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Topology optimization considering overhang constraints: Eliminating sacrificial support material in additive manufacturing through design

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Additively manufactured components often require temporary support material to prevent the component 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 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 constraints, and are shown to be dependent on the allowable magnitudes of these constraints.
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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
iNe
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=1eβμe(φ)+μe(φ)
φmax
eβφmax (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=iNe
Lφiw(xi¯
xe)
jNe
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)=tanhTT)+tanhTi
S(φ)T))
tanhTT)+tanhT(1T)) (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=jNi
SφjwS
mNi
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)veV
0ψiψi
max =1i(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(1ρe(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)=min +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=βeβμe(φ)+1
φmax
eβφmax ∂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)
jNe
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 iNj
Band i= j;
0 otherwise.
(16)
where ∂ρj
s
∂ψiis given as
∂ρj
s
∂ψi=
kNj
s
βTsech βTj
s)) T)
2
tanhTT)+tanhT(1T))
∂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
103, 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.
References
Bendsøe M (1989) Optimal shape design as a material distribution
problem. Struct Optim 1(4):193–202
Bourdin B (2001) Filters in topology optimization. Int J Numer
Methods Eng 50(9):2143–2158
Brackett D, Ashcroft I, Hague R (2011) Topology optimization for
additive manufacturing. Proceedings of the Solid Freeform Fabri-
cation Symposium, Austin, TX
Bruns T, Tortorelli D (2001) Topology optimization of non-linear elas-
tic structures and compliant mechanisms. Comput Methods Appl
Mech Eng 190(26-27):3443–3459
Cloots M, Spierings A, Wegener K (2013) Assessing new support
minimizing strategies for the additive manufacturing technology
SLM. Proceedings of the Solid Freeform Fabrication Symposium,
Austin, TX
Gao W, Zhang Y, Ramanujan D, Ramani K, Chen Y, Williams CB,
Wang CC, Shin YC, Zhang S, Zavattieri PD (2015) The status,
challenges, and future of additive manufacturing in engineering.
Comput Aided Des 69:65–89
Gaynor AT, Guest JK (2014) Topology Optimization for Addi-
tive Manufacturing: Considering Maximum Overhang Constraint.
15th AIAA/ISSMO Multidisciplinary Analysis and Optimization
Conference, AIAA Aviation
Gaynor AT, Meisel NA, Williams CB, Guest JK (2014) Multiple-
material topology optimization of compliant mechanisms cre-
ated via polyjet three-dimensional printing. J Manuf Sci Eng
136(6):061,015
Gorny B, Niendorf T, Lackmann J, Thoene M, Troester T, Maier
H (2011) In situ characterization of the deformation and fail-
ure behavior of non-stochastic porous structures processed by
selective laser melting. Mater Sci Eng A 528(27):7962–7967
Guest J, Pr´
evost J, Belytschko T (2004) Achieving minimum length
scale in topology optimization using nodal design variables and
projection functions. Int J Numer Methods Eng 61(2):238–254
Guest JK (2009a) Imposing maximum length scale in topology opti-
mization. Struct Multidiscip Optim 37(5):463–473
Guest JK (2009b) Topology optimization with multiple phase projec-
tion. Comput Methods Appl Mech Eng 199(1-4):123–135
Guest JK (2014) Projection-based topology optimization using dis-
crete object sets. ASME 2014 International Design Engineering
Technical Conferences and Computers and Information in Engi-
neering Conference, American Society of Mechanical Engineers,
Buffalo, NY, pp 1–8
Guest JK (2015) Optimizing the layout of discrete objects in struc-
tures and materials: A projection-based topology optimization
approach. Comput Methods Appl Mech Eng 283:330–351
Guest JK, Smith Genut LC (2010) Reducing dimensionality in topol-
ogy optimization using adaptive design variable fields. Int J Numer
Methods Eng 81(8):1019–1045
Guest JK, Zhu M (2012) Casting and milling restrictions in topology
optimization via projection-based algorithms. ASME International
Design Engineering Technical Conferences and Computers and
Information in Engineering Conference. 913–920
Guest JK, Asadpoure A, Ha SH (2011) Eliminating beta-continuation
from heaviside projection and density filter algorithms. Struct
Multidiscip Optim 44(4):443–453
Ha SH, Guest JK (2014) Optimizing inclusion shapes and patterns in
periodic materials using discrete object projection. Struct Multi-
discip Optim:1–16
Hu K, Jin S, Wang CC (2015) Support slimming for single material
based additive manufacturing. Comput Aided Des 65:1–10
Hussein A, Hao L, Yan C, Everson R, Young P (2013) Advanced lat-
tice support structures for metal additive manufacturing. J Mater
Process Technol 213(7):1019–1026
Jansen M, Lombaert G, Diehl M, Lazarov B, Sigmund O, Schevenels
M (2013) Robust topology optimization accounting for misplace-
ment of material. Struct Multidiscip Optim 47(3):317–333
Leary M, Merli L, Torti F, Mazur M, Brandt M (2014) Optimal
topology for additive manufacture: A method for enabling addi-
tive manufacture of support-free optimal structures. Mater Des
63:678–690
Mercelis P, Kruth JP (2006) Residual stresses in selective laser sin-
tering and selective laser melting. Rapid Prototyp J 12(5):254–
265
Mumtaz K, Vora P, Hopkinson N (2011) A method to eliminate
anchors/supports from directly laser melted metal powder bed
processes. Proceedings of the Solid Freeform Fabrication Sympo-
sium, Austin, TX
Poulsen T (2003) A new scheme for imposing a minimum length scale
in topology optimization. Int J Numer Methods Eng 57(6):741–
760
Rozvany G (1996) Some shortcomings in Michell’s truss theory. Struct
Optim 12(4):244–250
Rozvany G, Birker T (1995) Generalized Michell structures - exact
least-weight truss layouts for combined stress and displacement
constraints: Part I - general theory for plane trusses. Struct Optim
9(3-4):178–188
Rozvany G, Gollub W (1990) Michell layouts for various combina-
tions of line supports-I. Int J Mech Sci 32(12):1021–1043
Rozvany G (1998) Exact analytical solutions for some popular bench-
mark problems in topology optimization. Struct Optim 15(1):42–
48
Sigmund O (1997) On the design of compliant mechanisms using
topology optimization. Mech Struct Mach 25(4):493–524
Sigmund O (2007) Morphology-based black and white filters for
topology optimization. Struct Multidiscip Optim 33(4-5):401–424
Stolpe M, Svanberg K (2001) An alternative interpolation scheme for
minimum compliance topology optimization. Struct Multidiscip
Optim 22(2):116–124
Svanberg K (1987) Method of moving asymptotes - a new method
for structural optimization. Int J Numer Methods Eng 24(2):359–
373
Thomas D (2009) The development of design rules for selective laser
melting. University of Wales Institute, PhD thesis
Vandenbroucke B, Kruth J (2007) Selective laser melting of biocom-
patible metals for rapid manufacturing of medical parts. Rapid
Prototyp J 13(4):196–203
Wang F, Lazarov BS, Sigmund O (2011) On projection methods, con-
vergence and robust formulations in topology optimization. Struct
Multidiscip Optim 43(6):767–784
Xu C, Chai W, Huang Y, Markwald R (2012) Scaffold-free inkjet
printing of three-dimensional zigzag cellular tubes. Biotechnol
Bioeng 109(12):3152–3160
Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear den-
sity filter based on heaviside functions. Struct Multidiscip Optim
41(4):495–505
Zhou M, Rozvany G (1991) The COC algorithm, part II: topological,
geometrical and generalized shape optimization. Comput Methods
Appl Mech Eng 89(1-3):309–336
Zhou M, Lazarov BS, Wang F, Sigmund O (2015) Minimum length
scale in topology optimization by geometric constraints. Comput
Methods Appl Mech Eng 293:266–282
... 3D printing has capability to manufacture any geometry compared to other manufacturing processes. However, it is necessary for several parts to print support structures that guarantee structural stability and avoid the collapse or deformation of the material in the regions with overhangs in the manufacturing process [9], [14]. This support is eventually represented in waste material, additional costs [15], and possible defects on the surfaces [9]. ...
... This support is eventually represented in waste material, additional costs [15], and possible defects on the surfaces [9]. There are support materials that can be removed chemically, improving the result of the part [14]. However, it is an additional process that affects the manufacturing time and cost. ...
... Several authors propose strategies to obtain or approximate the support volume, such as the use of the kth nearest point algorithm [16], convex hull surface triangles method [13], or a Quadtree decomposition [8] to find the volume of support structures. Another essential aspect considered in the aforementioned strategies is the minimum self-support angle, which is suggested for direct metal laser sintering (DMLS) printed parts of 45 degrees [14]. ...
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In three-dimensional (3D) printing, due to the geometry of most parts, it is necessary to use extra material to support the manufacturing process. This material must be discarded after printing, so its reduction is essential to minimize manufacturing time and cost. An important parameter that must be defined before starting the printing process is the part orientation, which has repercussions on the quality, deposition path, and post-processing among others. Usually, the user sets up this parameter arbitrarily, so this paper takes advantage of it on optimization techniques and proposes an approximation of the volume be covered by the support material, which depends directly on the angle of the part to be printed and its geometry. Among mono-objectives optimization strategies, this work focuses on five of them. Their performance is compared by two metrics: support volume and execution time. Then, the best result is compared with commercial software.
... Tailoring the design process to the specific capabilities and limitations of the manufacturing process produces useful results and reduces the need for post-processing. The incorporation of manufacturing constraints into topology optimization has been an active area of research, particularly for AM constraints related to (for example) unsupported overhangs (Gaynor et al. 2014b;Gaynor and Guest 2016;Langelaar 2016Langelaar , 2017Mass and Amir 2017;Gaynor and Johnson 2018) and consideration of build orientation (Langelaar 2018;Wang and Qian 2020). ...
... Manufacturing primitives, similar to structuring elements in morphology, may consist of a shape adapted for a material or manufacturing process (Guest 2009b;Guest and Zhu 2012;Ha and Guest 2014;Guest 2015;Vatanabe et al. 2016;Ha et al. 2019), a morphable shape in which one or more parameter is variable (Ha and Guest 2014;Norato et al. 2015;Guo et al. 2017), or a surrogate representation of a manufacturing constraint (Gaynor and Guest 2016;Vatanabe et al. 2016). The primitive presented in (Carstensen 2020), for example, is aimed at nozzle-type AM and consists of an immutable core flanked by an adaptable bonding layer, resulting in features which are integer multiples of the nozzle diameter. ...
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Topology optimization (TO) is well suited to exploit the geometric freedom provided by additive manufacturing (AM), but only when the two technologies are properly integrated. Failure to account for the manufacturing process in the execution of the optimization formulation can lead to performance loss and increased production time and/or cost. This paper discusses a TO methodology motivated by the unique features of wire and filament based AM processes with high deposition rates where a constant thickness of deposited features is desired to manage heat flow and path planning during fabrication. In addition to typical manufacturing constraints such as minimum feature size and feature separation, the proposed approach utilizes discrete object projection to impose a constant thickness requirement on all structural features, including structural members and connection points (joints). The mathematical consistency of the developed framework enables the use of gradient-based optimizers, and tradeoffs between design freedom and computational cost are discussed. Although the technique was developed with a specific electron beam fabrication process in mind, it is readily extendable to other AM technologies with similar requirements as well as to create lattice-like designs. The approach is demonstrated on benchmark minimum compliance problems and is shown to successfully design structural components that are directly manufacturable.
... The primary objective of these sacrificial structures is to support and allow the realization of overhang surfaces that cannot support themselves, which have a critical angle of inclination relative to the manufacturing substrate [10]. Optimizing the positioning and orientation of the part on the production platform by taking into account this type of constraint is up-to-date [11]. The inclination angle of the surface also has a great influence on the roughness and can therefore also be taken into account in the step of placing the part on the production plate [12]. ...
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To obtain a functional part from additive manufacturing (AM) technologies, some surfaces require post-processing by machining. An approach is developed using additive manufacturing supports as a clamping device for the milling operation. A model combining an analytical approach to determine the cutting forces with a finite element model (FEM) to predict the dynamical response of the workpiece-supports system is proposed. The complex structure of the supports is homogenized with a simplified geometry with equivalent stiffness and mechanical properties. A case study from the biomedical field is proposed: the finishing operation of a custom-made maxillary reconstruction plate is simulated. A parametric study is proposed with: (1) two different lattice geometries used as support structures; (2) up and down milling; (3) different depths of cut.
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Simulation of AM products can capture a number of aspects. Apart from the traditional types of simulation of the end product, such as mechanical, thermal and fluid analyses, it is possible to simulate the AM build process. While simulating products intended for AM can sometimes be performed in exactly the same way as with products intended for traditional manufacturing, there are several aspects that may require a specialized workflow.
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In this chapter, general workflow in Additive Manufacturing process is shown, from preprocessing activities that include preparing appropriate CAD model, selecting required STL file resolution, up to setting processing parameters for AM process.
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In this chapter, three strategic domains of Additive Manufacturing application are presented: tool making, medicine and transportation, with main benefits and results obtained by application of AM. Chapter presents some of on-going or already finished project from mentioned AM application fields.
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The development of parameters for a certain additive technology is the key to increase the number of materials that are processed as well as the applications. This chapter shows the details to take into account for the development of parameters for various technologies.
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A standard is a published document that describes a technical specification or a list of guidelines in the form of rules, definitions, methods, vocabularies, or codes of practices. Standards provide a unified source of reference for specifying or representing products. Before the industrial revolution, manufacturers from different places used to compare and copy the dimensions and specifications of components to match those of a prototype.
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FGM is a special class of composite material that was first developed in Japan around 1984 for the propulsion system and airframe of space planes. The challenge was to create a thermal barrier that would be capable of withstanding a temperature of 1000 °C over a cross-section of 10 mm. A sharp interface between the matrix and the reinforcement in a traditional composite material would cause cracking in high temperatures. The cracks occur due to the generation of interfacial stress induced by the mismatch of thermal expansion between two different materials.
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The manufacturability model of complex structure is the premise of manufacturing, and it is necessary to consider material properties, structure design, manufacturing constraints, and so on. However, due to the inconsistent restrictions between design and manufacturing, it is not easy to obtain the manufacturable structure that matches its design performance using layer-wise manufacturing. This paper presents a topology optimization method for manufacturable form, which incorporates the self-supporting factors such as overhang angle and length based on the characteristics of the generic additive manufacturing process. The support relationship between the supporting and supported elements in self-supporting constraints is mapped to a cascade relationship between two adjacent layers. To avoid a low-density structure supporting multiple high-density ingredients, we establish a fabrication model using the smax and smin operators. Also, the sensitivity analysis and variable updating method are given under the Solid Isotropic Material with Penalization method. Furthermore, numerical examples are shown to validate the correctness and superiority of this proposed self-supporting structure design method.
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Metal powder bed AM processes have a significant drawback in that they require anchors/supports to hold overhanging features down during laser processing. This severely restricts the geometries that the processes can make, adds significant time and cost to production and reduces throughput as parts cannot be easily stacked in the build bed. A method to eliminate the need for these anchors/supports has been invented and will be described. Early parts made without anchors will be shown and next steps for research will be discussed.
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Compliant mechanisms are able to transfer motion, force, and energy using a monolithic structure without discrete hinge elements. The geometric design freedoms and multi-material capability offered by the PolyJet 3D printing process enables the fabrication of compliant mechanisms with optimized topology. The inclusion of multiple materials in the topology optimization process has the potential to eliminate the narrow, weak, hinge-like sections that are often present in single-material compliant mechanisms. In this paper, the authors propose a design and fabrication process for the realization of 3-phase, multiple-material compliant mechanisms. The process is tested on a 2D compliant force inverter. Experimental and theoretical performance of the resulting 3-phase inverter is compared against a standard 2-phase design.
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To successfully produce metal parts by SLM, additional support structures are needed to support overhanging surfaces in order to dissipate process heat and to minimize geometrical distortions induced by internal stresses. These structures are often massive and require additional post-processing time for their removal. A minimization of the extent to which support structures are needed would therefore significantly reduce manufacturing and finishing efforts and costs. A specific component segmentation strategy is developed. It allows the segmentation of critical areas of the component by applying a specific scanning strategy with appropriate energy input and optimized supporting strategies. The results indicate that the supporting effort can generally be reduced, e.g. overhang geometries with an angle to the horizontal of less than 35° can be manufactured without any support. The successful realization of the segmentation strategy in combination with optimized support structures allows the implementation of a stacking strategy, thereby using the available work space more efficiently.
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This paper presents an investigation into residual stresses in- Selective Laser Sintering and Selective Laser Melting. First, the origin of these stresses is explored, and a simple theoretical model is presented to predict the residual stress distribution. Nest, an experimental model is developed, which can be used to actual residual stresses. remaining in the ports produced. A set of test samples is then tested to investigate the influence of the parameters on the residual stresses being developed- At the last some methods to refuce the residual stress are presented .
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This paper gives an overview of the issues and opportunities for the application of topology optimization methods for additive manufacturing (AM). The main analysis issues discussed are: how to achieve the maximum geometric resolution to allow the fine features easily manufacturable by AM to be represented in the optimization model; the manufacturing constraints to be considered, and the workflow modifications required to handle the geometric complexity in the post optimization stages. The main manufacturing issues discussed are the potential for realizing intermediate density regions, in the case of the solid isotropic material with penalization (SIMP) approach, the use of small scale lattice structures, the use of multiple material AM processes, and an approach to including support structure requirement as a manufacturing constraint.
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Additively manufactured components often require temporary support material during the 3D printing process. In the case of polymer material process such as Fuse Deposition Modeling (FDM), the support material can be dissolved away. However in the case of metals in a selective laser melting (SLM) process, the support and component material are one in the same. Since the support structure adds both material cost and post-processing cost to every component printed, it is desired to limit or completely eliminate the need for such material. As such, it is proposed to take advantage of the maximum printable overhang angle (the angle at which the AM process requires no support material) by harnessing topology optimization as the design engine. This is accomplished through a topology optimization projection scheme, in which the angle constraint is imposed through a Heaviside projection and not applied as an explicit constraint. Solutions to two standard topology optimization problems are included and show good agreement with the overhang constraint.
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We look to expand the reach of continuum topology optimization to include the design of 'structures' that gain functionality or are specifically manufactured from discrete, non-overlapping objects. While significant advancements have been made in restricting the geometric properties of topologyoptimized structures, including restricting the minimum and maximum length scale of features, continuum topology optimization is still largely limited to monolithic structures. A wide variety of structures and materials, however, gain their stiffness or functionality from discrete objects, such as fiberreinforced composites. This work examines a recently developed method for optimizing the distribution of discrete objects (2d inclusions) across a design domain and extends the approach to variable shape and variable sized objects that must be selected from a designer-defined set. This essentially enables simultaneous optimization of object sizes, shapes, and/or locations within the projection framework, without need for additional constraints. As in traditional topology optimization, gradient-based optimizers are used with sensitivity information estimated via the adjoint method, solved using finite element analysis. The algorithm is demonstrated on benchmark problems in structural design for the case where the objects are stiff inclusions embedded in a compliant matrix material.
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Current topology optimization methodologies assume a monolithic, free form approach to design. Many engineered materials and structures, however, are composed of discrete, non-overlapping objects such as fiber or particle-based materials. Application of the topology optimization methodology to these types of materials therefore requires controlling the shape and interaction of designed features to ensure solutions are meaningful and physically realizable. Achieving such control on continuum domains is challenging as features form via the union of elements of like phase. A topology optimization approach is proposed herein for optimizing the size, shape, and layout of inclusion-like features in a continuum domain. The technique is based on the Heaviside Projection Method and uses multiple regularized Heaviside functions whose interaction is tailored so that the designer may restrict the minimum and maximum length scale of inclusions, and minimum spacing between inclusions. The technique is demonstrated on the design of material microstructures with enhanced elastic stiffness.
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A density-based topology optimization approach is proposed to design structures with strict minimum length scale. The idea is based on using a filtering-threshold topology optimization scheme and computationally cheap geometric constraints. The constraints are defined over the underlying structural geometry represented by the filtered and physical fields. Satisfying the constraints leads to a design that possesses user-specified minimum length scale. Conventional topology optimization problems can be augmented with the proposed constraints to achieve minimum length scale on the final design. No additional finite element analysis is required for the constrained optimization. Several benchmark examples are presented to show the effectiveness of this approach.