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Struct Multidisc Optim
DOI 10.1007/s00158-016-1551-x
RESEARCH PAPER
Topology optimization considering overhang constraints:
Eliminating sacrificial support material in additive
manufacturing through design
Andrew T. Gaynor1·James K. Guest2
Received: 13 April 2016 / Revised: 9 June 2016 / Accepted: 14 June 2016
© Springer-Verlag Berlin Heidelberg (outside the USA) 2016
Abstract Additively manufactured components often
require temporary support material to prevent the com-
ponent from collapsing or warping during fabrication.
Whether these support materials are removed chemically
as in the case of many polymer additive manufacturing
processes, or mechanically as in the case of (for example)
Direct Metal Laser Sintering, the use of sacrificial material
increases total material usage, build time, and time required
in post-fabrication treatments. The goal of this work is to
embed a minimum allowable self-supporting angle within
the topology optimization framework such that designed
components and structures may be manufactured without
the use of support material. This is achieved through a series
of projection operations that combine a local projection to
enforce minimum length scale requirements and a support
region projection to ensure a feature is adequately supported
from below. The magnitude of the self-supporting angle is
Preliminary results of this study were presented at the 15th
AIAA/ISSMO MAO Conference at Aviation 2014, June 16-20,
2014, Atlanta, Georgia, USA; and at WCSMO-11, June 7-11,
2015, Sydney, Australia.
Andrew T. Gaynor
andrew.t.gaynor2.ctr@mail.mil
James K. Guest
jkguest@jhu.edu
1Materials Manufacturing Technology Branch, Weapons
and Materials Research Directorate, U.S. Army Research
Laboratory, RDRL-WMM-D, Building 4600, APG,
Aberdeen, MD 21005, USA
2Department of Civil Engineering, The Johns Hopkins
University, 3400 N. Charles Street, Baltimore,
MD 21218, USA
process dependent and is thus an input variable provided
by the manufacturing or design engineer. The algorithm is
demonstrated on standard minimum compliance topology
optimization problems and solutions are shown to satisfy
minimum length scale, overhang angle, and volume con-
straints, and are shown to be dependent on the allowable
magnitudes of these constraints.
Keywords Additive manufacturing ·3D printing ·
Projection methods ·Anchors ·Design for additive
manufacturing ·Self-supporting ·Overhang features
1 Introduction
Additive manufacturing (AM) is a free-form manufacturing
technique in which a component is built in a layer-by-
layer manner. It has a demonstrated capability to produce
components that are far more complex than those that can
be created using more traditional manufacturing techniques
such as milling or casting. The free-form nature of topology
optimization, and its ability to discover novel, high perfor-
mance solutions, makes it a natural design tool for integra-
tion with AM processes. Yet while AM significantly opens
up the design space for engineers, manufacturing constraints
and limitations remain (Gao et al. 2015) and ultimately
must be tightly integrated within the topology optimization
methodology to fully leverage the capabilities and freedom
provided by AM processes. This paper focuses on develop-
ing a topology optimization algorithm capable of handling
one of the more challenging AM-specific constraints known
as overhang constraints.
Both polymer-based processes, such as Fused Deposi-
tion Modeling (FDM), and powdered metal based processes,
such as Direct Metal Laser Sintering (DMLS), require
A. T. Gaynor, J. K. Guest
support material in order to manufacture certain topologies.
In FDM, also known by the more generic term Fused Fila-
ment Fabrication (FFF), polymer filament is pushed through
a heated print head to deposit molten material on the solid-
ified layer below. This ‘structural’ material is typically sur-
rounded by a soluble support material that is printed around
the part boundaries to prevent the structural material from
distorting during the build process. Such distortions include
curling from residual stress buildup (from rapid cooling dur-
ing the solidification process) and sagging from expansive
unsupported regions, both of which can potentially result
in catastrophic collapse of the part during fabrication. Fol-
lowing fabrication, the support material is removed in a
post-print liquid bath. Although the removal process is rel-
atively straightforward, using support material increases the
total material consumed, increases print time, and requires
a chemical bath that must be refreshed based on usage.
Support material in metal AM processes, particularly
laser powder bed fusion processes such as DMLS, is a sig-
nificantly more complicated issue as described in Hussein
et al. (2013). In DMLS, a laser either selectively melts or
sinters a very thin layer of powder, typically on the order of
40 microns, in a build pattern defined by the part geometry
(Fig. 1) and the machine specific scan strategy. Interestingly,
the average particle size is typically only slightly smaller
than the layer height, so the melting often occurs on lay-
ers a single particle deep. Once a powder layer is fused,
the build platform moves down, the powder bed is recoated
with metal powder, and the process is repeated. The signif-
icant thermal gradients generated by this selective melting
and solidification process can lead to significant distortions
(curling, warping), and even cracking of the part, partic-
ularly in regions of the component having low stiffness
such as cantilever features (Vandenbroucke and Kruth 2007;
Mercelis and Kruth 2006; Gorny et al. 2011). As with poly-
mers, these effects are ultimately all due to residual stress
accumulation. Temporary support materials, referred to as
anchors, provide structural resistance against this behavior
by connecting the build platform to the part at various loca-
tions. Additionally, the un-sintered powder has relatively
low conductive capabilities, and thus these anchors provide
a high conduction path from the point of melting/sintering
to the typically thick build plate (usually one inch (2.54
centimeters) or greater), allowing heat to escape from the
system. As in FDM processes, the need to fabricate these
sacrificial support anchors increases material usage, build
time, and post-fabrication processing time. Unlike most
FDM processes, however, metal anchors must be removed
mechanically by machining, chipping or grinding them off
of the finished part, significantly increasing post-processing
time and equipment requirements and potentially degrading
surface finish.
As discussed in Thomas (2009), the need for metallic
anchors, and support materials in general, can be avoided
by preventing what are referred to as overhang features.
Put simply, these are features that rise in the build direc-
tion at a ‘shallow’ angle without supporting material below
them. For example, a simple unsupported cantilever fea-
ture would have a zero degree angle and thus be flexible
in bending and subject to a larger thermal gradient, as
the powder below the cantilever would be at a signifi-
cantly lower temperature. Such a feature would require an
anchor to prevent warping. This is in contrast to a feature
resembling a vertical column, which would have a direct
conductive path to the build plate and resist any thermal gra-
dients axially, without need for an anchor. Similar effects
are seen in polymers and biomaterials. In bioprinting, for
example, Xu et al. (2012) illustrated that overhang features
Fig. 1 Direct Metal Laser Sintering schematic. Image via Wikimedia Commons, courtesy of the author, Materialgeeza
Topology optimization considering overhang constraints
tended to fail in manufacturing due to high bending stresses
and droplet impact-induced crash. The actual angle above
which support material is not needed, referred to as the
self-supporting angle, is process-dependent. Thomas identi-
fied 45 degrees as a typical minimum self-supporting angle
in DMLS (Thomas 2009). Quantification of this angle is
outside of the scope of this paper, and thus the presented
algorithm assumes this magnitude is given as input by the
designer.
To date, the majority of design-related work has focused
on post-processing results to eliminate overhang features
(Mumtaz et al. 2011; Cloots et al. 2013). For example, Leary
et al. (2014) and Hu et al. (2015) proposed modifying a
given topology by changing the angles and shapes of fea-
tures to meet the overhang angle constraint, respectively.
This, of course, is undesirable from a design optimization
point-of-view as optimality and therefore part performance
will be eroded, potentially dramatically, through these re-
design operations. Brackett et al. (2011) proposed searching
for and penalizing angles that violated the maximum over-
hang constraint during the design evolution following every
design iteration. The penalization scheme, however, was not
included in the sensitivity analysis and thus design deci-
sions made by the optimizer did not directly account for
the overhang constraint, making the algorithm a heuris-
tic approach. Hussein et al. (2013) accepted the existence
of support material, but sought to minimize their material
volume through optimization.
Herein, we attempt to more thoroughly integrate over-
hang constraints into the topology optimization method-
ology. Following the original overhang projection-based
methodology presented by Gaynor and Guest (2014), we
embed a projection step associated with the overhang angle
constraint within the standard Heaviside Projection method-
ology (HPM) (Guest et al. 2004). In short, the former
ensures that structural features only form if they do not
violate the overhang rule, while the latter ensures features
obtain a minimum length scale as defined in the original
work on solid features (Guest et al. 2004) and/or void fea-
tures (Sigmund 2007;Guest2009b). Extensions to more
recent projection-based implementations, including those
using an alternate definition of local length scale (Wang
et al. 2011), considering manufacturing flaws (Jansen et al.
2013), using multiple materials (Gaynor et al. 2014), or
using features that are discrete objects (Guest (2015,2014);
Ha and Guest 2014), can also potentially fit within the
methodology, as can constraint-based restrictions on max-
imum (Guest 2009a) and minimum length scale (Poulsen
2003; Zhou et al. 2015). The combined algorithm there-
fore allows the designer to prescribe the minimum length
scale of features as well as the minimum allowable self-
supporting angle to avoid overhang violations. As with other
projection-based methodologies, overhang restrictions are
achieved without adding explicit constraints to the optimiza-
tion problem formulation.
2 Overhang projection
The need for sacrificial support material during the additive
manufacturing build process can be eliminated by design-
ing structures that are entirely self-supporting. That is, by
designing structures where all features rise at an angle that
is at least as large as the minimum allowable self-supporting
angle. Several researchers have suggested through experi-
mental testing that the minimum self-supporting angle for
DMLS printed parts is 45 degrees (Thomas 2009). However,
realizing that this angle is governed by the details of the pro-
cess, the proposed approach is general and may be extended
to any angle between 0 and 90 degrees. Figure 2illustrates
several different allowable overhang angles that will be
specifically illustrated in this paper, including 26.6 degrees
Fig. 2 Examples of different minimum allowable self-supporting
angles for satisfying overhang constraints. The blue region is imagined
to be built already while the green region indicates the minimum angle
at which features may be created without requiring support material.
The build direction is assumed upwards and the overhang feature may
be leaning to the right (as shown)ortotheleft
A. T. Gaynor, J. K. Guest
(arctan(1/2)), 45 degrees, and 63.4 degrees (arctan(2)). The
blue regions in these figures indicate a feature that is already
printed, while the green regions indicate the minimum
allowable self-supporting angle above which sacrificial sup-
port material would not be required.
Although one could imagine a series of local geometric
constraints to check for overhang violations, the approach
used herein is to embed the overhang restriction within the
projection methodology so that solutions naturally achieve
the overhang restriction. We follow the typical material dis-
tribution method where the design domain is discretized
with finite elements and the goal is to determine relative
density, or volume fraction, within each elemental domain.
Element relative density is denoted as ρewith a void being
ρe=0 and solid material being ρe=1. To allow the use
of gradient-based optimizers, the binary 0/1 relative den-
sity is relaxed and intermediate magnitudes between 0 and 1
are penalized to drive solutions back to binary distributions.
We have implemented the popular Solid Isotropic Material
with Penalization (SIMP) method (Bendsøe 1989; Zhou and
Rozvany 1991) as well as the Rational Material with Penal-
ization (RAMP) method (Stolpe and Svanberg 2001)for
interpolating the stiffness (Young’s modulus) of elements.
Both produced quality solutions, though we developed a
slight preference for RAMP due to its non-zero gradient at
ρe=0. This property was particularly useful as material
often had to ‘grow’ out of void space below a structural
feature to make said feature self-supporting. Similar bene-
fits were seen in Guest (2015) where it was necessary for
stiff inclusions (ρe=1) to ‘grow’ out of compliant matrix
material (ρe=0).
We now briefly explain the overhang projection logic
before discussing the numerical implementation and sen-
sitivity analysis. In the following sections, we assume the
build direction is in the vertical upwards direction, and thus
refer to the support material region being ‘below’ a given
point. Of course the build direction can occur in any other
direction and the center of the support region for a point
would be defined as 180 degrees from this build direction.
2.1 Projection concept
The idea behind the overhang projection is quite simple:
An element emay become a solid element if and only
if (i) the ‘local’ design variables indicate material should
be deposited into the element, and (ii) material exists in
the ‘supporting’ elements below esuch that eis not part
of a overhang feature. These two conditions correspond to
the standard minimum length scale Heaviside projection
(Guest et al. 2004) that imposes a minimum length scale on
designed features and the overhang condition first proposed
in Gaynor and Guest (2014) that ensures these features are
supported, respectively.
Ultimately these two conditions can be quantified as
0/1 (no/yes) and multiplied with a resultant of 1 indicating
that material may exist. We now introduce three variables
that will be used to create this algorithmic effect: (i)φis
a dependent variable, as will be explained, that is passed
through the Heaviside operator to create a circular (2D) or
spherical (3D) solid feature in the finite element space ρe,as
in standard Heaviside projection methods, (ii)ψis the new
independent design variable that indicates whether mate-
rial should be deposited at a given location; and (iii)ρSa
subset of variable φthat indicates whether material can be
deposited at the considered location (i.e., indicates whether
or not the overhang condition is violated). From these defi-
nitions, it can be understood that ψand ρSwill be combined
to form φ, which is then used to determine element volume
fraction ρe.
The independent design variables ψand thus dependent
variables φ(and thus, ρS) are located at the nodes of the
finite element mesh herein but in general can be located at
any points in space (e.g., Guest and Smith Genut (2010)).
2.2 Neighborhood sets
It is very useful to define the neighborhood sets that identify
lists of elements and design variables that are mathemati-
cally related. Herein we utilize two mappings: one neigh-
borhood set for the standard local projection operations, Ne
L,
and one set related to the overhang support conditions, Ni
S.
The local neighborhood set for an element eis composed
of all nodal variables within a distance rmin of the element
centroid. As in the original projection work (Guest et al.
2004), rmin corresponds to the minimum allowable radial
Fig. 3 The local neighborhood set for the starred element. This ele-
ment may have material projected onto it by any nodal design variable
φwithin the radius rmin
Topology optimization considering overhang constraints
length scale of a designed feature. The local neighborhood
set is shown in Fig. 3and is defined as
i∈Ne
Lif xi−¯
xe≤rmi n (1)
where xiis the location of design variable i,and ¯
xeis
the location of the centroid of the element e. We note this
neighborhood set takes the same form as when using linear
density filters (Bruns and Tortorelli 2001; Bourdin 2001)
and linear sensitivity filters (Sigmund 1997).
The overhang support neighborhood set relates a given
design variable ψito the support region below it, defined as
the region that must contain some material for the point to
be considered supported and not in violation of the overhang
constraint. We limit this set to those points within a distance
rsbelow the design variable i, within the region bounded by
the defined minimum self-supporting angle, thereby creat-
ing a wedge-like region in two dimensions as illustrated in
Fig. 4. Herein rsis set to 1.5rmin , however it was found
that using rsequal to rmin also resulted in quality solu-
tions. In three dimensions this region would be a cone-like
shape. The overhang support neighborhood set Ni
Scontains
all points (φ) within the support wedge appearing below a
givendesignvariablei.
For later convenience, it is also useful to consider the case
of rsbeing infinity, which essentially extends the wedge
to the build plate, creating a triangle shape in 2D (when
ignoring design domain boundaries) whose edges rise at
the minimum allowable self-supporting angle. All points φ
located within this triangle will be defined as the set Ni
B.
2.3 Projection functions
To develop the projection functions for the overhang con-
straint, we begin with the standard projection function for
imposing minimum (radial) length scale on designed fea-
tures. The regularized Heaviside function (Guest et al. 2004)
relates variable φto element relative densities ρethrough
the following function:
ρe=1−e−βμe(φ)+μe(φ)
φmax
e−βφmax (2)
where βis the regularization parameter dictating the aggres-
siveness of the Heaviside approximation (Guest et al. 2011),
φmax is the maximum magnitude of φ(φmax =1 herein),
and μeis the averaged or filtered design variables in the
neighborhood set Ne
L:
μe=i∈Ne
Lφiw(xi−¯
xe)
j∈Ne
Lw(xj−¯
xe)(3)
where wis the weighting function, which for distance-based
weighting is
w(xi−¯
xe)=1−(xi−¯
xe)
rmin
(4)
As in the case of imposing minimum length scale, if
avariableφiachieves a magnitude greater than zero, all
elements whose centroid lies within a distance rmin of φi
will have a magnitude of μegreater than zero, and conse-
quently a ρethat will approach one, indicating the element
is to contain material. This exactly follows the typical HPM
logic with the key difference now being that φis no longer
an independent design variable, but is now a dependent
variable.
As described in Section 2.1, material is deposited into
an element if and only if the independent, local design
variables ψindicate material should be deposited and the
support dependent variables ρSindicate it can be deposited
(i.e., is supported below so as not to violate the overhang
condition). As a dependent variable of φ,ρSserves as a
pseudo density, which ultimately becomes a true density
after a Heaviside projection. This concept is consistent with
past works that use multiple projections (Guest (2009b,
2015)). The strict rule is enforced through a multiplication
scheme as follows:
φi=ψiρi
S(5)
This expression ensures that if the independent variable
ψior support material variable ρi
Shave a magnitude of
zero then no material will be projected from location i,
Fig. 4 Support neighborhood sets Ni
sfor various minimum allowable self-supporting angle
A. T. Gaynor, J. K. Guest
whereas if both achieve nonzero magnitudes then some level
of material will be deposited.
The support variable ρi
Sis computed via a projection
from φvariables located within the support neighborhood
set Ni
S(Fig. 4). Herein we use the thresholding projection of
Jansen et al. (2013), a modified version of Xu et al. (2010),
to perform this projection:
ρi
S=HT(φ)=tanh(βTT)+tanh(βT(μi
S(φ)−T))
tanh(βTT)+tanh(βT(1−T)) (6)
where βTis the thresholding Heaviside parameter, Tis the
threshold value, and μi
Sis simply the average of φvariables
in the support neighborhood set of variable i:
μi
S=j∈Ni
SφjwS
m∈Ni
SwS
(7)
where wSis the support region weighting function, cho-
sen here a uniform weighting (wS=1). The threshold
Tindicates the magnitude of the filtered variable μi
Sat
which the point iis considered moderately supported. The
thresholding (6) is plotted in Fig. 5for T=0.1.
As for the magnitude of the threshold parameter T,it
should be chosen such that there is adequate support mate-
rial but not so large that the majority of the wedge must be
filled with material, as this would be more restrictive than
the actual prescribed self-supporting angle. We therefore set
Therein such that a point is considered supported if one half
of one of the sides of the support wedge have φ=1. This
can be approximated through the following equation:
T=180
2π(90 −θ)
h
rS
(8)
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
supported
unsupported
Fig. 5 Thresholding Heaviside function for threshold T=0.1
where θis the minimum allowable self-supporting angle in
degrees and his the size of an element in the finite element
mesh.
It should be noted that one could alternatively use the
standard projection function (2) instead of the threshold
function (possibly with an additional exponent parameter to
penalize under-supported cases), or use ρein Ni
Sin place of
φin the overhang projection scheme. The latter in particu-
lar would make sense as it is a direct indicator as to whether
material is contained in the elements below the considered
design point. However, the overhang constraint is a severe
design space restriction and our numerical testing suggested
the algorithm regularly used intermediate relative densities
to satisfy the overhang condition when using these alterna-
tives. The critical issue with using ρeto indicate support is
that regions along structural boundaries composed of inter-
mediate relative densities would be projected to a magnitude
of one and thus count as a supporting feature. The algorithm
was found to take advantage of this property and utilize
significant volumes of intermediate densities, essentially
creating a computational version of sacrificial support mate-
rial. Although it is possible this issue could be overcome
with parameter tuning, our studies have indicated the thresh-
olding function, which generally requires multiple φin Ni
S
to achieve a non-zero magnitude, more consistently created
support material that was very close to unit magnitude and
thus providing a crisper representation of topology.
From the preceding equations, it should be clear that
whether material can be projected from a point φionto the
elemental domain is entirely dependent on the magnitudes
of other φvariables below this point, and thus the algo-
rithm must proceed in a layer-by-layer manner, essentially
mimicking actual AM processes. We also note that the first
layer is supported by the build plate, or can be assumed sup-
ported by a support structure below the part (as is commonly
done to facilitate part removal from the build plate, usu-
ally through wire electrical discharge machining (EDM)).
Thus the overhang support projection is not needed for
these elements and ρi
Scan be fixed at one, making material
placement entirely dictated by ψ.
In summary, elemental relative densities ρeare computed
by (2) and are a function of dependent variable φ;φare
computed by (5) and are a function of the independent opti-
mization variables ψand the dependent support indicator
variable ρS;andρSare computed by (6) and are a function
of φin the support wedge below the considered point, i.
This last dependency means the algorithm must proceed in
a layer-by-layer manner.
This approach has been constructed so that solutions may
achieve binary distributions and so that structural features
(i) attain a minimum radial length scale as defined in (Guest
et al. 2004)and(ii) achieve a minimum self-supporting
Topology optimization considering overhang constraints
angle constraint to eliminate the need for support material.
The magnitudes of both the minimum radial length scale
and minimum self-supporting angle constraints may be dic-
tated by design specifications and/or processing capabilities
and are thus prescribed by the designer.
The disadvantages of the proposed approach are that it
progresses in a layer-by-layer manner and that we have
embedded projections, amplifying, in a sense, the nonlin-
earity of the governing functions. The former is undesirable
from a computational point of view as it inhibits efficient
parallel processing, although it should be recognized the
actual AM process operates in this exact layer-by-layer
manner. With regard to the ‘amplified’ nonlinearity, it is
possible the algorithm will be more likely to converge to
low quality local minima. Subsequent sections propose a
continuation scheme to help alleviate this issue.
2.4 Problem formulation
The proposed algorithm will be demonstrated in the con-
text of minimum compliance (maximum stiffness) topology
optimization. The goal is to optimize the distribution of
material, ρe, across the design domain so as to minimize
external work funder a volume of material constraint. The
resulting optimization formulation takes on the following
well-known form:
min
ψf(ψ)=FTd
subject to:
K(ψ)d=F
e∈
ρe(ψ)ve≤V
0≤ψi≤ψi
max =1∀i∈(9)
where Fis the vector of applied nodal loads, dis the vec-
tor of nodal displacements, Kis the global stiffness matrix,
ρeis elemental volume fraction of element e,veis the
volume of element e,Vis the total allowable material vol-
ume, and is the design domain. For later use, let us also
define the variable vf as the total allowable volume frac-
tion, or the ratio of Vto the total volume of the design
domain. We emphasize here that there is not an explicit con-
straint on overhang features, but rather overhang constraints
are achieved through the proposed projection methodology.
Although other works have proposed penalty functions for
overhang constraints (Brackett et al. 2011), we note that
these to date have been heuristic as the sensitivity analysis
does not account for this penalty function.
Using RAMP, the element stiffness Kethat is assembled
into Kis given as (Stolpe and Svanberg 2001):
Ke=ρmin +ρe(ψ)
1+η(1−ρe(ψ)) Ke
0(10)
where ηis the penalty parameter, ρmin is a small positive
number to maintain positive definiteness of the global stiff-
ness matrix, and Ke
0is the stiffness matrix of a solid element
multiplied by 1
(1+ρmin ).
For completeness, we note that the element stiffness
matrix is given as the following when using SIMP (Bendsøe
1989; Zhou and Rozvany 1991):
Ke(ψ)=(ρmin +ρe(ψ)η)Ke
0(11)
where ηis again a penalty parameter.
2.5 Sensitivity analysis
The derivatives of the objective and constraint functions in
(9) with respect to the independent design variables ψare
required to guide the gradient-based optimization and can
be computed from the chain rule. In vector form, this can be
written as:
∂f
∂ψ=∂f
∂ρe
∂ρe
∂φ
∂φ
∂ψ(12)
The first term is found via the adjoint method and is
well known for minimum compliance with RAMP to be the
following:
∂f
∂ρe=− η+1
(η(ρe(ψ)−1)−1)2deTKe
0de(13)
where deis the vector of displacements associated with
element e.
The second term in (12) is simply:
∂ρe
∂φi=βe−βμe(φ)+1
φmax
e−βφmax ∂μe(φ)
∂φi(14)
for all iin Ne
L(and is zero otherwise), where the deriva-
tive of μewhen using a distance-based weighting function
is given as:
∂μe
∂φi=w(xi−¯
xe)
j∈Ne
Lw(xj−¯
xe)(15)
The last term in (12), ∂φ
∂ψ , is at the core of the overhang
projection scheme and is complicated by the fact that a sin-
gle φjvariable is a function of all ψappearing below it
within the the region bounded by the overhang angles; i.e.,
A. T. Gaynor, J. K. Guest
Fig. 6 MBB beam definition
(L=120 units, H=20 units)
L
L
2
H= L
6
F
all ψin set Nj
B. Examining (5)and(6), it can be shown that
this derivative can be written as:
∂φj
∂ψi=⎧
⎪
⎨
⎪
⎩
ρi
sif i=j;
ψj∂ρj
s
∂ψiif i∈Nj
Band i= j;
0 otherwise.
(16)
where ∂ρj
s
∂ψiis given as
∂ρj
s
∂ψi=
k∈Nj
s
βTsech βT(μj
s(φ(ψ)) −T)
2
tanh(βTT)+tanh(βT(1−T))
∂μj
s
∂φk
∂φk
∂ψi
(17)
The final term reveals the embedded nature of the derivative
and, when combined with (16), reveals that each derivative
of φwith respect to ψimust be propagated in the build
direction from i. Although this seems somewhat compli-
cated, implementation is relatively simple and solved by
starting from iand computing ∂φj
∂ψione layer at a time. These
individual terms can be computed on the fly or stored if
sufficient memory is available.
2.5.1 Implementation
The optimization problem (9) is solved with the gradient-
based Method of Moving Asymptotes (MMA) optimizer
(Svanberg 1987). Anticipating that the embedded Heavi-
side function and the φi=ψiρi
Srelations may cause
the algorithm to become more susceptible to low qual-
ity local minima, we have taken a conservative approach
in the implementation of the optimization algorithm. This
includes the common practice of a continuation method on
the RAMP exponent η, as well as a continuation method on
the Heaviside parameter β. Unless otherwise mentioned, the
penalization parameter ηis initially set to 10 and increased
by 2 every continuation step until reaching a maximum
penalization of 18, while the Heaviside parameter βis ini-
tially set to 5 and increased by 5 each continuation step until
a maximum magnitude of 25. The threshold parameter βT
is held constant at 25. We note the initial RAMP penalty
magnitude is larger than typically used but the algorithm
exhibited worse and/or slower design progression when
using no penalization in the initial steps, likely due to the
fact that continuous, unpenalized solutions do not resem-
ble converged binary solutions due to the very strict nature
of the overhang restriction (similar observations were made
in Guest (2015)). Continuation steps occur at convergence,
defined as a change in the objective function of less than
10−3, but at no more than 500 iterations. We also used tight-
ened asymptote parameters of 0.05/(β +1)as the initial
setting of s0, 1.15 as the asymptote increase parameter, and
0.6 as the tightening parameter (see Reference Guest et al.
(2011) for related discussion although we note the parame-
ter magnitudes used here are tighter than used in that work).
The continuation scheme and relatively tight asymptotes
resulted in predominantly smooth convergence and qual-
ity solutions for a wide range of tested problems. It should
be emphasized, however, that the author’s focus here is on
producing quality solutions, and not on tuning these algo-
rithmic parameters, and thus the above should be viewed as
a conservative implementation. With this in mind, a more
aggressive case is also presented.
3 Examples
The proposed algorithm is tested on two well-known mini-
mum compliance topology optimization problems (Rozvany
1998): the Messerschmidt-B¨
olkow-Blohm (MBB) beam
showninFig.6and the cantilever beam shown in Fig. 7.
All problems are meshed using four node quadrilateral ele-
ments with 1:1 aspect ratio. The MBB problem employs
L
F
H
2
Fig. 7 Cantilever beam definition (L=40 units, H=25 units)
Topology optimization considering overhang constraints
Fig. 8 MBB beam minimum compliance solution without overhang constraint, f=97.65, fη=0=91.74. The small circle at the left of the beam
indicates the minimum feature size of radius rmin
symmetry and is meshed with 240 by 80 elements over
half of the domain. The cantilever beam problem is meshed
with 160 by 100 elements. The minimum radial length scale
rmin is set to 0.8 units for all examples and the minimum
self-supporting angle is varied. The allowable volume frac-
tion vf is set to 50 % for all examples unless otherwise
noted. All examples use a uniform initial distribution of ψ
such that the volume constraint is exactly satisfied. This ini-
tially uniform distribution of the design variable creates a
graded initial distribution of material, ρealong the build
direction, such that the largest ρeappear at the build plate
and smallest appear at the surface farthest from the build
plate in the build direction. This graded distribution essen-
tially results from the fact that material must “grow” from
the build plate for any feature at the opposite side of the
domain to exist. The authors have also investigated non-
uniform ψand found the difference in design progression
to be negligible since the material must still ‘grow’ from
bottom up.
3.1 MBB example
Figure 8shows the topology-optimized MBB beam solution
considering the minimum length scale constraint, with-
out implementing the overhang constraint. This solution
features a near binary (solid-void) distribution of ρeand
exhibits many properties resembling the analytical solutions
for the truss MBB problem, including tension and compres-
sion paths that are near orthogonal, or so-called Michell-like
structures (Rozvany 1996,1998; Rozvany and Gollub 1990;
Rozvany and Birker 1995) (noting that the truss analytical
solutions do not require a length scale constraint, and thus
are not directly comparable). For reference, the solution in
Fig. 8exhibits a penalized compliance of f=97.65.
We now consider an overhang constraint, and begin
by assuming the structure is built from the bottom up
(Fig. 9) and that the minimum self-supporting angle (over-
hang angle) is 45 degrees, the general rule-of-thumb for
DMLS processes. Figure 10a shows the solution from Fig. 8
with the bottom surfaces of features that would be subject
to an overhang constraint highlighted. The solid white and
dashed white lines in this image indicate regions that would
satisfy and violate a self-supporting angle of 45 degrees,
respectively. That is, the solid white lines rise at angles
greater than 45 degrees and are thus self-supporting, while
the dashed white lines rise at angles less than 45 degrees
and would thus need support structures. Of course this
diagram could be redrawn for different allowable magni-
tudes of the self-supporting angle but would likely always
contain regions that violate an overhang constraint since
features near the top of the beam are nearly horizontal
(nearly zero degrees).
Figure 11b shows the optimized topology using the
proposed overhang projection scheme with a minimum
self-supporting angle of 45 degrees and the same mini-
mum length scale constraint. The overhang restriction has
led to substantial topological changes in the beam and an
increase in compliance to f=116.11. The Michell-like
features connecting the top and bottom of the beam have
been replaced with vertical features rising at an angle of
45 degrees or more so as to satisfy the overhang con-
straint. As these features approach the top “chord” of the
beam they split and branch out in opposing directions at
45 degrees. Looking more closely at the top chord of
this beam, we see the inner bottom surface is now regu-
larly supported from below with the branches from these
structural features, thereby eliminating the (near) horizon-
tal features highlighted by dashed white lines in Fig. 10a.
Fig. 9 Upward build direction
for the MBB beam problem
L
L
2
H= L
6
F
Build Plate
Build Direction
A. T. Gaynor, J. K. Guest
Fig. 10 MBB beam solutions
(a) without an overhang
constraint and (b) with overhang
constraint assuming a minimum
self-supporting angle of 45
degrees. The dashed white and
solid white lines in (a) indicate
features that would and would
not violate a 45 degree overhang
constraint, respectively. The
buildplateisassumedatthe
bottom of the domain and the
build direction is upwards as in
Fig. 9
In a sense, the algorithm is designing features that both
carry the prescribed load and also serve as (permanent)
support structures for the top chord. The beam depth also
decreases steadily moving away from midspan. Although
both of these changes lead to reduced structural efficiency
and an increased compliance of 19 %, they are required to
satisfy the overhang constraint and volume constraint.
Solutions found using different magnitudes of the mini-
mum self-supporting angle are shown in Fig. 11. The least
restrictive of these constraints is 26.6 degrees and the corre-
sponding solution (Fig. 11a) bears the closest resemblance
to the solution without an overhang constraint giving the
impression of ‘curved’ members with shallow angles in cer-
tain regions. This is in contrast to the solution found under
the most restrictive of these cases (63.4 degrees, Fig. 11c)
which features a large number of tightly packed, nearly ver-
tical posts connecting the top and bottom chords, each of
which split into two branches as the top chord is approached.
This dense network of posts is required to continuously
support the top chord, which prefers to maintain a shallow
overall angle and thus needs constant support from internal
structural features. We note the outer edge of the top chord
is nearly a line connecting the location of the point load
and the locations of support. This is in contrast to the other
cases where the trace of the outer edge is concave, creating
a (more efficient) larger beam depth near midspan.
As expected, the elimination of sacrificial support mate-
rial (and thus improved manufacturability) comes at the
cost of reduced beam efficiency and thus increased struc-
tural compliance. Figure 12 displays the compliances for
each of the presented solutions, including the free-form
solution without an overhang constraint. The penalized and
unpenalized compliance magnitudes refer to the compliance
computed for the final topology using RAMP parameter
Fig. 11 MBB beam solutions
considering overhang
constraints with various
magnitudes of the minimum
allowable self-supporting angle
Topology optimization considering overhang constraints
Fig. 12 Comparison of compliance values of various self-supporting
angles for the MBB beam problem
magnitudes of η=18 and η=0, respectively. As the plot
clearly shows, compliance increases with increasing restric-
tion of the design space, ranging from a 12 % increase for
a self-supporting angle of 26.6 degrees to a 79 % increase
for a self-supporting angle of 63.4 degrees. The designer
therefore has a tool to explore various options and trade-offs
between manufacturability and performance, with the ‘best’
solution likely being application dependent (Fig. 12).
We also note that the solution using 63.4 degree self-
supporting angle has the most intermediate densities of the
three cases. We believe this is due to the fact that the
top chord needs nearly continuous support via nearly ver-
tical supporting features, each of which must satisfy the
minimum length scale constraint. The allowable volume
constraint seems to prevent these supporting features from
becoming fully dense, leading to a very minor blurring
effect around the edges of these features. Further parameter
tuning will likely fix this minor issue.
To explore the interplay between the volume and
overhang (and minimum length scale) constraints, the
MBB problem is solved with a minimum allowable self-
supporting angle of 45 degrees and allowable volume frac-
tions vf of 25 %, 40 %, 50 %, and 60 %. The optimized
Fig. 13 MBB beam solutions
considering a 45 degree
overhang constraint with various
magnitudes of allowable
material volume fraction vf
A. T. Gaynor, J. K. Guest
Fig. 14 Comparison of compliance values of various volume fractions
for the MBB beam problem
solutions are shown in Fig. 13. Interestingly, the vf =
40 % solution (Fig. 13c) has strong similarities with the
solution in Fig. 11c found using a minimum allowable self-
supporting angle of 63.4 degrees and vf =50 %. These
similarities include a shallower beam with the outer edge
of the top chord nearly being a line connecting the location
of the point load and the locations of support. This line is
shallow and thus requires a large number of interior posts
connecting the top and bottom chords rising at an angle
greater than 45 degrees. It should be noted that several of
these posts bend along their length, particularly the posts
near midspan. This is unexpected, as such curvature would
lead to the development of internal bending moments in
these posts, and thus we believe this solution is a local min-
ima. Nevertheless, the overhang constraint is satisfied and
the resemblance to Fig. 11c is interesting. As vf increases,
the top and bottom chords become thicker, overall beam
depth increases away from midspan, leading to a less shal-
low top chord and thus less internal support structures, each
of which starts to approach the allowable minimum over-
hang angle. Most interesting is the solution for vf =25 %.
As can be seen, the algorithm is clever in creating a shal-
low beam at the bottom of the design domain (location of
build plate) and creates two vertical compressive struts to
carry the load from the point of application to the beam
(Fig. 13d).
As shown in Fig. 14, the objective function values for
the various volume fractions, vf , exhibit the expected trend
of increased allowable volume fraction leading to reduced
compliance values, the same trend seen in any free-form
topology optimization. As can be seen in Fig. 13, the 60 %
vf solutions achieves a compliance of f=93.84, which
is better than the freeform topology optimization solution at
50 % vf . Alternatively, the 40 % vf solution exhibits a com-
pliance of f=222.52, which, despite the noted similarities
to the 63.4 self supporting solution (Fig. 11c), exhibits a
much larger compliance. As expected, the 25 % vf solution
exhibited significantly worse performance (f=1780.39),
but did manage to adhere to the overhang constraints by
creating a non-intuitive shallow beam with nearly vertical
load-transfer struts.
As a final test for the MBB beam example we consider
an alternative build orientation of building the beam upside-
down. This can be visualized by placing the build plate at
the top of the domain and defining the build direction as
downwards, as shown in Fig. 15. The optimized solution for
a minimum self-supporting angle of 45 degrees is shown in
Fig. 16. The overhang constraints for this case are clearly
satisfied and, as one would expect for the 45 degree over-
hang case, the solution resembles an upside down version of
the topology built in the upwards direction (e.g., Fig. 13b).
The exception to this similarity is at the ends of the beams,
where the downward build direction has resulting in two
vertical posts connecting the ends of the beam to the sup-
ports. These beam end vertical post features were also seen
in topology optimized structures presented in (Guest and
Zhu 2012) considering machining (milling) constraints in
the vertically upwards direction. Therefore, interestingly,
Fig. 15 Downward build
direction for the MBB problem
L
L
2
H= L
6
F
L
2
L
Build Plate
Build Direction
Topology optimization considering overhang constraints
Fig. 16 MBB beam solution
considering downward build
(Fig. 15) with 45 degree
overhang constraint,
f=134.79, fη=0=124.24
an additive process progressing in the downwards direc-
tion created beam end features that resembled those found
when using a subtractive process progressing in the upwards
direction.
Critical to additive manufacturing is deciding on the
‘optimal’ build direction for a part. In comparing the com-
pliance values achieved in building from various directions,
it is clear that building bottom up, f=116.11 creates a
more structurally efficient solution than building top-down,
f=134.79, for this design example.
Fig. 17 Cantilever beam solution without overhang constraints. (a)
The optimized topology and (b) the optimized topology with overhang
region highlighted with dashed white and solid white lines indicat-
ing features that would and would not violate a 45 degree overhang
constraint went built upwards, respectively
3.2 Cantilever beam example
The cantilever problem of Fig. 7was also solved subject to a
minimum length scale constraint, allowable volume fraction
vf of 50 %, and various minimum allowable self-supporting
angles. Figure 17 shows the optimized solution when not
considering an overhang constraint. The overhang regions
are highlighted in Fig. 17b such that dashed white and solid
white regions indicate features that would and would not
violate a 45 degree self-supporting angle constraint, respec-
tively, when the beam is built from bottom up (Fig. 18).
As the dashed white lines indicate, this problem is actu-
ally challenging from an overhang constraint perspective as
nearly horizontal overhang regions can be seen near the top
of the beam and load transfer from the tip of the cantilever
progresses at a relatively shallow angle.
Figure 19 displays topology-optimized solutions found
when using the overhang projection approach for mini-
mum allowable self-supporting angles of 26.6, 45, and 63.4
degrees. The solutions show that the overhang constraint is
satisfied in these cases, and the trends follow those seen
in the MBB design example. That is, the solution found
when using a steeper, more restrictive self-supporting angle
requirement leads to a dense network of steep posts connect-
ing the top and bottom chords of the structure, essentially
serving a dual purpose of permanent support structures and
load-carrying elements. However, there also tends to be a
slight blur effect along the internal feature boundaries as
these areas are less structurally efficient. Additionally, for
the 63.4 and 45 degree self-supporting angle cases, we note
a small region of intermediate density along the build plate
L
F
H
2
Build Direction
Build Plate
Fig. 18 Upward build direction for cantilever beam problem
A. T. Gaynor, J. K. Guest
Fig. 19 Minimum compliance solution to cantilever beam problem
with various magnitudes of the overhang constraint
near the end of the cantilever. This is most likely due to the
fact that the algorithm must decide between either (i) creat-
ing an extremely inefficient load path by carrying the load
downwards at the self-supporting angle before the load can
move in the direction of the cantilever supports or (ii) cre-
ating a more shallow bottom chord (as in the unconstrained
solution) and designing anchors that reach down to the build
plate to support that member, but that offer no structural
benefit post-fabrication despite counting against the volume
constraint. As the self-supporting angle is relaxed and made
more shallow, the number of inner (and less structurally effi-
cient) features decreases and the structure becomes more
similar to solutions without an overhang constraint. Finally,
we note these solutions again do not resemble analytical
Fig. 20 Comparison of compliance values of various self-supporting
angles for the cantilever problem
solutions derived without the overhang or minimum length
scale constraints, and further that the overhang constraint
leads to asymmetric cantilever solutions (Fig. 19).
The free-form topology optimization solution (Fig. 17a)
achieved a compliance of f=39.31, compared to compli-
ance values of 41.86, 47.04 and 64.21 for self supporting
angles of 26.6, 45, and 63.4 degrees, respectively – see
Fig. 20 for comparison. This equates to increases of 6 %,
19 %, and 63 %. Interestingly, the increase in compliance
is relatively minor in the case of solution for a 26.6 degree
Fig. 21 Design evolution of MBB beam with 63.4 degree self-
supporting angle and more aggressive algorithmic parameters
Topology optimization considering overhang constraints
Iteration
0 100 200 300 400 500
Objective Function
0
200
400
600
800
Fig. 22 Convergence plot for MBB problem with a 63.4 degree self-
supporting angle with more aggressive algorithmic parameters
self-supporting angle, indicating that producing an entirely
self-supported (or sacrificial support-free) topology comes
at little cost in this case. It is also note-worthy that for the 45
degree case, the 19.4 % increase in compliance was almost
exactly the same increase in compliance seen in the MBB
beam case (18.9 %). As previously discussed, one should
expect the compliance to increase as the problem becomes
more restricted, although the relative magnitudes of these
increases should be problem dependent.
4 Concluding remarks
This paper proposes using topology optimization to design
additively manufactured components that are subjected to
overhang and minimum length scale constraints. By design-
ing components and structures whose features rise in the
build direction at an angle that is greater than a process-
specific minimum allowable self-supporting angle, sacrifi-
cial support material, be it anchors in metallic AM processes
or polymer support materials, is eliminated from the design
and fabrication process, saving material, build time, and
time in post-fabrication processing treatments at the cost of
structural efficiency. The proposed method utilizes a series
of projection methods such that the overhang constraint
may be imposed without adding explicit constraints to the
optimization problem.
The algorithm was illustrated in the context of well-
known MBB and cantilever beam design problems.
Although these problems are somewhat academic, particu-
larly in 2D, the solutions were shown to satisfy minimum
length scale and overhang constraints, and topology-
optimized solutions were highly dependent on the mag-
nitude of the allowable self-supporting angle and, for the
MBB problem, the selected build direction. The proposed
approach thus relies on additive manufacturing engineers to
identify a minimum self-supporting angle and a minimum
feature length scale, both of which are input into the design
algorithm. This provides the designer an effective design
tool for exploring the cost-performance tradeoffs related to
manufacturing restrictions.
Despite this preliminary success, the proposed approach
does have two primary disadvantages. First, the projec-
tion scheme to determine topology, as well as sensitivity
analysis, must proceed in a layer-by-layer manner, as ele-
ments in a given layer are dependent on the distribution of
material in the layer(s) below. Although this directly mim-
ics the layer-by-layer nature of actual AM processes, it is
generally inefficient from a computational point of view
as it inhibits the utilization of efficient parallel process-
ing. Second, the topological variables, ρe, are a function
of multiple embedded nonlinear functions, which may lead
to convergence issues for more difficult design problems.
To combat this issue herein, the authors used conservative
MMA parameters and thus smaller step sizes than typical
implementations of topology optimization. This was found
to be an effective strategy as it produced quality solutions
for a wide range of problems (beyond those presented here),
but tuning of the algorithmic parameters to improve conver-
gence speed was not a focus of this work and surely can
be improved. For example, Fig. 21 shows the design evolu-
tion and Fig. 22 the convergence history plot for the MBB
problem with self-supporting angle of 63.4 when using a
more aggressive parameterization (η=[10,20,30]and
β=[10,25,25]for maximum iterations of [50, 350,100],
with typical MMA parameter magnitudes of 0.5/(β +1)
for the initial asymptotes, and 1.2 and 0.7 for the increase
and decrease parameters, respectively). As can be seen, the
algorithm quickly finds a good, feasible solution featuring
a beam with solid web, and then slowly introduces holes
until the final solution resembling the previously reported
solution (Fig. 11c) is reached. The iteration history shows
several moderate jumps in compliance where certain sup-
porting features become disconnected but then reattach in
subsequent iterations. Tuning the parameters to provide a
proper balance of computational efficiency, smooth conver-
gence, and solution quality is thus still an open question.
Acknowledgments The authors dedicate this paper to George
Rozvany, Founder President of the International Society for Structural
and Multidisciplinary Optimization (ISSMO) and Founding Editor
of Structural and Multidisciplinary Optimization, for his tremen-
dous, pioneering research in the fields of structural and topology
optimization, and for his friendship and support of the senior author.
This research was partially supported by an appointment of the
first author to the Postgraduate Research Participation Program at
the U.S. Army Research Laboratory (USARL) administered by the
Oak Ridge Institute for Science and Education through an intera-
gency agreement between the U.S. Department of Energy and USARL,
and partially supported by the US National Science Foundation under
A. T. Gaynor, J. K. Guest
award 1462453. The authors also thank Krister Svanberg for kindly
providing the MMA optimizer code. Any opinions, findings, and con-
clusions or recommendations expressed in this article are those of the
authors and do not necessarily reflect the views of the National Science
Foundation, the Department of Energy, or the Army Research Lab.
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