ARL-CR-0815 ● JUNE 2017

US Army Research Laboratory

Progress toward Topology Optimization (TO) 

for Additive Manufacturing (AM) and Fatigue

prepared by Terrence E Johnson

Oak Ridge Associated Universities (ORAU) 

4692 Millennium Drive, Suite 101 

Belcamp, MD 21017

under contract W911NF-16-2-0093

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ARL-CR-0815 ● JUNE 2017

US Army Research Laboratory

Progress toward Topology Optimization (TO) 

for Additive Manufacturing (AM) and Fatigue

prepared by Terrence E Johnson

Oak Ridge Associated Universities (ORAU) 

4692 Millennium Drive, Suite 101 

Belcamp, MD 21017

under contract W911NF-16-2-0093

Approved for public release; distribution is unlimited.
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Progress toward Topology Optimization (TO) for Additive Manufacturing 

(AM) and Fatigue

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W911NF-16-2-0093

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Terrence E Johnson

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Oak Ridge Associated Universities (ORAU)

4692 Millennium Drive, Suite 101 

Belcamp, MD 21017

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ARL-CR-0815

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US Army Research Laboratory

Vehicle Technology Directorate ATTN: VTD-TVM

Aberdeen Proving Ground, MD 21005-5066

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

This report summarizes research progress to date in topology optimization (TO) for additive manufacturing (AM) and fatigue.

The effort has been divided into 2 parts: TO for AM and TO for fatigue of metals. The 2-D and 3-D solutions are presented for 

TO for AM and fatigue. The proposed design approaches have produced promising results and suggest further investigation. A 

path forward is suggested in the conclusion.

15. SUBJECT TERMS

topology optimization, fatigue, additive manufacturing, TO, AM

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Terrence E Johnson

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Contents

List of Figures iv

List of Tables iv

Acknowledgments v

1. Introduction 1

2. Overhang Projection and Void Projection 2

3. Problem Formulation (TO for AM) 4

4. Examples (TO for AM) 5

4.1 Overhang Projection 5

4.2 Void Projection 7

5. Clustered Fatigue Stress Measure and Stress Life Calculation (TO for 

Fatigue) 10

6. Problem Formulation (TO for Fatigue) 11

7. Examples (TO for Fatigue) 12

7.1 Cantilever in Shear 12

7.2 MMB 14

8. Summary 15

9. Conclusions 16

10. References 17

Distribution List 19
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List of Figures

Fig. 1 3-D search design domain..................................................................... 3

Fig. 2 3-D cantilever beam.............................................................................. 6

Fig. 3 TO solution for 3-D cantilever beam.................................................... 6

Fig. 4 3-D Column with AM base plate .......................................................... 7

Fig. 5 Solution of 3-D column with AM base plate ........................................ 7

Fig. 6 2-D simply supported beam definition ................................................. 8

Fig. 7 2-D short beam definition ..................................................................... 8

Fig. 8 3-D cantilever beam definition ............................................................. 8

Fig. 9 Solution for simply supported beam in Fig. 6....................................... 9

Fig. 10 Solution to short beam in Fig. 7............................................................ 9

Fig. 11 Solution to cantilever beam in Fig. 8 .................................................... 9

Fig. 12 Reorientation of solution in Fig. 11 .................................................... 10

Fig. 13 Initial geometry for shear problem ..................................................... 12

Fig. 14 Density distribution for geometry given in Fig. 13............................. 12

Fig. 15 Stress distribution for geometry given in Fig. 14 ............................... 13

Fig. 16 Initial MMB, half-symmetry............................................................... 14

Fig. 17 Density distribution for half-symmetry MMB.................................... 15

Fig. 18 Stress distribution for half-symmetry MMB....................................... 15

List of Tables

Table 1 Weight vs. cycles-to-failure comparison for cantilever in shear ........ 13

Table 2 Weight vs. cycles-to-failure comparison for MMB............................ 14
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Acknowledgments

The Topology Optimization (TO) for Additive Manufacturing (AM) work was 

done in collaboration with Dr Andy Gaynor of the US Army Research Laboratory’s 

Weapons and Materials Research Directorate. Andy used his TO code to produce 

the solutions shown in Figs. 3, 5, and 9–12. 
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INTENTIONALLY LEFT BLANK.
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1. Introduction

Additive manufacturing (AM) is a layer-by-layer manufacturing method that builds 

parts from the bottom up. This way of manufacturing is relatively new, first 

appearing in the 1980s. AM is different from the traditional subtractive 

manufacturing process, which cuts away material from a starting substrate to get a 

final component. Topology optimization (TO) is an extremely powerful free-form 

rigorous design method that was developed for designing structures. TO can 

produce efficient designs for prescribed objectives and constraints that make it ideal 

for AM. AM has the capacity to realize TO designs that are not realizable by 

traditional manufacturing processes due to cost, tool-path constraints, or operator 

limitations. While AM significantly widens the design space for TO, manufacturing 

constraints and limitations remain1 and should be addressed in the design process.

An objective of this work is to consider manufacturing constraints, such as 

overhangs and enclosed pores, within the TO methodology for structural design.

Fatigue is a fundamental mode of failure in vibrating structures such as rotorcraft.

Current US Army rotorcraft maintenance schedules are costly as they can keep 

aircraft grounded for an extended time period and require manual labor to survey 

the structure for possible damage caused by fatigue. Structures with improved 

fatigue properties could therefore reduce maintenance costs. Traditional TO 

formulations for lightweighting (i.e., without consideration of fatigue) often 

produce designs with stress concentrations or singularities that cause a reduction in 

fatigue life. Manual adjustments or additional structural optimization are needed to 

fulfill engineering stress requirements. Therefore, a second objective of this work 

is to consider/couple fatigue within TO design for the lightweighting of structures.

This report presents progress that has been made toward designing lightweight, 

fatigue-constrained topologically optimized structures using AM. To date, this 

research effort has been divided into 2 distinct components: 1) TO for AM, 2) TO 

for fatigue. Future plans call for these component algorithms to be combined and 

demonstrated as TO for AM and fatigue. 

The goal of TO for AM is to design structures that account for AM limitations 

within the design. The limitations of interest in this work are the production of 

support material and enclosed pores. Both limitations will be separately considered 

in the TO formulation. 

The goal of TO for fatigue is to codify and experimentally demonstrate a TOcapability to maintain or enhance fatigue life. This will be done by selecting a 

published methodology for handling stress within TO and use it to develop 3-D TO 
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solutions. The solutions will be realized/proven/demonstrated using traditional 

manufacturing techniques and fatigue-testing protocols. Though a non-AM process 

will be used here, the novelty of this second effort is fatigue testing of a TOdesigned structure. This work will verify the selected stress handling methodology 

and produce test results that will serve as a benchmark for TO for stress and fatigue 

efforts. 

This report is outlined as follows: Section 2 gives an overview of the overhang and 

void-projection schemes used to account for AM constraints. Section 3 describes 

the TO problem formulation that is used to generate solutions for the presented 

projection schemes. Section 4 presents the solution to the problem formulation 

given in Section 3. Section 5 presents the approach selected for accounting for 

stress within a TO structure. Section 6 presents the TO problem formulation used 

to demonstrate the methodology presented in Section 5. Section 7 presents 

examples that solve the TO problem formulation given in Section 6. The report 

ends with a summary and concluding remarks in Sections 8 and 9.

2. Overhang Projection and Void Projection

An overhang is a solid feature that rises in the build direction, at a shallow angle to 

horizontal, with supporting material below it. In extrusion-based processes, support 

material is generated during fabrication to hold-up soft overhang material as it 

hardens to prevent part distortion. Following fabrication, support material removal 

can result in surface damage, trapped material, increased part time and fabrication 

cost. In metal powder bed fusion processes, support material is generated during 

fabrication to prevent solid feature distortion from residual stress accumulation 

because of thermal gradients within the sintered solid and to produce a conductive 

path from the point of melting/sintering to the build plate. Following fabrication, 

support material removal can lead to degraded surface finish, trapped material, 

increased part time, and fabrication cost. Therefore, several research efforts aim to 

remove overhangs from designed structures.1–11 In this work, overhangs are 

removed by making structures self-supporting.

The proposed approach to make a structure self-supporting is to account for 

overhangs in the TO design formulation. The proposed method is 3-D and an 

extension of work done by Gaynor and Guest.6 In that work, overhangs were totally 

eliminated by a so-called overhang projection scheme. The idea behind the 

overhang projection scheme is simple: an element (e) may become a solid element 

if and only if the following occur: 1) the local design variables indicate material 

should be deposited into the element12 and 2) sufficient material exists in the 

supporting elements below e such that e does not violate the defined overhang 
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angle. If e becomes solid, it will not contribute to an overhang. Three variables are 

used to create this effect: 1) the φ is the vector of dependent variables that is passed 

through the Heaviside Projection Method (HPM) to create a spherical6 solid feature 

in the finite element space, 2) the ψ is the vector of independent design variables 

that indicates whether material should be deposited at a given location, and 3) the 

ρS is a subset of variables φ that exist in the neighborhood below the point of 

interest (Fig. 1) and indicate whether material can be deposited at the considered 

location (i.e., indicate whether or not the overhang condition is violated). The Ψ

and ρS will be combined to form φ, which is then used to determine element volume 

fraction ρe

: φi = Ψi * ρS

i

. Whether material can be projected from a point φφii onto 

the elemental domain is entirely dependent on the magnitudes of other φφ variables 

below this point, and thus the algorithm must proceed in a layer-by-layer manner, 

essentially mimicking actual AM processes. The overhang support neighborhood 

that relates a given Ψi to the support region below it is shown in Fig. 1. In Fig. 1, 

the blue dots represent locations within the finite element domain that have an 

assigned variable Ψi. The green dots represent locations below Point i that represent 

the neighborhood where ρS is calculated. The Ψ and ρS determine if material should 

be placed at Point i.

Fig. 1 3-D search design domain

The density of an element is defined by Eq. 1. If ρe = 1, then the element is solid.

If ρe = 0, then the element is void. The β is the regularization parameter dictating 

the aggressiveness of the regularized Heaviside function.

12 The μe is the proximity 

weighted-average of design variables defined by φs in a spherical neighborhood 

with a prescribed radius (rmin) from an element of interest e. The φmax is the 

maximum magnitude of φ (herein = to 1).

ρρee = 1 − ee−ββμμee(φφ) + μμee(φφ)

φφmmmmmm

ee−ββφφmmmmmm (1)

To recap, the elemental densities ρe are computed by Eq. 1 and are a function of 

dependent variable φi

. The φi is a function of the independent optimization variable 

Ψi and the dependent support indicator variable ρS

i

. The ρS

i is computed using a 
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relationship given in Gaynor and Guest6 and are a function of φ in the support 

neighborhood below the considered point, i. This last dependency means that the 

algorithm must proceed in a layer-by-layer manner starting from the bottom.

Lastly, this underlying algorithm may be slightly adjusted to solve a completely 

different problem in TO for AM: the enclosed pore solution. Enclosed pores are 

hollow sections of a volume, which should be filled only with air. However, in 

metal powder bed fusion processes, these hollow sections will trap powder that is 

very difficult (if not impossible) to remove without severely damaging the 

structure. This may also be an issue in extrusion-based processes where support 

material can get trapped. This work proposes a simple approach to removing 

enclosed pores in TO design: void projection. Void projection can be accomplished 

in TO using Eq. 2:

ρρee = −ee−ββμμee(φφ) + μμee(φφ)

φφmmmmmm

ee−ββφφmmmmmm . (2)

The idea being that instead of projecting solid material from the design variables 

(which are located at the nodes), the algorithm project voids. When this is coupled 

to the layer-by-layer bottom-up approach discussed for overhang projections, the 

result is structures with void pathways that ensure material removal. This is further 

discussed in Section 4.2.

3. Problem Formulation (TO for AM)

The proposed scheme for TO for AM is demonstrated using the well-known13

minimum compliance (maximum stiffness) problem. The goal of the optimization 

will be to simultaneously satisfy the objective function and constraints while 

searching for the location within the design space where the gradient of the 

objective function is zero. The location within the designs space where the gradient 

is zero can be a local or global minimum or maximum to the objective function. To 

achieve its goal, the optimization process determines the material distribution, ρe

, 

of the elements within the design domain. The optimization formulation takes on 

the following form:

�

minFFTTdd

ss.tt.� KK = FF,

∑ ρρeevvee ≤ VV

, (3)

where FFTTdd (compliance) is the objective function, KK = FF (equilibrium), and 

∑ ρρeevvee ≤ VV is the volume constraint.

F is the vector of applied nodal loads, d is the vector of nodal displacements, K(ψ)

is the global stiffness matrix, ρe

(ψ) is elemental volume fraction of element e, ve is 
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the volume of element e and V is the total allowable material volume. The Ψi varies 

between 0 and 1. The element density ρe affects structural equilibrium through the 

global stiffness matrix. The global stiffness definitions of interest in this work are 

RAMP and SIMP and RAMP defined in Eqs. 4 and 5, respectively13–15:

KKee = �ρρmm mm + ρρee

1+γγ(1−ρρee)

� KK0

ee (4)

KKee = �ρρmm mm + ρρeeγγ

�KK0

ee . (5)

Both produce quality solutions, yet a preference has been developed for RAMP 

because of its nonzero gradient at ρe = 0. This property was useful as material grew 

out of void space below a structural feature to make said feature self-supporting.

Similar benefits are seen in Guest16 where it was necessary for stiff inclusions to 

grow out of compliant matrix material.

4. Examples (TO for AM)

The proposed methodologies for overhang and void projection are demonstrated 

using simple 2-D and 3-D structures described in Sections 4.1 and 4.2. The radial 

length scale (rmin) prescribed in the 3-D overhang projection examples is 2 times 

the element length. The allowable volume fraction is 50% and the initial 

distribution of ψ is a uniform 50%. The SIMP exponent 5 is chosen to further make 

densities that are less than 1 inefficient by reducing their stiffness value. The β used 

in the density definition is 25. Selecting a value of 25 further increases the 

algorithm’s ability to eliminate intermediate density elements at the structure’s 

boundary. In all examples, the optimization process was run for several iterations 

until convergence was obtained.

4.1 Overhang Projection

The first example used to demonstrate the proposed 3-D solid projection selfsupporting scheme is the 3-D cantilever beam (Fig. 2). The mesh is composed of 

4-node brick elements that are 1:1:1 in aspect ratio. The angular self-support 

condition is 45° from horizontal. The solution (Fig. 3) shows that the imposed 

angular restriction of 45° has not been exceeded. This fundamentally demonstrates 

that the optimization process can build structures from the bottom up without 

violating the imposed constraint on the feature angle. The cliff that exist at the rightmost edge is needed to support the external load. The removed sections of the beam 

beneath the cliff and in the interiors result from the optimization process working 

to satisfy the volume fraction constraint of 50% and minimize compliance.

Constraining the volume of the final solution is effectively how the optimization 
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process lightweights the structure. The obtained solution is a local minimum in a 

nonconvex design space.

Fig. 2 3-D cantilever beam

Fig. 3 TO solution for 3-D cantilever beam

The second example is a 3-D tall rectangular box that sits on a base (Fig. 4). The 

base represents the base plate in an AM machine—such as a Stratasys FDM printer.

The printing direction is in the vertical direction from the base plate. This example 

is intended to show that the algorithm actively works to ensure that the structure is 

self-supporting by building the structure from the baseplate. It can be seen (Fig. 5)

that no part of the structure has angular features less than 45°. The solution is 

multicolored, which means it is not physically realizable. A physically realizable 

solution would be an all-solid color (e.g., all red). Thus, interpretive work must be 

done by the designer to realize the solution (through thresholding out low-density 

elements, thus defining the solid structure by those finite elements above the 

threshold).
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Fig. 4 3-D Column with AM base plate

Fig. 5 Solution of 3-D column with AM base plate

4.2 Void Projection

The void projection scheme is demonstrated using 2-D (Figs. 6 and 7) and 3-D 

beam (Fig. 8) problems. In 2-D, the elements are 4-node quadrilaterals that have an 

aspect ratio of 1:1. The rim in the 2-D void projection example is 2 times the 

element length. The beam in Fig. 6 represents half of a Messerschmidt-BolkowBlohm (MBB) beam. Since the load, geometry, and boundary conditions (3-point 

bending) are symmetric about the center of the beam, only half of the beam needs 

to be topologically optimized. The half solution can then be mirrored about the 

vertical line of symmetry of the beam. The half beam is used in solving the problem.
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The solution in Fig. 9 shows a full beam with a cutout section. Blue represents void 

(no material). Red represents solid (material). The cutout section defined by the 

optimization process is beneath the solid section because the void projection 

scheme will always opt to remove material from the bottom up unless doing so will 

violate the design constraints or equilibrium. The resulting physical topology is one 

that does not trap material. This is also seen in the topologically optimized solutions 

in Figs. 10–12.

Fig. 6 2-D simply supported beam definition

Fig. 7 2-D short beam definition

Fig. 8 3-D cantilever beam definition
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Fig. 9 Solution for simply supported beam in Fig. 6

Fig. 10 Solution to short beam in Fig. 7

Fig. 11 Solution to cantilever beam in Fig. 8
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Fig. 12 Reorientation of solution in Fig. 11

The short beam problem (Fig. 7) has a slanted load and is intended to demonstrate 

how the optimization process adjusts for load orientation. The solution in Fig. 10 is 

particularly interesting because a feature is extruded from the bulk solid section to 

support a slanted load (see Fig. 7 for the initial geometry and problem setup). The 

optimization process is formulated to remove at least 50% of the material from the 

bottom up and does so as seen in the large blue region. The remaining solid section 

is used for directing the load to the boundaries.

In 3-D (Fig. 11), the same phenomenon is seen: the optimization process removes 

material from the bottom up to satisfy the volume constraint. The prescribed 

volume constraint is 50%. Solid material is removed from sections that have less 

impact on the load path from the load to the cantilever boundary conditions.

Figure 11 shows a hole in the beam near the boundary. Figure 12 shows that the 

hole extends through to beam’s cross section. This hole is a discontinuity in the 

beam and therefore causes stress concentrations. This denotes the need to introduce 

stress constraints within the optimization process. Overall, the 2-D and 3-D 

examples show that the proposed void projection scheme produces solutions that 

are not likely to trap material.

5. Clustered Fatigue Stress Measure and Stress Life Calculation 

(TO for Fatigue)

Fatigue life can be enhanced or maintained while lightweighting by accounting for 

stress. That is, there is a direct correlation between stress reduction and prolonged 

fatigue life. As a result, the handling of stress is of primary importance. Within a 

finite-based TO realm, stress can be handled locally, globally, or in clusters. The 

approach of interest in this work is the clustered approach proposed by Holmberg

et al.

17 Clusters are groupings of stress from elements in the FE domain. The 
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number of groups can vary from 1 (i.e., globally: all elemental stresses are grouped 

into 1 set) to the maximum number of stresses calculated from the finite element 

system (i.e., locally: there is 1 cluster per integration point). The clustered stress 

definition is as follows:

σσii

PPPP = � 1

ii

∑ �σσaa

vvvv(xx)�

pp

aa∈ωωii �

1

pp . (6)

The von Mises stress measure, σσaa

vvvv, is calculated at an integrate point within a finite 

element. The Ni is the number of stress points considered in a cluster. It acts as a 

built-in scaling of σσii

PPPP, which is beneficial for convergence in optimization 

problems. The p is called the p-Norm exponent and is set equal to 8.

17 Von Mises 

is a function of σσaa given in Eq. 7. The σσaa is a function of penalization factor δδss,

which is used to increase the stress values of element with intermediate density,

thereby making them inefficient and candidates for elimination from the final 

design domain. The optimal final density distribution within the design domain is 

0–1. Intermediate densities require further analysis and interpretation by the 

designer.

σσaa(xx) = δδss(ρρee(xx))σσ�aa(xx) (7)

6. Problem Formulation (TO for Fatigue)

The optimization problem formulation used to demonstrate fatigue life 

enhancement and preservation is handled using stress constraints. The formulation 

is as follows:

�

mmmmmmxx ∑ mmeeρρee (xx)

ss.tt. � σσii

PPPP(xx) ≤ σσ�, = 1, ... , cc

xxee ≤ xxee ≤ �xx�ee�, ee = 1, ... , ee

, (8)

where ne is the number of design variables and me is the solid element mass related 

to design variable e. The e:th variable8 is denoted ρe (x) and xe is the e:th design 

variable, limited by the box constraint limits xe = 1 and xe = e, where e is a small 

positive number used to avoid the stiffness matrix becoming singular. The stress 

measure is the modified P-norm based on von Mises stresses, which for cluster 

number i is denoted σi

PN(x). The number of clusters, or equally, the number of stress 

constraints, is denoted nc and σσ� is the stress limit. In the previous formulation, the 

equilibrium equation is not used as a constraint. Instead, the displacement vector 

results from the calculation of u=K-1(x) F within the algorithm (i.e., nested 

formulation). The stiffness penalization is SIMP as described previously in 

Section 4.
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7. Examples (TO for Fatigue)

The examples used to demonstrate Holmberg’s stress constraint formulation are the 

cantilever with shear load and the MMB beam. The elements used in the finite 

element domain are 2-D plane elements. The element type is 4-node quadrilaterals.

The number of elements in the initial design domain is 300 (10 × 30). The solutions 

were run to 500 iterations. The topologically optimized solutions in Figs. 14 and 17

show the density distributions.

7.1 Cantilever in Shear

The design domain for the problem is given in Fig. 13. The density and stress 

distributions are given in Figs. 14 and 15. The external load is distributed over 

several nodes to relieve stress concentrations that result from applying point loads.

The initial distribution of φ is uniform at 2%. The optimization algorithm used to 

solve this problem is Matlab’s18 “fmincon”. The derivatives were provided to 

fmincon following the formulation in Holmberg.17 The Heaviside projection12

method was used to calculate density.

Fig. 13 Initial geometry for shear problem

Fig. 14 Density distribution for geometry given in Fig. 13
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Fig. 15 Stress distribution for geometry given in Fig. 14

The density distribution (Fig. 14) varies from 0 (void) to 1 (solid). The results show 

a solid-like structure with angled features that meet and connect with a rectangular 

base region. The base is needed to support the external shear load. The inner regions 

of the features are light red and surrounded by dark white-colored elements. This is 

a product of the optimization formulation to satisfy the given objectives and 

constraints. Since the objective of the problem is to minimize weight, the 

optimization process is formulated to satisfy this objective by inserting intermediate 

density elements that weigh less than full-density solid elements. As seen in Eq. 7, 

intermediate elements also carry less stress than full-density elements, which helps 

in satisfying the stress constraint. The intermediate solids can be pushed toward 

0–1 by increasing β in the Heaviside projection method or by artificially 

prescribing all elements with a density between 0 and 0.5 to 0, and prescribing all 

elements with a density between 0.1 and 1 to 1 during postprocessing. Figure 15 

shows the stress distribution. The von Mises stress in every element is less than the

prescribed stress constraint.

Table 1 shows what happens to cycles-to-failure19 as weight is reduced. Cycles-tofailure reduces by almost 100% as weight is drastically reduced. This highlights the 

need to constrain cycles-to-failure as weight is reduced. The cycles-to-failure 

calculation is a rough estimate because the density distribution is composed of 

intermediate elements. Making the solution fully solid would result in a better 

estimate of cycles-to-failure.

Table 1 Weight vs. cycles-to-failure comparison for cantilever in shear

Initial Final Reduction (%)

Weight 300 86 71

Cyclesto-failure 8 × 1014 3 × 1012 99.6
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7.2 MMB

A second example that further demonstrates lightweighting and fatigue life is the 

2-D MMB beam. The initial distribution of φ is uniform at 99%. The gradient-based 

optimization process used to solve these problems is “fmincon”. The derivatives 

were provided to fmincon following the formulation in Holmberg et al.

17 Instead of 

distributing the load across several nodes, the method of element exclusion 

suggested by Holmberg17 for reducing stress near the point of application was 

implemented. The density and stress distribution plots are given in Figs. 17 and 18, 

respectively. The density plot shows regions of full, intermediate, and void 

elements. The stress distribution plot shows that the maximum von Mises stress is 

near the application of the point load. However, the von Mises stress has not been 

exceeded. As in Table 1, Table 2 also shows the penalty on cycles-to-failure as 

weight is reduced.

Table 2 Weight vs. cycles-to-failure comparison for MMB

Initial Final Reduction (%)

Weight 300 134 55

Cycles-tofailure 4 × 104 1 × 104 75

Fig. 16 Initial MMB, half-symmetry
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Fig. 17 Density distribution for half-symmetry MMB

Fig. 18 Stress distribution for half-symmetry MMB

8. Summary

This report documents current progress made toward developing a methodology for 

AM structures that are TO when considering fatigue. To address this topic, the 

research has been split into 2 separate efforts: TO for AM and TO for fatigue. The 

TO for AM efforts are focused on accounting for AM constraints within design: 1) 

removal of support material and 2) the elimination of enclosed pores. To date, the 

3-D projection method produces topologies with features that do not violate the 

overhang constraint and are therefore self-supporting. Also, the void projection 

scheme produces topologies with outlets for trapped material. The TO for fatigue 

effort is focused on using a published methodology to design structures considering 

fatigue. The selected methodology is Holmberg’s clustered approach.17 The

methodology was successful in producing topologies that do not violate imposed 

stress constraints. However, it was also shown that reducing weight penalizes 

fatigue life.
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9. Conclusions

The methodology for developing self-supporting AM structures is currently 

programmed to design structures with various angles, but this report showed 

examples of the typical design situation of 45°. Any subsequent shape or size 

optimizations should account for the angular overhang constraint if the structure is 

to remain self-supporting. The void projection method produces orifices to release 

trapped material. It is currently programmed to develop orifices from the bottom 

up. Therefore, the structures that will likely be produced are those that will allow 

material to escape from the bottom. It may be useful to implement an aesthetic 

constraint that minimizes the size of gaping orifices that are currently being 

produced while maintaining structural properties such as stiffness, buckling, and 

eigen frequency.

In considering TO for fatigue, the current solutions were developed with no 

attention given to the structure’s fatigue life before optimization. The result was a 

worsening of fatigue properties. This effect can be mitigated by first assessing the 

fatigue life of the initial structure, then setting the fatigue constraint to this value 

while reducing mass. This would ensure that mass is decreased while maintaining 

fatigue life. In addition to maintaining fatigue life, it is proposed that TO can also 

be used to improve fatigue life. This can be done be allowing mass redistribution 

instead of mass removal. Thus, the TO for fatigue problems can be described as 

follows: 1) keep mass constant and change fatigue properties or 2) keep fatigue 

properties constant and change mass. Addressing these problems will be the focus 

of future work.
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10. References

1. Gao W, Zhang Y, Ramanujan D, Ramani K, Chen Y, Williams CB, Wang CC, 

Shin YC, Zhang S, Zavattieri PD. The status, challenges, and future of additive 

manufacturing in engineering. Comp Aided Des. 2015;69:65–89.

2. Qian X. Undercut and overhang angle control in topology optimization: a 

density gradient based integral approach. Int J Num Meth Eng.

3. Guo, X, Zhou J, Zhang W, Du Z, Liu C, Liu Y. Self-supporting structure 

design in additive manufacturing through explicit topology optimization. 

Comp Meth App Mech Eng. 2017.

4. Langelaar M. Topology optimization of 3D self-supporting structures for 

additive manufacturing. Add Mfg. 2016;12:60–70.

5. Langelaar M. An additive manufacturing filter for topology optimization of 

print-ready designs. Struct Multidiscip Opt. 2016:1–13.

6. Gaynor AT, Guest JK. Topology optimization considering overhang 

constraints: Eliminating sacrificial support material in additive manufacturing 

through design. Struct Multidiscip Opt. 2016;54.5:1157–1172.

7. Mirzendehdel AM, Krishnan S. Support structure constrained topology 

optimization for additive manufacturing. Comp Aided Des. 2016;54.5:1–13.

8. Li Q, Chen W, Liu S, Tong L. Structural topology optimization considering 

connectivity constraint. Struct Multidiscip Opt. 2016;54(4):971–984.

9. Wu J, Wang CC, Zhang X, Westermann R. Self-supporting rhombic infill 

structures for additive manufacturing. Comp Aided Des. 2016;80:32–42.

10. Fernandez-Vicente M, Canyada M, Conejero A. Identifying limitations for 

design for manufacturing with desktop FFF 3D printers. Int J Rapid Mfg. 

2015;5.1:116–128.

11. Papadrakakis M, Papadopoulos V, Stefanou G, Plevris V. Topology 

optimization for additive manufacturing with controllable support structure 

costs.

12. Guest JK, Prevost JH, Belytschko T. Achieving minimum length scale in 

topology optimization using nodal design variables and projection functions. 

Int J Num Meth Eng 2004;61.2:238–254.

13. Bendsoe M. Optimal shape design as a material distribution problem. Struct 

Opt. 2001;1(4):193–202.
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14. Stolpe M, Svanberg K. An alternative interpolation scheme for minimum 

compliance topology optimization. Struct Multidiscip Opt. 2001;22(2):116–

124.

15. Zhou M, Rozvany G. The COC algorithm, part II: topological, geometrical and 

generalized shape optimization. Comp Meth Appl Mech Eng. 1991;89(1–

3):309–336.

16. Guest JK. Optimizing the layout of discrete objects in structures and materials: 

a projection-based topology optimization approach. Comp Meth Appl Mech 

Eng. 2015;283:330–351.

17. Holmberg E, Torstenfelt B, Klarbring A. Stress constrained topology 

optimization. Struct Multidiscip Opt. 2013;48.1:33–47.

18. Matlab. R2016a. 2016. [accessed June 2017] 

https://www.mathworks.com/products/matlab.html.

19. Schijve J. Fatigue of structures and materials. Dordrecht (NL): Kluwer 

Academic; 2001.
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