid}}^{\text{\tiny{AM}}}\\\
\mu^{\mathrm{ero}}&=0.75\\\ \mu^{\mathrm{int}}&=0.50\\\
\mu^{\text{\tiny{AM}}}&=0.20\;.\\\ \end{split}$ (21)

where the factor 1.11 comes from the equations $r_{\mathrm{fil}}$ =
$2r_{\mathrm{min.Solid}}^{\text{int}}$ and $t_{\mathrm{dil}}$ =
$0.80r_{\mathrm{min.Solid}}^{\text{int}}$, which are obtained from Figs. 9(b)
and 9(a), respectively. The reason why $\mu^{\text{\tiny{AM}}}=0.20$ performs
better than $\mu^{\text{\tiny{AM}}}=0.25$ is described in the following. It is
recalled that in the illustrative example, the minimum size of the solid phase
($r_{\mathrm{min.Solid}}^{\text{\tiny{AM}}}$) has been set equal to 5 finite
elements. When using a projection $\mu^{\text{\tiny{AM}}}=0.25$, the dilation
distance is $t_{\mathrm{dil}}=0.6\times 0.5\times 1.25\times
r_{\mathrm{min.Solid}}^{\text{\tiny{AM}}}=1.88$ finite elements. On the other
hand, if $\mu^{\text{\tiny{AM}}}=0.20$ is used, the dilation distance is
$t_{\mathrm{dil}}=0.8\times 0.5\times 1.1\times
r_{\mathrm{min.Solid}}^{\text{\tiny{AM}}}=2.20$ finite elements. Numerically,
1.88 results in one finite element (0.88 elements of rounding error), and 2.2
results in two finite elements (0.2 elements of rounding error). That is, a
projection using $\mu^{\text{\tiny{AM}}}=0.20$ induces a smaller rounding
error and produces a dilation distance one finite element larger than the one
obtained with $\mu^{\text{\tiny{AM}}}=0.25$, which results in a better bonding
between members in the dilated projection. The reader intending to modify the
projection thresholds or imposed length scale is recommended to make use of
the numerical method provided in (Trillet, Duysinx, and Fernández 2021), as it
entails smaller rounding errors and thus less work in parameter tuning
compared to the analytical method provided in the cited work.

Now that the formulation of the topology optimization problem has been
presented and the projection and filtering parameters have been defined, the
proposed method is demonstrated on the cantilever beam presented in the
previous section. As the minimum size of the solid phase is set to twice the
size of the nozzle, $r_{\mathrm{fil}}=1.11\>\times 2\times
r_{\mathrm{nozzle}}=5.6$ mm. The eroded, intermediate and AM designs, along
with the deposition paths are shown in Table 4.

Similar to the maximum-size-constrained designs reported for the first method,
the AM-constrained designs do present small unfilled gaps, which are mostly
situated at the intersection of solid members. In terms of printability, the
designs perform similarly as well. However, the AM-constrained designs exhibit
better performance on the face of increased design freedom. The difference is
most noticeable when large volumes of material are allowed in the topology,
which highlights the need to prefer the AM-constrained optimization problem
over the maximum-size-constrained one.

Table 4: Optimized designs provided by the AM-constrained topology optimization problem. The difference in compliance with respect to the reference design is indicated in parentheses. The percentage next to the symbol *B represents the printability index defined in Eq. (12). |  $V^{\text{\tiny{AM}}}=30\%$ |  $V^{\text{\tiny{AM}}}=40\%$ |  $V^{\text{\tiny{AM}}}=50\%$
---|---|---|---
$\bm{\bar{\rho}}^{\mathrm{ero}}$ |  |  |
$\bm{\bar{\rho}}^{\mathrm{int}}$ |  |  |
$\bm{\bar{\rho}}^{\text{\tiny{AM}}}$ |  |  |
|  c($\bm{\bar{\rho}}^{\text{\tiny{AM}}}$)=132.3 (+6.4$\%$) |  c($\bm{\bar{\rho}}^{\text{\tiny{AM}}}$)=101.8 (+14.6$\%$) |  c($\bm{\bar{\rho}}^{\text{\tiny{AM}}}$)=80.2 (+9.6$\%$)
Deposition Path |  |  |

For a small amount of material, such as $V=30\%$, the reference (Table 1), the
size-constrained (Table 2) and the AM-constrained (Table 4) optimization
problems generate similar topologies. However, for large volumes of material,
the topologies can be very different and therein lies the benefit of the AM-
constrained formulation. For example, for $V=50\%$, the
$\bm{\bar{\rho}}^{\text{\tiny{AM}}}$ design presents zones where two material
beads are placed next to each other to form a bulky zone. As the volume of
material is increased in the design space, more parallel material beads merge,
as shown in the fourth column of Table 4.

Table 5: AM-constrained designs for different nozzle sizes and $V^{\text{\tiny{AM}}}=60\%$. The firs row shows the optimized designs. The second row shows the eroded designs superimposed on the AM-designs. The third row shows the deposition paths obtained with PrusaSlicer (v.2.2.0). The percentage next to the symbol *B represents the printability index defined in Eq. (12). |  $r_{\mathrm{nozzle}}=2.5$ mm |  $r_{\mathrm{nozzle}}=4.1$ mm |  $r_{\mathrm{nozzle}}=5.5$ mm
---|---|---|---
$\bm{\bar{\rho}}^{\text{\tiny{AM}}}$ |  |  |
$\bm{\bar{\rho}}^{\text{\tiny{AM}}}$ , $\bm{\bar{\rho}}^{\mathrm{ero}}$ |  |  |
Depos. Path |  |  |

Table 5 shows three AM-constrained results obtained with varying nozzle sizes,
and a volume fraction of $V^{\text{\tiny{AM}}}=60\%$. The Table includes the
eroded designs superimposed on the optimized designs and the deposition path
obtained with PrusaSlicer. Note that the deposition paths are obtained with a
nozzle size
$r_{\mathrm{nozzle}}=0.5r_{\mathrm{min.Solid}}^{\text{\tiny{AM}}}$, which
places at least 2 parallel paths per solid member. If the deposition nozzle
were chosen twice the size, i.e.
$r_{\mathrm{nozzle}}=r_{\mathrm{min.Solid}}^{\text{\tiny{AM}}}$, then each
solid member would consist of at least one bead. In such a case, the eroded
and intermediate fields could be read as the deposition trajectory, since due
to the projection strategy, they are positioned exactly at the center of the
design intended for manufacturing, as shown in the second row of Table 5.
Therefore, the deposition path provided by the eroded or intermediate design
is expected to leave fewer unfilled zones than the one proposed by
PrusaSlicer.

An interesting observation can be made from the results depicted in Table 5,
where the AM-constrained designs present material beads, at first sight,
oriented with the major principal stress direction. Surely, this pattern is
due to the compliance cost function that is evaluated for a single load case,
yet, it might be beneficial when considering an orthotropic elasticity of the
material bead, for instance, in the WAAM process (Laghi et al. 2020). This
observation, however, is not addressed in this article, hence it is left open
for future research.

As the design intended for manufacturing corresponds to a dilated one, it is
not possible to guarantee the minimum size of the cavities in
$\bm{\bar{\rho}}^{\text{\tiny{AM}}}$. In addition, the presence of small
cavities can be a consequence of utilizing large material beads. Note that in
practice it is difficult to achieve high accuracy in the material bead width
for some LSAM processes, as for instance WAAM (Xia et al. 2020). In such
instances, some dimensional uncertainty is present that could fill the voids
present in the design. Such beneficial uncertainty could potentially be
considered in the topology optimization formulation by using perturbation
techniques on the length scale (Lazarov, Schevenels, and Sigmund 2012). This
topic, however, is outside the scope of this work.

The approach that defines Method 2 is based on constraining the maximum size
of a design and dilating it according to the prescribed deposition nozzle
size. This results in a dilated design with improved printability, and in an
eroded design that is placed at the center of the dilated members resembling a
structural skeleton that can be interpreted as the deposition path. It should
be noted that the structural skeleton is a by-product of constraining the
intermediate design, and is not explicitly computed using edge detection
algorithms (Zhou et al. 2015). This is the main advantage of the proposed
Method 2 in comparison to other methods that could eventually be used for
improving the printability of the designs, since, to date, the edge detection
algorithms for topology optimization are not robust in the density method,
mainly due to the fact that intermediate densities complicate boundary
detection. For this reason, in the density method, length scale control
algorithms based on the explicit calculation of the structural skeleton are
applied almost at the end of the optimization process (Zhou et al. 2015), when
the topology has already been defined, i.e. they operate as post-processing
tools. That being said, Liu (2019) proposes a functional for level set
topology optimization that allows to define a variable length scale within the
design, which would also allow generation of designs adapted to the deposition
nozzle, similar to Method 2. However, the cited method has not been validated
in the density framework nor in 3D design domains, these two aspects being a
major challenge for algorithms based on the explicit calculation of the
skeleton. The following section demonstrates that the proposed Method 2 can be
easily applied to 3D geometries, either considering AM processes with a fixed
printing direction or with variable printing direction, such as AM processes
assisted by robotic arms.

##  4 3D example and discussion

The two methods proposed in the previous sections were presented using a two-
dimensional design case, where the deposition path is defined in a 2D plane.
This section assesses and compares the methods using a three-dimensional
design case. To this end, the cantilever beam shown in Fig. 10 is used. The
design domain is discretized into 2.76 million cubic elements of 1 mm edge
length. The optimized design is intended for an AM process equipped with a
deposition nozzle of $r_{\mathrm{nozzle}}=5$ mm. It is assumed that the AM
process generates layers orthogonal to the build direction, as shown in Fig.
10. In addition, it is assumed that each layer is 2 mm thick, so the final
component is made up of 60 layers. As each layer of the optimized design must
conform to the size of the deposition nozzle, the maximum size restriction is
applied in the plane, as shown in Fig. 10.

As in the previous section, the finite element model considers an isotropic
linear elastic behavior for both the deposited material and interlayer bond.
Undoubtedly, this consideration may not be representative of some materials
and LSAM processes, such as stainless steel in WAAM (Kyvelou et al. 2020),
where material anisotropy should not be disregarded during topology
optimization (Zhang, Liu, and To 2017). However, as this work focuses on the
nozzle size constraint, we keep the material modeling part general assuming an
isotropic behavior instead of using a specific anisotropic model (see, for
instance, Jantos, Hackl, and Junker 2020).

Figure 10: In (a), the 3D cantilever beam for compliance minimization. In (b), the layer and the local region $\Omega_{i}$ where the local maximum size restriction is formulated. The design domain is discretized with $240\times 120\times 96$ finite elements. Table 6: 3D cantilever beams obtained from the Reference, the Maximum-size-constrained and the AM-constrained topology optimization problems. The design domain is shown in Fig. 10. The optimized designs are sliced using PrusaSlicer (v.2.2.0). The percentage of the design that is filled with deposited material is indicated next to the sliced design. The second and third rows show the least and most filled layers, respectively. The fourth row shows two representative layers for the deposition paths obtained with PrusaSlicer. |  Reference |  Max.-Size-Constrained |  AM-Constrained
---|---|---|---
Sliced Design |  |  |
|  c($\bm{\bar{\rho}}^{\mathrm{int}}$)=1.00 |  c($\bm{\bar{\rho}}^{\mathrm{int}}$)=1.21 |  c($\bm{\bar{\rho}}^{\text{\tiny{AM}}}$)=1.16
Least filled |  |  |
|  Layer: 45 , *B 86.6$\%$ |  Layer: 57 , *B 96.6$\%$ |  Layer: 2 , *B 95.2$\%$
Most filled |  |  |
|  Layer: 22 , *B 97.6$\%$ |  Layer: 42 , *B 100.0$\%$ |  Layer: 15 , *B 97.5$\%$
Representative Layers |  |  |
Layer: 53 , *B 93.0$\%$ |  Layer: 6 , *B 98.7$\%$ |  Layer: 42 , *B 97.4$\%$
|  |
|  Layer: 6 , *B 89.5$\%$ |  Layer: 24 , *B 99.1$\%$ |  Layer: 30 , *B 97.0$\%$

Three designs are reported, the Reference, the Maximum-size-constrained and
the AM-constrained designs. The optimization problems are solved for a volume
restriction of $15\%$ and with a minimum size of $r_{\mathrm{min.solid}}=5$
mm. The optimized results are sliced in PrusaSlicer, and the obtained
deposition paths are shown in Table 6. The first row of the table shows the
component after all the layers have been deposited and the averaged
printability of all layers combined. The remaining rows show 4 selected layers
and the associated deposition path. These layers correspond to the least
filled, the most filled, and two representative of the average. The layers are
shown inside the projected perimeter of the complete structure to provide
guidance on the location of the layer with respect to the complete design.

From the printability indices of Table 6, it is clear that as in the 2D
examples, the maximum-size-constrained design contains the least amount of
unfilled zones. Also, since two material beads are placed per solid member
width, no T-joints or path crossovers are obtained. In addition, due to the
uniform size of the solid members, the size-constrained component resembles a
design composed of walls of uniform width. As a number of AM processes are
proficient in the production of walls, such as WAAM, it is expected that the
size-restricted component will be the easiest to manufacture among the three
reported designs.

The small unfilled gaps (1.1$\%$) in the maximum-size-constrained design are
mainly zones where the local maximum-size constraint is violated, as those
pointed out by arrows in Table 6. This drawback caused by the smoothed
aggregation function could be reduced, but not completely eliminated, by using
a finer mesh or larger aggregation exponents (Fernández et al. 2019), i.e. at
the cost of increasing computational expense. Another approach to improve
printability is to post-process the optimized designs using the geometric
constraints proposed by Zhou et al. (2015), which, in our experience, enforce
length scale more strictly.

Although the size-constrained design is the best candidate for manufacturing,
it exhibits the lowest performance, since the maximum size restriction
prevents bulky zones that are required to increase the structural stiffness.
This could be even more detrimental in applications involving stress or
fatigue constraints (Collet, Bruggi, and Duysinx 2017). Therefore, it is
expected that for cases involving mechanical constraints, the AM-constrained
problem will be a better choice for yielding high performance designs suitable
for AM processes equipped with large deposition nozzles.

It was observed in the 2D examples that the eroded and intermediate designs
can be interpreted as the structural skeleton or the deposition paths in the
AM-constrained formulation. To illustrate this in the 3D design case, Table 7
shows the AM-design and 3 representative layers of the component. Each layer
displays the eroded design in red and the AM layer in yellow. The table
contains 3 optimized solutions obtained with the same nozzle size but with
different volume restriction.

Table 7: AM-constrained designs. The layers show the eroded design in red and the AM design in yellow. |  $V^{\text{\tiny{AM}}}=15\%$ |  $V^{\text{\tiny{AM}}}=20\%$ |  $V^{\text{\tiny{AM}}}=30\%$
---|---|---|---
$\bm{\bar{\rho}}^{\text{\tiny{AM}}}$ |  |  |
Layer 2 |  |  |
Layer 20 |  |  |
Layer 40 |  |  |

From the eroded design it is rather intuitive to conceive a deposition path
for the layer shown in Table 7. However, there are some places that could
hinder the path interpretation, as those pointed out by arrows in the Layer 2
depicted in Table 7. These places are in violation of the local volume
restriction that imposes the maximum size and, therefore, it is not possible
to ensure that these zones will be filled during the LSAM process. Due to
these zones in conflict with the local maximmum size restriction, the
optimized design will likely need to be post-processed to conform these
details to the resolution of LSAM process, for instance, using more strict
geometric constraints on the optimized designs (Zhou et al. 2015).

Table 8: Design details of the 3D cantilever beams shown in Fig. 11. |  Maximum-Size-Constrained |  AM-Constrained
---|---|---
Optimized Design |  |
|  c($\bm{\bar{\rho}}^{\mathrm{int}}$)=1.15 |  c($\bm{\bar{\rho}}^{\text{\tiny{AM}}}$)=1.08
$\bm{\bar{\rho}}^{\mathrm{ero}}$ |  |
Section A-A |  |
Section B-B |  |
Section C-C |  |
Section D-D |  |

(a) Size-Constrained

(b) AM-Constrained

Figure 11: Optimized 3D cantilever beam for compliance minimization.

Notably, the 3D examples shown in this section assume a fixed printing
direction. Nevertheless, there are Large-Scale AM processes that are able to
change the stacking direction of the layers, either by the rotation of the
baseplate or of the deposition tool. Interestingly, the methods proposed in
this paper can be easily adapted to such processes. For this purpose, the
local $\Omega_{i}$ region where the maximum size restriction is evaluated
should be defined as a sphere instead of a two-dimensional disk, as presented
in (Fernández et al. 2020). For instance, Fig. 11 shows the size-constrained
and AM-constrained designs obtained by using a spherical $\Omega_{i}$ region.
The volume constraint is set as $V=15\%$ and the length scale is defined
identical to the previous examples. The inner details of these designs are
summarized in Table 8. The first row of the table contains the optimized
designs and the sections planes that display the working layer. The second row
shows the eroded design and the printing direction devised intuitively. The
remaining rows display the sections, where the eroded design is shown in red
and the cross-section in yellow.

The designs in Table 8 are considerably different from their counterparts
obtained with a fixed vertical printing direction (Table 6). The designs
having the printing direction freedom possess two thick bars at the loaded
side (see also Figure 11) that connect the upper part of the design with the
load transmission zone. These bars are also present in the reference design
(Table 6), which would explain the high performance of these solutions (Table
8) with respect to those obtained using a fixed building orientation (Table
6). Interestingly, these two curved bars present in the designs of Table 8
possess different geometry depending on the formulation used. As can be seen
in section D-D, these bars are flat in the size-constrained design, while they
are round in the AM-constrained design. Although both geometries are suited to
the size of the deposition nozzle, the round bar offers a higher area moment
of inertia and hence a higher bending stiffness.

As before with the fixed printing direction, the AM-constrained design shows
better performance than the maximum-size constrained one, presumably because
it allows thicker members where this is beneficial.

##  5 Other test cases and TO problems

The proposed Method 2, based on dilating a design constrained in maximum size,
was introduced in Eq. (14) using the compliance minimization problem subject
to a volume constraint. This formulation was chosen for the sake of
simplicity, but in fact, Method 2 is applicable to any topology optimization
problem where the maximum size constraint (Eq. (9)) has been validated.
Moreover, in our experience, the maximum size constraint, and hence Method 2,
can be applied in most of fields where the robust formulation based on the
eroded, intermediate and dilated designs has been implemented. For the sake of
demonstration, in this section we therefore present results of other topology
optimization problems.

In the following, the proposed Method 2 is considered under compliance
minimization subject to a volume and stress constraints. The topology
optimization problem can be written as:

$\displaystyle\begin{split}{\min_{\rho}}&\quad\alpha\>c(\bm{\bar{\rho}}^{\text{\tiny{AM}}})+(1-\alpha)\>c(\bm{\bar{\rho}}^{\mathrm{ero}})\\\
\mathrm{s.t.:}&\quad\mathbf{v}^{\intercal}\bm{\bar{\rho}}^{\text{\tiny{AM}}}\leq
V^{\text{\tiny{AM}}}\\\
&\quad\mathrm{G_{ms}}(\bm{\bar{\rho}}^{\mathrm{int}})\leq 0\\\
&\quad\mathrm{max}(\bm{\sigma}(\bm{\bar{\rho}}^{\text{\tiny{AM}}}))\leq\sigma^{*}\\\
&\quad 0\leq{\rho_{i}}\leq 1\;,\end{split}$ (22)

where $\bm{\sigma}(\bm{\bar{\rho}}^{\text{\tiny{AM}}})$ represents the
equivalent Von Mises stress computed in the dilated field and $\sigma^{*}$ is
the user-defined stress limit. The max($\cdot$) function is computed using a
p-mean approximation, following exactly the same implementation as Verbart,
Langelaar, and Van Keulen (2017). The equivalent Von Mises stress of the
dilated field is computed as described in (da Silva, Beck, and Sigmund 2019).
For the sake of brevity, we omit the numerous equations defining the stress
constraint and interested readers are referred to the cited works.

The classical L-beam with a volume constraint of $40\%$ is considered. For
this test case, three topology optimization problems are solved, the reference
(Eq. 6), the AM-constrained (Eq. 14), and the AM-stress-constrained one (Eq.
22). In the latter, the imposed stress limit ($\sigma^{*}$) is 0.8 times the
maximum stress computed in the reference design. The three optimized designs
are shown in the first row of Table 9.

Table 9: L-Shaped beam optimized for compliance minimization. 200 finite elements discretize the long edge of the beam. The minimum size of the solid phase is defined by a circle of 4 finite element radius. In these problems, the maximum Heaviside projection exponent ($\beta$) is set to 20, and the SIMP exponent is fixed at 3.0. |  Reference |  AM-Constrained |  AM-Stress-Constr.
---|---|---|---
Optimized Design |  |  |
|  c($\bm{\bar{\rho}}^{\mathrm{int}}$)=1.00, max($\bm{\sigma}$)=1.00 |  c($\bm{\bar{\rho}}^{\text{\tiny{AM}}}$)=1.09, max($\bm{\sigma}$)=1.01 |  c($\bm{\bar{\rho}}^{\text{\tiny{AM}}}$)=1.24, max($\bm{\sigma}$)=0.86
Stress field $\bm{\sigma}$ |  |  |
$\bar{\bm{\rho}}^{\mathrm{ero}}$ and Design field |  |  |
Perimeter path |  |  |

The two AM-constrained results in Table 9 (second and third columns) clearly
reach the intended purpose, which is the production of designs adapted to the
deposition nozzle size. In addition, in such designs, the eroded field can be
interpreted as the deposition path, as shown in the third row of Table 9.
However, the introduction of stress constraints produces a different
arrangement of material beads. The stress-constrained design places the
material beads mostly around the vicinity of the L-beam curvature, which,
along with the smooth curvature of the design, reduces the maximum stresses in
the structure compared to the other two designs. This solution demonstrates
the feasibility of implementing the proposed Method 2 in problems involving
stress constraints.

Although the combination of Method 2 with stress constraints is demonstrated,
there are still aspects that can be improved. For example, it can be seen that
the AM-stress-constrained design does not accurately meet the imposed stress
constraint. This is probably due to the p-mean function, which underestimates
the maximum stress value for the sake of a differentiable approximation,
therefore other relaxation strategies should be examined (da Silva, Beck, and
Sigmund 2019). Also, observations regarding small gaps in the deposition paths
hold as before, which could result in potential stress concentrations. A more
in-depth study of these aspects is identified as a direction for future
research.

To close this section, we include results for the well-known force inverter
benchmark test. The AM-constrained formulation of the force inverter can be
written as follows:

$\displaystyle\begin{split}{\min_{\rho}}&\quad\mathrm{max}(c(\bm{\bar{\rho}}^{\mathrm{ero}}),c(\bm{\bar{\rho}}^{\mathrm{int}}))\>(1-\alpha)+c(\bm{\bar{\rho}}^{\text{\tiny{AM}}})\>\alpha\\\
\mathrm{s.t.:}&\quad\mathbf{v}^{\intercal}\bm{\bar{\rho}}^{\text{\tiny{AM}}}\leq
V^{\text{\tiny{AM}}}\\\
&\quad\mathrm{G_{ms}}(\bm{\bar{\rho}}^{\mathrm{int}})\leq 0\\\ &\quad
0\leq{\rho_{i}}\leq 1\;,\end{split}$ (23)

where $c$ represents the output displacement of the corresponding design
(considered negative due to the adopted reference system). The force inverter
is modeled under linear elasticity, as is done in many other works in the
literature (Sigmund 1997; Bendsoe and Sigmund 2004), so further formulation
and implementation details are omitted for the sake of brevity and the reader
is referred to the cited works.

Table 10: Half force inverter design solutions. Boundary conditions are illustrated in the reference design. There, the red arrow represents the external force and the blue arrow the output displacement to maximize. The orange dashed line is the symmetry line. The design domain is discretized using 200$\times$100 finite elements. The minimum size of the solid phase is defined by a circle of 4 finite element radius. $c$ represents the output displacement of the corresponding design. In these problems, the maximum Heaviside projection exponent ($\beta$) is set to 38, and the SIMP exponent is fixed at 3.0. |  Reference |  Size-Constrained |  AM-Constrained.
---|---|---|---
Design |  |  |
|  c($\bm{\bar{\rho}}^{\mathrm{int}}$)= $-$1.00 |  c($\bm{\bar{\rho}}^{\mathrm{int}}$)= $-$0.94 |  c($\bm{\bar{\rho}}^{\text{\tiny{AM}}}$)= $-$0.97
$\bar{\bm{\rho}}^{\mathrm{ero}}$ and Design |  |  |
perimeter path |  |  |

Here, once again, the reference, the maximum size-constrained, and the AM-
constrained problems are solved. The force inverter design solutions for a
volume contraint of $30\%$ are sumarized in Table 10, along with the eroded
fields and the deposition paths (note that due to symmetry, only half the
domain is modeled). These results exhibit the same pattern observed in
previous examples, namely, the maximum size-constrained design achieves better
printability than the reference design by improving conformability to the
width of the material beads. However, the maximum size constraint could
introduce a significant performance reduction, as it provides low design
flexibility by imposing a uniform size of solid members throughout the design.
On the other hand, the AM constrained design besides providing a good
printability, offers more design flexibility since it allows the creation of
voluminous zones that improve performance. In addition, as a by-product,
Method 2 yields an eroded design that can be interpreted as the deposition
path.

##  6 Conclusions and Perspectives

This work aims to propose topology-optimized designs suitable to Large-Scale
Additive Manufacturing (LSAM) processes. The design limitation considered here
is the small resolution of LSAM processes resulting from depositing large
volumes of material to reduce printing time. The process resolution, which is
typically defined by the size of the deposition nozzle, is introduced in the
topology optimization problem through a geometric control of structural
features. Specifically, two methods based on minimum and maximum size control
have been proposed. The first method imposes the minimum and maximum size of
the solid members in proportion to the size of the deposition nozzle. This
results in designs with structural features of uniform width, which matches a
discrete number of material beads arranged in a parallel fashion. The method
has been evaluated in 2D and 3D design spaces, showing the capability to
produce optimized designs with improved manufacturability. Interestingly, when
a fixed building orientation is chosen for a 3D design problem, the method
generates structures composed of walls, which are suited for fabrication by
most LSAM processes. However, this method can lead to substantial reduction of
performance due to the splitting of structural members as enforced by the
maximum size restriction. The second method instead dilates a maximum-size
constrained design which, due to a specific choice of length scales, resembles
a structural skeleton or the deposition path. The dilation operation of this
skeleton produces a design tailored to the deposition nozzle size, allowing
for an efficient arrangement of the material beads within the design domain.
Numerical results indicate that this second approach also produces designs
that conform to the specified resolution, and, in addition, it comes with a
lesser performance reduction compared to the first method. This is caused by
the fact that the second method allows for thicker members to be created using
parallel beads, in regions where this is beneficial.

In light of the numerous design guidelines that must be taken into account
when designing for LSAM, it is reasonable to think of combining the proposed
topology optimization methods with others existing in the literature. For
example, the 3D designs obtained for a fixed printing direction (Table 7)
could be difficult to manufacture with certain LSAM processes due to the
presence of extreme overhang angles. When the use of additional support
structures is not considered or feasible, the introduction of overhang angle
constraints (Langelaar 2016) in the topology optimization problem could be
considered. Also, depending on the application, it may be necessary to post-
machine a component manufactured by WAAM for instance, to improve surface
finish and mechanical performance. As illustrated in Fig. 12, the post-
machining operation slims the WAAM component, similar to an erosion process of
the robust formulation. Therefore, it is likely feasible to consider post-
processing in the adopted formulation. Certainly, the access of the machining
tool should also be considered, as in the topology optimization methods
presented in (Li et al. 2016; Langelaar 2019; Gaynor and Johnson 2020). These
ideas and the combination of the proposed methods with other Topology
Optimization methods are subjects of future research.

(a) $\bm{\bar{\rho}}^{\mathrm{dil}}$

(b) Machining

(c) $\bm{\bar{\rho}}^{\mathrm{int}}$

Figure 12: Post machining of a WAAM component. (a) The component obtained
after the WAAM process. (b) The component undergoing machining. (c) The final
component. Pictures are courtesy of Fronius International Gmbh
(www.fronius.com).

## Acknowledgements

The corresponding author acknowledges the research project FAFil (Fabrication
Additive laser par dépôt de Fil ), funded by INTERREG V A Grande Région and
the European Regional Development Fund (ERDF). Computational resources have
been provided by the Consortium des Équipements de Calcul Intesif (CÉCI),
funded by the Scientific Research Fund of Belgium (F.R.S.-FNRS)

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