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Topology optimization through stiffness/weight ratio analysis for a three-

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COTech

IOP Conf. Series: Materials Science and Engineering 700 (2019) 012012

IOP Publishing

doi:10.1088/1757-899X/700/1/012012

1

Topology optimization through stiffness/weight ratio analysis

for a three-point bending test of additive manufactured parts

A A Garcia-Granada1, *, J Catafal-Pedragosa1,2 and H G Lemu2

1 GEPI-IQS Grup Enginyeria Producte Industrial, Universitat Ramon Llull, Via

Augusta, 390. 08017, Barcelona, Spain.

2 Department of Mechanical and Structural Engineering and Materials Science,

University of Stavanger, Norway.

* Corresponding author: andres.garcia@iqs.url.edu

Abstract. Topology Optimization (TO) is a technique that allows for increasingly efficient

designs and its objective is to maximize the performance of mechanical systems or structure in

a variety of fields. Attempts to employ TO for parts manufactured with conventional methods

such as casting, forging, injection moulding, CNC machining and the like could not lead to

desired optimum results due to the existing manufacturing constraints regarding geometrical

complexity. Currently, additive manufacturing (AM) techniques allow the fabrication of more

complex shapes which in principle will lead to improved performances through application of

the TO concept. This study focuses on structural optimization of additive manufactured parts of

thermoplastic parts based on analysis of the stiffness/weight (mass) ratio for a beam subjected to

a three-point bending load. The experimental work is done on optimization of parts manufactured

by Fused Deposition Modelling (FDM) technology and finally compared with an identical model

manufactured using Polyjet 3D printer. Different TO software are compared to conduct the

optimization, and a module of SolidWorks 2018 from Dassault Systems is chosen for the

topology optimization for the final experiment. The study focuses on the results on stiffness/mass

ratios, paying attention to the influence of different printing parameters on the test results. An

increase of stiffness/weight ratio of 31.7% was predicted by software while experiments showed

an increase of just 27.04%.

1. Introduction

Topology optimization is a type of structural optimization that is used as a tool or technique in areas that

need reduction of weight in a component by optimal distribution of mass (weight) and hence leads to

improved stiffness to weight ratio. The optimization technique produces shapes by removal of materials

from regions where the component shows low levels of stress under loading conditions. Such shapes

can be complex and hence difficult to manufacture using the traditional manufacturing techniques. With

the current opportunities provided by additive manufacturing, however, the difficulty of fabrication of

complex shapes is not an issue, and hence topology optimized design can be realized. As additive

manufacturing is a technology that is not yet fully mature, it requires detailed studies of several aspects

to understand the material behavior under loading conditions. In this introduction, a brief literature

review is reported to understand the previous work on material characterization of additive

manufactured parts and also on topology optimization.

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In order to perform a proper topology optimization, the optimization software needs to rely on

material properties verified for each additive manufacturing strategy. In Akessa et al. [1], for instance,

a study was conducted to characterize the mechanical properties of ABS-M30 material using rectangular

samples that were subjected to 3-point bending tests. The samples were manufactured by varying the air

gap, raster width and raster angle. A similar study was reported by Gebisa and Lemu [2] in which the

effects of varying the FDM process parameters on the flexural properties of ULTEM 9085 were

investigated. The objective of this study was to consider all possible combinations of parameters; air

gap, raster width, raster angle, contour number and contour width.

According to other recent studies carried out by Domingo-Espin et al. [3], anisotropic material

properties should be considered when using FEA simulation of FDM parts exceeding the elastic region

limit. From this study, conclusions were achieved using tests and simulations of an “L” shaped cantilever

beam with bending and torsion for polycarbonate materials. Furthermore, dynamic properties were

studied as described by Domingo-Espin et al. [4], where a simple prismatic part was loaded using a

dynamic mechanical analysis (DMA). Results showed that the building parameters, namely nozzle

diameter, number of contours and distance between rasters can control the elastic behavior of the FDM

manufactured part, being the number of contours the most influential parameter. Test parameters, such

as amplitude, frequency and temperature, showed a great influence on the damping capacity of the part.

Creep behaviour of polycarbonate (PC) parts manufactured using FDM process were studied by

Salazar-Martín et al. [5] using experimental method focusing on the effect of three process parameters:

(1) part build orientation, (2) raster to raster air gap, and (3) number of contours. The study was

conducted on the primary and secondary creep behaviour. It was found that increasing the density of the

sample, by increasing number of countours and reducing air gap, causes creep strain to decrease. The

study also shows the significance of arranging the deposited filaments in the same direction the sample

is loaded. The influence of FDM manufacturing parameters on mode I fracture properties has been

recently studied by Sedigi et al. [6] to explore how a part can hold deformation energy beyond elastic

limits, taking into account plasticity and crack locations.

Other additive manufacturing techniques have been studied to understand the influence of

manufacturing parameters on material properties. For example, Morales-Planas et al. [7] studied the

influence of different manufacturing parameters such as part orientation on mechanical properties of

Multi Jet Fusion PA12 focused on achieving the right design for watertightness, strength and tolerances.

Once material properties are well studied, a topology optimization can be performed taking into account

these values and a literature review on topology optimization is provided.

A review on topology optimization was performed by Hassani et al. [8] already in year 1998 and is

continuously reviewed due to the growth of software and hardware developments. For example,

Campbell et al. [9] provided a review of numerical optimization techniques for meta-device design for

optical materials. In this paper, the literature review focused on lightweight design considering topology

optimization needs for the best stiffness to weight ratio. Gebisa and Lemu [10] reported a case study on

topology optimized design for additive manufacturing. An engine bracket was topologically redesigned

to reduce its weight considering fabrication in AM. The study results show that topology optimization

is a powerful technique to reduce the weight of a structural product while maintaining the design

requirements if additive manufacturing is considered.

Faskhutdinov et al. [11] reported a study done on the topology optimization of a jet engine part with

Selective Laser Melting (SLM) technology where the process of TO is described. The optimization is

the process of choice of the best option imaginable. That decision is done based on some dependent

values (design data) and a target function. The values of design data are found at which the target

function has a minimum. While there may be a number of targets, one will have to have priority on

others, as not all of them may be compatible. Then, topology optimization allows finding an optimal

material distribution in a given design space under the certain loads and boundary conditions. Recently,

Wang et al. [12] worked on lightweight design for robots by integrating topology optimization and

parametric system optimization using TOSCA software. Their target was to maintain the deformations

of the end-effector of a serial painting robot reducing the mass of components.

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The objective of the work reported in this article is to conduct topology optimization of 3D printed

parts of thermoplastic materials for improved stiffness to mass ratio using both experimental test (3-

point bending test) and optimization software.

2. Materials and Methods

2.1. Three-point bending test

A three-point bending test is defined as a starting point for the topology optimization reported herein.

This is because tensile tests are not adequate as they provide a uniform stress distribution across the

section of the specimen and therefore the optimization is limited to a reduction of cross section. Three-

point bending test is relatively simple to conduct in a common laboratory facility. It creates different

stress values across the thickness of the specimen and provides room for topology optimization.

In this study, the stiffness to mass ratio is used as a parameter of optimization. Stiffness for a constant

section specimen of a simple supported beam under transverse load is theoretically calculated as follows

based on pure bending:

=

=48

3 (1)

where k is stiffness, F is applied force, y is displacement in loading direction, E is Young modulus,

I is section inertia and L is span length. For the same beam loading, maximum stress (

σ

) is obtained at

the middle of the specimen and theoretically expressed as follows:

=ℎ

8 (2)

where h is section height of the specimen.

Theoretical equation for stress does not include stress concentrations near supports but test is defined

to obtain fracture on lower parts of specimen. Therefore, in classical engineering optimization,

increasing inertia will improve stiffness and at the same time reduce maximum stress avoiding plasticity

and fracture with the exception of stress concentration effects if new shapes involve sharp edges.

2.2. Simulation software for topology optimization

Software for topology optimization is growing very fast together with the rapid increase in

computational speed and hardware capacity. As a result, many modelling and simulation tools are

incorporating topology optimization modules in their software package. In this study, a module of

SolidWorks 2018 from Dassault Systems is chosen for the topology optimization. The software allows

the definition of boundary conditions similar to the three-point bending test setup and provides design

rules to select surfaces that should be defined as design features and non-design features. The design

features are subjected to material removal if not contributing to load sharing while the non-design

features should be kept as they are in initial design, regardless of the stress level acting on them. The

software also allows the definition of other rules such as the optimization criteria. The optimization

criteria in this case is defined based on the best stiffness to mass ratio with a load of 50 N, considering

the material anywhere in the beam remains within the elastic region.

2.3. Additive fabrication and testing machine

Fortus 450mc machine from Stratasys (Figure 1(a)), FDM technology, is chosen to fabricate original

and optimized parts using ABS – M30 material. According to the material data, the density is around

1040 kg/m3, Young’s modulus between 2180 and 2230 MPa and yield strength between 26 and 31 MPa

with elongation at break between 2 and 7%, depending on the orientation of the part (Table 1) [13]. For

model slicing and machine control, the pre-processor software of the machine, Insight® 12.2, was used.

Raster and contour width were set to 0.4064 mm with just one contour and 0 mm air gap between

contours. Two orientations were chosen, flat and edge and for each orientation two angles 0º and 45º

were chosen.

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Optimized parts required support material for overhanging areas. The FDM process of the machine

uses support material referred to as SR20, which dissolves at 70±3 °C and removed from the

manufactured part. The solvent is a water solution with additions of sodium hydroxide and sodium

carbonate. Two sample designs, i.e. rectangular samples and optimized samples were printed and tested.

Four variations with different parameters where designed for each sample, and three specimens for each

variation were tested. In other words, a total of 12 + 12 = 24 ABS specimens were tested.

Table 1. Properties for ABS- M30 and Verowhite (from [13] and [14]).

Density

[kg/m

3

]

Young’s modulus

[MPa]

Yield stress

[MPa]

Elongation

at break [%]

ABS-M30

1040

2180-2230

26-31

2-7

Verowhite

1170

2000-3000

50-65

10-25

In order to compare the results with another printing technology, the identical designs were

manufactured using an Objet30 Prime machine from Stratasys (Figure 1 (b)) which is based on Polyjet

technology. This machine cures acrylic liquid by using ultraviolet lamps which can provide high

accuracy with layers of 0.015 mm. The material used for this study was VeroWhite [14]. According to

the material manufacturer, this material has the material properties given in Table 1. A total of 10 + 10

= 20 samples from Verowhite material were manufactured with the same manufacturing conditions to

check repeatability. Finally specimens where tested using three-point bending test on an Instron 5985

(Figure 1 (c)) where displacement and force where recorded. Span length for the beam supports was set

to 100 mm with a velocity of 1mm/min.

Figure 1. Machines used for additive manufacturing machine and testing (a) Fortus 450mc

FDM machine (b) Objet 30 Prime Polyject machine and (c) Instron 5985 tensile test machine.

3. Results and discussion

The original geometry considered for this project is shown in Figure 2, where two small extensions were

provided to a rectangular beam to avoid falling from the three-point bending supports. Theoretical

calculation are provided with equation (1) and equation (2) ignoring these extensions. From theoretical

calculations, the following values are obtained: mass, m = 20.8 g, stiffness, k = 640 N/mm (at E = 2000

MPa) and stress, σ = 1.875 MPa (for F = 50 N), which is below 26 MPa for the lowest yield stress and

without considering stress concentration effects.

(a)

(b)

(c)

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Figure 2. Original geometry to be optimized for a three point bend.

Once the samples are defined, topology optimization was performed to achieve an improvement in

the stiffness/mass ratio. Figure 3 (a) shows the optimized model, i.e. after removing unwanted material

but with sharp edges. Then, a smooth optimized geometry is generated (Figure 3 (b)) upon generating

soft cure transitions. In this optimization process, the mass is reduced from 20.8 g down to 9.1 g. Since

the section is not constant or of regular shape, theoretical calculations are not possible for stiffness and

strength.

Figure 3. Optimized part (a) with rough surfaces as optimized and (b) smoothed surface

geometry created base on optimized shape.

The original and the optimized parts are then manufactured with both machines. Figure 4 (a) shows

the specimen manufactured using ABS-M30 material and SR20 support. After the support material is

removed, the specimen is three-point bending tested as illustrated in Figure 4 (b).

Figure 4. Optimized part (a) as manufactured with material support material and (b) under

three-point bending test.

The same procedure was repeated for the specimens fabricated from Verowhite material. Object30

Prime allows fabrication of parts which are closer to CAD geometry as it allows much smaller layer

thickness. This means, it is possible to get a smoother surface in this case compared with that of Fortus

450mc machine. As shown in Figure 5(a), the fracture started from the top connection between front

and bottom face but low friction led to sliding of the test sample (Figure 5(b)) with conditions dissimilar

to the case in topology optimization, which is defined with ideal conditions where the load is placed in

the middle and lower supports are always in the same place. Figures 6(a) and (b) show von Mises stress

distribution for the original design and the optimized shape, respectively. In both figures, stress is much

higher at the load point (middle of the beam) and at the support points (locations of boundary

conditions), thus attention is paid to points in the lower part of the middle section of the specimen where

fracture is expected to happen.

(a)

(b)

(a)

(b)

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Figure 5. Optimized structure tested with unstable sliding support.

Figure 6. von Mises stress for 1 mm displacement for (a) original shape and (b) optimized shape.

Finally, Figure 7 shows results for all ABS-M30 tests for both original and optimized designs for

each fabrication condition. The difference in stiffness to mass ratio for all orientations ranged from

18590 to 20830 N/m/g for original part and from 23460 to 26960 N/m/g for optimized parts. Comparison

of all scenarios is provided in Table 2.

Figure 7. Test results for several printing directions for Fortus 450mc.

Analyzing all scenarios, it can be observed that the increase in stiffness to mass (ky /m) ratio varies

from 12.61% increase in the worst case to 45.03% in the best case. The achieved average increase in

stiffness to mass ratio is 27.04%. This is very important since the layer orientation of real complex 3D

components cannot be controlled for each location and therefore it is important to expect that

improvement is achieved for any combination of orientations. Table 3 gives summarized values of the

average mass, loads, displacements and stiffness.

0

500

1000

1500

2000

2500

0 2 4 6 8 10 12 14 16

Load (N)

Displacement (mm)

3-point Bending Test

(a)

(b)

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Table 2. Stiffness comparison of ABS-M30 for different scenarios.

Stiffness to mass (ky /m) ratio (N/m/g)

Average

value

Lowest

value

Highest

value

Worst case

optimization

Best case

optimization

Rectangular

1.975E+04

1.859E+04

2.083E+04

2.083E+04

1.859E+04

Optimized

2.509E+04

2.346E+04

2.696E+04

2.346E+04

2.696E+04

k

y

/m increase after

optimization

27.04%

26.19%

29.43%

12.61%

45.03%

Table 3. Average values for each sample design.

Mass

(g)

Max. load

(N)

Displ. at max.

load (mm)

Load at 1 mm

displ. (N)

k

y

(N/m)

k

y

/m

(N/m/g)

Rectangular

19.71

1847

10.06

389.2

3.892E+05

1.975E+04

Optimized

8.87

906

3.43

222.6

2.226E+05

2.509E+04

Furthermore, fracture is also analyzed for each specimen and fabrication orientation. Figure 8 shows

fractured parts from edge fabrication ((a) and (b)) and flat fabrication ((c) and (d)). For each sample

shown, the upper part is 0º and the lower part is 45º. Original parts are shown on the left and optimized

parts on the right. Fracture is complete for flat orientations while fracture is produced on the bottom of

the specimen (as expected) for edge manufactured components.

Figure 8. Fracture of original samples (left a and c) and optimized samples (right b and d).

Results are compared with predictions made by optimization software. In the simulations for

optimization, a Young´s modulus of 1526 MPa was used for a better fitting to test data. 3D tetrahedral

elements of side length 1 mm were used to obtain 12 elements per each mm3 with 4 Jacobian points for

each element. Therefore, for the original shape a total of 130087 nodes and 187218 elements were used

to be eliminated during the optimization process and to define the light weight structure. However, the

importance of optimization simulations is the reduction of weight using reliable material data. As given

(a)

(b)

(c)

(d)

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in Table 2, this varies depending on the material orientation. Comparison of the stress results of the

optimized model with the original model is given in Table 4. This comparison shows that optimization

software (FEA) resulted a 31.7% improvement in stiffness to mass ratio (with the same Young’s

modulus) while average results for the 3-point bending tests gave a 27.04% improvement. These

percentage improvements between the simulation approach and experimental test approach have no

significant difference.

Table 4. Comparison of FEA simulations with tests.

Stiffness to mass (ky /m) ratio (N/m/g)

FEA

(E =1526 MPa)

3-point bending

test (average)

Difference

Rectangular 2E+04 1.975E+04 1.25%

Optimized

2.634E+04

2.509E+04

4.75%

Improvement of ky /m

after optimization 31.7% 27.04%

The work was repeated using another printer having better resolution. In this case, 10 samples for

original design and 10 samples for optimized design were manufactured using Verowhite material and

an Object30 Prime printer. For the last samples of Verowhite material, the support was not mechanically

removed after solvent exposition for 24 hours in order to avoid possible damage to the structure. The

weight of components is shown in Figure 9. Average values of each printing direction are taken for

ABS-M30 samples. The weight of the Verowhite original design is very close to the computer aided

design weight of 24.10 g considering density of 1170 kg/m3 while ABS-M30 designs do not reach the

expected density due to accuracy in layers and trajectories.

Figure 9. Mass comparison for Verowhite and ABS-M30.

0

5

10

15

20

25

30

12345678910

Mass (g)

Verowhite-original Verowhite-optimized

ABS30-original ABS30-optimized

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On the other hand, Verowhite optimized shapes do not show any improvements in the stiffness to

mass ratios as shown in Figure 10. A possible reason for this can be the sling of the Verowhite

components that place the load in a point far from the center of the specimen due to a surface that looks

like a polished surface. If focused on the original structures, the curves are very similar for ABS-M30

and Verowhite if the weights are normalized. The ABS 30 model, however, has shown improvement in

stiffness to mass ratio. In both cases, for ABS-M30 and Verowhite, the optimized shape can take less

deformation and less energy when compared to original designs.

Figure 10. Comparison Verowhite to ABS-M30.

4. Conclusions

In this paper, a study focusing on topology optimization of additive manufactured part using Fortus

450mc (FDM technology) and Polyjet Object30 Primeis machines is reported. The selected part is tested

using three-point bending test. The optimization targets maximum (improved) stiffness to mass ratio. It

is observed that the obtained stiffness to mass ratio varies with printing orientation. For the ABS-M30

FDM manufactured parts, the topology optimized part under three-point bending test resulted in an

average improvement in stiffness to mass ratio of 27.04%, worst case improvement of 12.61% and best

case improvement of 45.03%. Similar design geometry was used to compare the optimized part performs

by using Verowhite material printed in a Polyjet Objet30 Prime machine. It is observed that the part

manufactured by Polyjet Objet30 Prime machine gives better accurate geometry compared with ABS-

M30 using FDM Fortus 450mc. The part manufactured using Verowhite showed no improvement in

stiffness to mass ratio. One reason for this lack of improvement is the glossy surface of the manufactured

part when the support material is removed. The sliding of the test specimen under the three-point

bending supports due to the glossy surface led to inaccuracies in the test results. It has also been observed

that the optimized shapes provide a lower fracture displacement and absorb less energy when compared

to original shapes.

Acknowledgements

This work has been performed within the Ris3Cat PlastFun (COMRDI16-1-0018) project (Plastic with

functionalized surfaces), funded by ERDF through the Programa Operatiu de Catalunya 2014-2020.

-20

0

20

40

60

80

100

120

-2 0 2 4 6 8 10 12 14 16 18

Normalised force (N/g)

Displacement (mm)

Verowhite-original1 Verowhite-optimized1

ABS-M30-original1 ABS-M30-optimized1

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