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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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