topology optimization - rwth aachen university...topology optimization has experienced considerable...

Seminar Thesis

Topology Optimization

using the SIMP method

Submitted by

Daniel Löwen

Supervisor

Dipl.-Ing. Anna-Lena Beger

Chair and Institute for Engineering Design

RWTH Aachen University

Aachen, 17. November 2016

This paper was presented at the

Center for Computational Engineering Science

- mathematics division - RWTH Aachen

Prof. Dr. Manuel Torrilhon


RWTH Aachen – Univ.-Prof. Dr.-Ing. Georg Jacobs 2

Abstract

Topology optimization has experienced considerable publicity and growth in the past few dec-

ades with many successful implementations especially in the aviation and automotive sector.

This thesis presents an introduction to topology optimization as a structural optimization tool,

showing the underlying theory and considerations. In a case study, an iPad stand, a common

consumer product, is developed using topology optimization and a prototype is produced with

additive manufacturing. The major benefits and constraints of additive manufacturing are ana-

lyzed with a focus on topology optimization, yielding a new set of opportunities and drawbacks

for the integration of additive manufacturing and topology optimization as a current field of

study, and revealing significant potential for numerous future applications.



Table of contents

1 Introduction ........................................................................................................................ 7

2 Theory ................................................................................................................................. 8

Structural Optimization Methods ................................................................................ 8

2.1.1 Sizing Optimization .................................................................................... 8

2.1.2 Shape Optimization .................................................................................... 8

Topology Optimization ................................................................................................ 9

2.2.1 Topology .................................................................................................... 9

2.2.2 Discrete Formulation ................................................................................ 10

2.2.3 SIMP ........................................................................................................ 11

2.2.4 Process .................................................................................................... 12

2.2.5 Regularization .......................................................................................... 12

2.2.6 Physical Significance of the SIMP Method ............................................... 14

3 Case Study ....................................................................................................................... 15

Used Tools ............................................................................................................... 15

Case Modeling ......................................................................................................... 15

Topology Optimization .............................................................................................. 16

Results Analysis ....................................................................................................... 17

3.4.1 Volume Fraction Study ............................................................................. 17

3.4.2 Feature Thickness Study ......................................................................... 18

Conclusion ................................................................................................................ 19

4 Additive Manufacturing ................................................................................................... 20

Introduction ............................................................................................................... 20

4.1.1 Material Properties ................................................................................... 20

4.1.2 Support Structures ................................................................................... 20

Additive Manufacturing and Topology Optimization ................................................. 21

4.2.1 Case Study Prototype .............................................................................. 21

5 Conclusion and Outlook ................................................................................................. 22



Table of figures

Figure 1: Use case example for a topology optimized part (Swerea 2009) ...................... 7

Figure 2: Different methods of structural optimization (Bendsoe 2004) ........................... 8

Figure 3: Set of equal forms ............................................................................................. 9

Figure 4: Circle-point topologies in different spaces ........................................................ 9

Figure 5: Removal of one point in a topology ................................................................. 10

Figure 6: Penalization factor graphs (Günther 2014) ..................................................... 11

Figure 7: Iterative process of a topology optimization (cf. Bendsoe 2004) ..................... 12

Figure 8: MBB-Beam with singular load, two supports and a symmetry

condition (Bendsoe 2004) ............................................................................... 12

Figure 9: Setting an Iso-value in post-processing (Bendsoe 2004) ................................ 14

Figure 10: Problem modeling: Load case (top left), iPad Pro properties (top

right), model right view (bottom left), model front view (bottom

middle), model iso view (bottom right) ............................................................ 16

Figure 11: Discretized design domain and parts (right view) ........................................... 16

Figure 12: Result of the topology optimization with different Iso-values .......................... 17

Figure 13: Results of the topology optimization with a target volume fraction of

5% (top left), 20% (top right), 50% (bottom left) and 95% (bottom

right) ................................................................................................................ 17

Figure 14: Results of the topology optimization with different feature thicknesses .......... 18

Figure 15: Areas of qualitative difference ......................................................................... 19

Figure 16: iPad stand comparison to the case study result (Left: Mirco 2017,

Right: Bluelounge 2017) .................................................................................. 19



List of tables

Table 1: Parameter study of topology optimizations for the MBB-Beam setup

with varying filter sizes and mesh refinement, results of the Matlab

implementation (cf. Sigmund 2001) ................................................................ 13



Summary

The paper begins with a motivation for using topology optimization and the introduction of a

basic definition of topology. The theory of topology optimization and the difference to other

structural optimization techniques are shown with a focus on the SIMP method and solutions

to arising issues. The physical significance of the SIMP material model is explained.

A case study is presented with an overview of used tools, the modeling of the problem with

load case and conditions for the entailing topology optimization, followed by a results analysis

and a conclusion. The case study is carried out with the hypothesis of the optimization yielding

an unconventional design that is shown to be the optimum for the given loadcase.

Additive manufacturing methods, benefits and constraints are outlined and analyzed with re-

spect to producing topology optimized parts. The subsequent opportunities but also the draw-

backs of the combination of additive manufacturing and topology optimization are presented.

Promising directions for research and potential fields of application are highlighted in the con-

clusion and outlook.



1 Introduction

Topology optimization is used for many applications including product development, concept

design or part optimization. In the automotive and aviation industry, a vehicle’s or plane’s per-

formance and fuel consumption is affected for instance by rolling resistance and acceleration,

both of which are dependent on mass. As topology optimization results in weight reduction of

parts, the effects are fuel savings, emission reduction and better product performance as the

material is used in a more efficient way (Figure 1).

The conventional development cycle of a product includes the modeling of an idea, mostly

computer aided, manufacturing, and validation by simulation or in a testing facility. Depending

on the validation results the part often must be redesigned and subsequently manufactured

and validated again. These iterations in the product development process lead to long devel-

opment times and high cost. Topology optimization can be utilized as a starting point for prod-

uct development or concept design. Iterations after validation by simulation or testing of a re-

sulting part are reduced or prevented because the part is optimized for the specific load case.

In concept design, product forms can be found by detaching the topology of the part from the

influence of the designers’ or engineers’ ideas. Furthermore, results of topology optimizations

often look very unconventional.

Figure 1: Use case example for a topology optimized part (Swerea 2009)



2 Theory

In this section the topology optimization is differentiated from other structural optimization

methods. The underlying theory is introduced and solutions to emerging issues are pre-

sented. The physical relevance of the material model is demonstrated at the end of the sec-

tion.

Structural Optimization Methods

There are several methods available that can be used to optimize the form or the elements of

a structure while specific constraints are met (Figure 2).

Figure 2: Different methods of structural optimization (Bendsoe 2004)

2.1.1 Sizing Optimization

In sizing optimization, the domain of the design model and the state variables are known a

priori and stay fixed throughout the optimization process.

The sizing optimization of a truss structure as seen in Figure 2, a) takes elements of the

structure as design variables, for instance the diameter of a rod or the thickness of a beam,

with constraints such as equilibrium constraints and optimizes for an objective that can be,

amongst others, mean compliance, peak stress, or deflection (cf. Bendsoe 2004).

2.1.2 Shape Optimization

The goal of shape optimization is finding the optimal shape of a domain with a known topol-

ogy and the domain itself as design variable. This domain is defined by boundary curves or

boundary surfaces and the optimal form of these boundaries is found by shape optimization

(Figure 2, b))




Ever since Bendsoe and Kikuchi presented the topology optimization method in 1988 (Bend-

soe 1988) it has become a popular method for material distribution with several open source

and commercial software packages implementing it as a design tool. It is not yet seen as a

self-sufficient process with results being taken as the final product but rather as a starting

point for any structural design process (cf. Bendsoe 2004).

The goal of topology optimization is “to find the optimal lay-out of a structure within a speci-

fied region (Bendsoe 2004).” Known quantities are applied loads, support conditions, struc-

ture volume and additional design restrictions such as location and size of holes or solid ar-

eas that are exempt from the design domain. However, physical size, shape, and connectiv-

ity of the structure are unknown (Figure 2, c)).

2.2.1 Topology

Topology is the mathematical study of properties that are preserved through invertible defor-

mations, twisting, and stretching of objects. However, tearing or gluing is not allowed. It can

be used to abstract the inherent connectivity of objects, ignoring their detailed form (cf. Weis-

stein 2016).

If two objects have the same topological properties, they are called homeomorphic. All forms

in Figure 3 are homeomorphic with a possible physical representation of a rubber band, that

can be stretched into each of these forms.

Figure 3: Set of equal forms

Topologies are dependent on the space, for instance the two objects in Figure 4 cannot be

continuously transformed into another in 2D as tearing and subsequent gluing would be re-

quired, whereas in 3D in form B one could move the point out of the plane of the circle and

lower it again inside the circle, meaning topological equivalence.

Figure 4: Circle-point topologies in different spaces



On the other hand, a figure eight curve formed by two circles touching each other at a spe-

cific point is topologically distinct from a circle. Removal of a point in the circle preserves con-

nectivity while removing the connection point of the figure eight results in two separate forms

(Figure 5). Here, the number of connected forms is the non-preserved property of the topol-

ogy.

Figure 5: Removal of one point in a topology

2.2.2 Discrete Formulation

The given design domain is discretized for a finite element analysis. For each element in the

discretized design domain the problem is whether there is material or not. The aim of topol-

ogy optimization is to minimize an objective function such as the compliance of the structure:

min𝜌

: Φ(𝜌, 𝑈(𝜌))

Compliance is defined as “the work done by the external loads (Reiss 1976).” In this prob-

lem, the compliance is dependent on the density vector ρ and the displacement vector U(ρ)

out of the finite element analysis and is minimized under a volume constraint:

𝑠. 𝑡. ∶ ∑ 𝑣𝑒𝜌𝑒 = 𝑣𝑇𝜌 ≤ 𝑉∗

𝑁

𝑒=1

where V* is the target volume, a fraction of the initial volume. Instead of a volume constraint,

geometry, stress, or other constraints are possible. The density is set as constant in each el-

ement and can take the values “0” for void and “1” for material:

𝜌𝑒 = { , 𝑒 = 1, … , 𝑁10

This satisfies the equation

𝐾(𝜌)𝑈 = 𝐹

with K being the stiffness matrix, U the displacement vector and F the load vector.



2.2.3 SIMP

The discrete topology optimization problem is ill-posed and very computationally expensive

(cf. Bruns 2005). For that reason, Bendsøe and Kikuchi moved from a discrete formulation to

a gradient based approach, where continuous design variables are introduced. The density,

the design function, is allowed to vary between a small number and one (cf. Bendsoe 1988):

0 < 𝜌𝑚𝑖𝑛 ≤ 𝜌 ≤ 1

The assumption for the relation between the stiffness of an element and the density is that

stiffness is linearly dependent on density. The “power law approach”, also called “Solid Iso-

tropic Material with Penalization” (SIMP) method, uses this assumption:

E(𝜌𝑒) = 𝜌𝑒𝑝

𝐸0 , 𝑝 > 1

Here, 𝐸0 is the stiffness of solid material, E(𝜌𝑒) the stiffness of element e and p is the penali-

zation factor, penalizing the density 𝜌𝑒 in element e. In Figure 6, left, the relation between

densities on the abscissa and the stiffness on the ordinate is shown for increasing penalization

factors. 𝑥𝑖 is the design variable 𝑥𝑖 = (𝜌𝑖

𝜌𝑖0).

For the commonly used penalization factor 𝑝 = 3 the right graph in Figure 6 shows the effect

of the penalization. For an exemplary density of 0.5 a low stiffness value is obtained, thus

having intermediate densities in the optimal design is uneconomical as it does not give much

stiffness for the amount of available material. Subsequently the topology optimization algorithm

will redistribute the material of the design domain such that the solution is penalized towards

0 and 1 as the stiffness values in elements.

Figure 6: Penalization factor graphs (Günther 2014)

For a penalization factor 𝑝 = 1 the problem has a convex form. Therefore, independent from

the starting point, the solution will converge to the global optimum. The approach to finding a

good optimal design that is not only locally optimal is to find the global optimum with 𝑝 = 1

and gradually increase the penalization factor.



2.2.4 Process

A topology optimization follows an iterative process that converges to an optimal design. This

process is shown in Figure 7.

Figure 7: Iterative process of a topology optimization (cf. Bendsoe 2004)

The first step is defining the design domain, load case, boundary conditions and a objective

function. The design domain is then discretized and a finite element analysis is carried out.

The results of the FEA are used in a sensitivity analysis to compute the gradients of the ob-

jective function, for instance the compliance. To resolve the complications of mesh-depend-

ency and checkerboarding, that are further discussed in section 2.2.5, a regularization filter is

applied. After filtering, the material in the design domain is redistributed using the “Method of

Moving Asymptotes” and the design variables in the elements are correspondingly updated.

The topology optimization algorithm iterates until the densities in the elements converge.

2.2.5 Regularization

The previously mentioned complications are demonstrated with the MBB-Beam setup in Fig-

ure 8.

Figure 8: MBB-Beam with singular load, two supports and a symmetry condition (Bendsoe 2004)

𝐾𝑈 = 𝐹

Method of Moving

Asymptotes

no

yes

Initialize FEM

Finite Element Analysis

Sensitivity Analysis

Regularization (filtering)

Optimization

(material redistribution)

Plot results

STOP

Update Design variables

ρe converged?



Table 1: Parameter study of topology optimizations for the MBB-Beam setup with varying filter sizes and

mesh refinement, results of the Matlab implementation (cf. Sigmund 2001)

Mesh-dependency

“The SIMP topology optimization problem […] lacks existence of solutions in its general con-

tinuum setting (Bendsoe 2004: 28-30).” This is due to the fact that the introduction of more

holes at a constant volume will generally increase efficiency of a structure. Subsequently, a

structure does not converge to a global result with mesh refinement (cf. Bendsoe 2004).

Qualitatively different optimal solutions are reached for different mesh-sizes and discretiza-

tions. This dependence of the solution on the refinement of the mesh is illustrated in Table 1,

where a finer mesh results in a more detailed structure, smaller features and more holes.

Bendsoe noted that mesh-refinement should ideally lead to a better finite element modelling

of the same, globally optimal structure and a better boundary description, not a different

structure.

Checkerboarding

Checkerboard patterns are areas where densities alternate between 0 and 1 between neigh-

boring elements and occur in the results of the topology optimizations in the first row of Table

1. This problem mainly appears when optimizing a domain discretized with 4-node Q4 ele-

ments (cf. Bendsoe 2004). These areas have a high stiffness so the optimization problem in-

cludes the checkerboards in the optimal design. However, this stiffness is artificial as mate-

rial only being connected on the edge has no physical stiffness. Thus, these patterns are not

optimal and should be prevented in the optimal solution.



Filtering

Both presented issues are considered numerical instabilities. Filtering of sensitivities or den-

sities is a filter-based method solving mesh-dependency as well as the checkerboard effect.

The method introduces a neighborhood

𝑁𝑒 = {𝑖 | ||𝑥𝑖 − 𝑥𝑒|| ≤ 𝑅}

with a defined radius R and updates the sensitivity (density) of an element as a weighted av-

erage of the sensitivities (densities) of the elements inside the neighborhood of that element:

Sensitivity filtering:

𝜕Φ̃

𝜕𝜌𝑒=

∑ 𝐻(𝑥𝑖)𝜌𝑖𝑖∈𝑁𝑒

𝜕Φ𝜕𝜌𝑖

𝜌𝑒 ∑ 𝐻(𝑥𝑖)𝑖∈𝑁𝑒

Density filtering:

𝐸𝑒(𝜌) = �̃�𝑒𝑝

𝐸0, 𝜌�̃� =∑ 𝐻(𝑥𝑖)𝜌𝑖𝑖∈𝑁𝑒

∑ 𝐻(𝑥𝑖)𝑖∈𝑁𝑒

Depending on the filter size the method yields a length scale for feature sizes in the optimal

design by blurring edges and preventing the occurrence of very small features (cf. Bendsoe

2004). In Table 1 the effects of the filtering can be seen, as increasing the filter radius results

in qualitatively similar optimized structures and the removal of checkerboard patterns.

2.2.6 Physical Significance of the SIMP Method

A common point of criticism has been the physical significance of the stiffness to density re-

lation, the intermediate densities in the SIMP method, as this can seem like an artificial mate-

rial model. Following the transition from a discrete formulation to a gradient based approach

there are no longer sharp boundaries between material and void areas. The greyscaled ele-

ments at the boundary are transformed to a sharp boundary by introducing an Iso-value as a

border for density values (Figure 9). All densities from 0 to the border are set to 0 (void), all

densities from the border to 1 are set to 1 (material). Through variation of the Iso-value, con-

nectivity of the resulting design structure can be ensured. Setting the Iso-value to eliminate

intermediate densities is done in post-processing.

Figure 9: Setting an Iso-value in post-processing (Bendsoe 2004)



3 Case Study

The goal of the case study is optimizing the topology of an iPad Pro 12,9” stand that is later

physically prototyped using additive manufacturing. The research question explores the ben-

efits of using topology optimization as a concept design tool with the hypothesis that the re-

sulting form of the iPad stand is very unconventional. For the case study two parameter stud-

ies are conducted, a variation of the volume fraction and a variation of the smallest feature

thickness.

Used Tools

For the topology optimization, the commercial software “SolidThinking Inspire (Version 2016.2

6160)” is used as it is straightforward to model the given case and sufficient options are avail-

able to carry out all aspects of the planned study. The implemented solver is “OptiStruct”, which

uses the SIMP method for topology optimization.

The printer used for additive manufacturing is a “Ultimaker 2” with the software “Cura (Version

2.1.2)” to prepare the model for printing (Ultimaker 2016).

Case Modeling

The case is modeled with the iPad standing at a 20-degree angle with the weight force of 7N

applied in the center of gravity that is in the middle of the iPad. A touch force, applied orthog-

onal on top of the iPad screen, 11 mm under the top edge, is modeled as a moment of 1,06

Nm around the bottom edge, with the weight force contributing to the moment (Figure 10, top

left). The touch force amounts to 3,8 N as the “Force Touch” sensor included in the iPad’s

touchscreen does not weigh beyond a maximum weight of ~385 g (cf. Defauw 2015).

The design domain is modeled large to not restrict the resulting topology in any way (Figure

10, bottom). The iPad is modeled with the sizes shown in Figure 10, top right, and a ground

plane is modeled larger than the design domain. All contact areas are modeled as glued to-

gether.

The design domain material is set to PLA (Formfutura 2015), for iPad and ground plane the

existing material aluminum is chosen.



Figure 10: Problem modeling: Load case (top left), iPad Pro properties (top right), model right view (bottom

left), model front view (bottom middle), model iso view (bottom right)


Meshing setup and creation in Inspire has no user interaction possibilities. The domain is dis-

cretized using tetrahedral elements (Figure 11).

The optimization target is maximizing stiffness, which is the same as minimizing compliance

as compliance is the reciprocal of stiffness, under the constraint of a given volume fraction.

Two parameter studies are carried out, the first study varying the target volume fraction be-

tween 5% and 95% in 5% steps with the steps 75% and 95% as control points for conver-

gence with a fixed feature thickness of 0,03 m. In the second study the minimum feature

thickness is varied between 0,015 m and 0,06 m at a fixed volume fraction of 20%.

Figure 11: Discretized design domain and parts (right view)



In the case study, all Iso-values are set to 0,5 to ensure comparability of results (Figure 12).

Figure 12: Result of the topology optimization with different Iso-values

Results Analysis

3.4.1 Volume Fraction Study

Figure 13 shows a selection of the results of the topology optimization for the volume fraction

study. Structural connectivity with an Iso-value of 0,5 is lost under a volume fraction of 15%.

Figure 13: Results of the topology optimization with a target volume fraction of 5% (top left), 20% (top right),

50% (bottom left) and 95% (bottom right)



In diagram 1 the compliance of the structure with varying volume fractions is shown. As ex-

pected, compliance increases as the volume fraction decreases. At the beginning the in-

crease is very slowly as the lost material volume has no considerable influence on the com-

pliance of the structure. After a volume fraction of 0,2 the increase rate begins to grow faster

meaning a loss of bearing material. This can be interpreted as the algorithm lacking material

to distribute to loaded areas resulting in high compliance values in the structure.

Diagram 1: Compliance values for different volume fractions

3.4.2 Feature Thickness Study

In section 2.2.5 the regularization for mesh dependencies was introduced, because a control

over the minimum thickness of features in the resulting design was needed. To check the

iPad-stand topology for dependency on feature thickness a study is conducted with the set-

tings given in 3.3. The resulting topologies look qualitatively similar (Figure 14) with smaller

feature size emphasizing certain areas, for instance the front part being more developed with

a curve at the front base (Figure 15, A). The connecting volume (Figure 15, B) is also devel-

oped narrower in height and broader in width. No additional holes are introduced into the to-

pology.

Figure 14: Results of the topology optimization with different feature thicknesses

0

0,0002

0,0004

0,0006

0,0008

0,001

0,0012

0% 20% 40% 60% 80% 100%

Co

mp

lian

ce

[m

/N]

Volume

Volume study - Compliance



Figure 15: Areas of qualitative difference

Topology optimizations for even smaller minimum feature thicknesses could not be carried

out as the optimization time for the 0,015 m thickness already exceeded ten hours.

Conclusion

The goal of the case study, developing an iPad stand with an optimized topology, has been

achieved. As a consumer product, a buyer usually chooses an iPad stand for the design or

additional features. The resulting topology looks, in compliance with the hypothesis, very un-

conventional when compared to existing iPad stands, that tend to be slim, smooth, and mini-

malistic (Figure 16, left and middle). The overall form is bulky with an irregular, rough surface

finish that depends on the Iso-value (Figure 16, right). Post-processing could smooth out the

surface and provide a better surface quality.

In both parameter studies results are qualitatively similar. For concept design this is an un-

satisfying result as designers would rather choose from a selection of different design pro-

posals considering that aesthetics is a deciding factor in the fields of customer products and

product design. This could be achieved by modifying the load case, nevertheless resulting in

a lot of additional work.

Figure 16: iPad stand comparison to the case study result (Left: Mirco 2017, Right: Bluelounge 2017)



4 Additive Manufacturing

In this section a general overview of methods, benefits, and constraints of additive manufac-

turing (AM) is given with the focus on producing topology optimized parts.

Introduction

A vast number of methods and associated materials are presently available ranging from

“Material Extrusion” processing polymers, “Binder Jetting” processing ceramics or “Powder

Bed Fusion” techniques processing metals. There are methods processing biological and

bio-compatible materials or even producing edible items.

Most methods produce a physical object from digital information in ways such as layer-by-

layer, piece-by-piece, or line-by-line without the need of intermediate shaping tools. The

method determines the objects geometry and material properties. Market factors such as

shorter product development cycles and an increasing demand for customized or personal-

ized products can be fulfilled using AM. These products, especially in the prototyping stage,

can be produced in minutes or hours, not weeks. The resources needed for AM are often

fewer, as is the need for special training (cf. Thompson 2016).

When using AM, various constraints must be considered regarding build and surface quality,

material property control and production cost, that also depend on the used AM method, ma-

terial, and machine capabilities. A cost analysis for AM is not carried out in this thesis.

4.1.1 Material Properties

Part quality and consistency depends on the chosen method. The boundaries between lay-

ers in AM parts are rarely seamless due to creating new material on already existing mate-

rial. Often, material properties resulting out of AM processes are anisotropic. This issue can

be addressed by modifying part orientation while printing to minimize the effect or post pro-

cessing the part. The characterization of mechanical and optical material properties and the

verification of internal features are not yet fully resolved issues with AM (cf. Thompson 2016).

4.1.2 Support Structures

During manufacturing, the part must be able to resist forces such as gravitational loads or in-

ternal forces from thermal and residual stresses. Sufficient resistance can be obtained by ori-

enting the part or by adding support structures that support sections of the part with for in-

stance an overhang structure, if the part is not self-supporting. Removal of support structures

must be considered beforehand as it may be impossible in internal voids (cf. Thompson

2016).



Additive Manufacturing and Topology Optimization

The typical workflow to produce a part by AM starts with a digital model that is prepared for

printing, the resulting digital model of a topology optimization can be considered as this start-

ing point. As opposed to conventional manufacturing methods, AM is able to produce most

parts without the need of extensive post-processing, if the constraints of AM such as the criti-

cal angle of an overhang structure are met.

Advantages

Using topology optimization and AM, the product development cycle can be reduced signifi-

cantly leading to cost savings and lower risk. The regular thinking of designers and engineers

imposed by conventional manufacturing methods and past expertise is overcome by a de-

sign proposal optimized for a specific use case, which can lead to better product perfor-

mance.

As seen in section 2.2.6, micro-structures can be designed to give the SIMP method of topol-

ogy optimization physical significance and manufactured considering that AM gives internal

geometrical freedom in parts. Alternatively, sponge-like material can also be produced by AM

leaving air filled pores in the material. It would be possible to reduce the penalization factor of

the SIMP method to converge to the global optimum while assigning densities to micro-struc-

tures or sponge-densities, meaning that optimal designs are producible. Similarly, it can be

achieved using different materials with disparate properties in the same part.

Drawbacks

Topology optimized parts often include small, overhanging features that prevent AM without

support structures. It is possible that the support structures, especially with internal voids,

cannot be removed. The manufacturing restriction of a critical angle must be included into

the topology optimization algorithm.

The properties of materials produced by AM processes must be considered as they can af-

fect the final part. Optimization carried out with an isotropic material model will not result in

the same material properties when manufactured with an AM process that produces aniso-

tropic structures.

4.2.1 Case Study Prototype

One of the hypothesis for the combination of methods is the manufacturing of topology opti-

mized parts without the need for post-processing. To show this, the iPad stand prototype is

exported as a STL file in the condition just after optimization with an Iso-value of 0.5. This file

is then imported into Cura and scaled down to 20% due to the build volume of the Ultimaker

2 restricting a print of the model at full size. Support structures under the middle connection

part must be added to account for the overhang. The model is printed using Formfutura

EasyFil™ PLA (Formfutura 2015) and is shown in Figure 16, right.



5 Conclusion and Outlook

This thesis has demonstrated topology optimization to be a powerful tool for designers and

engineers. The method can be used as a design tool in product development, resulting in un-

conventional designs that are detached from the influence of the user and his previous expe-

riences. It is implemented in several open source and commercial software packages, mak-

ing it available to designers and engineers for numerous applications.

The integration of topology optimization and additive manufacturing provides new and prom-

ising opportunities for product development. The method could be used in highly demanding

environments such as aviation or automotive, providing a method of weight reduction and

producing better performing parts. A small overview of possible uses in other fields includes

medical implants or prosthetics, home furnishing or product casings. There are numerous re-

search opportunities in this field of study showing great potential to boost the growth of both

methods.



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