Topology Optimization of Fatigue-Constrained Structures805142/FULLTEXT01.pdf · TRITA-MAT-A 2015:04 ISRN KTH/MAT/A-15/04-SE ISBN 978-91-7595-509-4 Department of Mathematics Royal

Topology Optimization of Fatigue-ConstrainedStructures

HENRIK SVÄRD

Doctoral ThesisStockholm, Sweden 2015

TRITA-MAT-A 2015:04ISRN KTH/MAT/A-15/04-SEISBN 978-91-7595-509-4

Department of MathematicsRoyal Institute of Technology

SE-100 44 Stockholm, Sweden

Akademisk avhandling som med tillstånd av Kungl Tekniska Högskolan framläg-ges till offentlig granskning för avläggande av teknologie doktorsexamen, fredagenden 22 maj 2015 klockan 14.00 i rum F3, Lindstedtsvägen 26, Kungl TekniskaHögskolan, Stockholm.

© Henrik Svärd, 2015

Print: Universitetsservice US AB, Stockholm, 2015

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Abstract

Fatigue, or failure of material due to repeated cyclic loading, is one of themost common causes of mechanical failures. The risk of fatigue in a loadcarrying component is often lowered by adding material, thereby reducingstresses. This increases the component weight, reducing the performance ofthe component and increasing its manufacturing cost. There is thus a need todesign components to be as light as possible, while keeping the risk of fatigueat a low enough level, i.e. there is a need for optimization of the componentsubject to fatigue constraints.

This thesis deals with design against fatigue using topology optimiza-tion, which is a form of structural optimization where an optimal design issought by using mathematical programming to decide which parts of a designdomain should be filled with material, and which should not.

To predict fatigue, accurate representation of the geometry and accuratestress computation are of utmost importance. In this thesis, methods for im-posing constraints such as minimum inner radii and minimum member sizesin the form of four new density filters are proposed. The filters are able togenerate a very sharp representation of the structural boundary. A methodfor improving the accuracy of stress results at the structural boundary is alsoproposed, based on extrapolation of results from the interior of the structure.The method gives more accurate stresses, which affects the resulting struc-tures when solving optimization problems.

A formulation for fatigue constraints in topology optimization is pro-posed, based on the weakest link integral. The formulation avoids the prob-lem of choosing between accurate but costly local constraints, and efficientbut approximate aggregated constraints, and gives a theoretical motivationfor using expressions similar to the p-norm of stresses.

For verifying calculations of the fatigue probability of an optimized struc-ture, critical plane criteria are commonly used. A new method for evaluatingsuch criteria using optimization methods is proposed, and is proved to giveresults within a user given error tolerance. It is shown that compared to exist-ing brute force methods, the proposed method evaluates significantly fewerplanes in the search of the critical one.

Keywords: topology optimization, fatigue constraints, stress constraints,density filters, restriction methods, weakest link theory, critical plane criteria.

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Sammanfattning

Utmattning, eller brott på grund av upprepad cyklisk belastning, är en avde vanligaste felorsakerna i mekaniska komponenter. Risken för utmattningi lastbärande komponenter kan ofta reduceras genom att mängden materialökas, vilket leder till lägre spänningar i strukturen. Den ökade mängden ma-terial leder dock till ökad vikt, och därmed försämrade produktegenskaperoch ökad produktionskostnad. Det finns därför ett behov av att konstruerakomponenter med så låg vikt som möjligt, samtidigt som risken för utmatt-ningsbrott bibehålls på en tillräckligt låg nivå. Med andra ord finns ett behovav optimering av komponenten med utmattningsbivillkor.

Denna avhandling behandlar konstruktion mot utmattning med hjälp avtopologioptimering, vilket är en form av strukturoptimering där en optimalkonstruktion söks genom att använda matematiska metoder för att bestämavilka delar av en konstruktionsdomän som skall fyllas med material, respek-tive inte.

För att konstruera mot utmattning krävs en noggrann representation avgeometrin, och noggrann spänningsutvärdering. I denna avhandling presen-teras metoder för att påtvinga geometriska bivillkor såsom minsta inre radieoch minsta strukturstorlek, i form av fyra nya densitetsfilter. Filtren kan ge-nerera mycket tydliga strukturella ränder. En metod för förbättring av nog-grannheten i spänningsutvärdering vid strukturens rand föreslås också, sombaseras på extrapolation av resultat från strukturens inre. Metoden ökar nog-grannheten i spänningsutvärderingen, vilket påverkar de resulterande struk-turerna vid lösning av optimeringsproblem.

En formulering för utmattningsbivillkor i topologioptimering föreslås,baserad på weakest link-integralen. Formuleringen gör att problemet med attvälja mellan noggranna men kostsamma lokala bivillkor, och beräkningsef-fektiva men approximativa globala bivillkor undviks, och ger en teoretiskmotivering för användande av uttryck såsom p-normen av spänning.

För detaljerad utvärdering av risken för utmattning är det vanligt att an-vända så kallade kritiskt plan-kriterier. En ny metod för utvärdering av såda-na kriterier föreslås. Metoden ger resultat inom en tolerans, som sätts fritt avanvändaren. Det visas att jämfört med tidigare föreslagna metoder utvärderarden nya avsevärt färre plan, i sökandet efter det kritiska.

Nyckelord: topologioptimering, utmattningsbivillkor, spänningsbivillkor, den-sitetsfilter, restriktionsmetoder, weakest link-teori, kritiskt plan-kriterier.

Acknowledgments

This thesis is the result of research carried out at the Department of Mathematics ofKTH Royal Institute of Technology, and Scania between January 2011 and March2015. A number of people have contributed in different ways to making these fouryears memorable. Space constraints prohibit me from thanking all of you in person,however, there are a few persons whose contributions have been significant, and towhom I would like to express my gratitude.

I would like to start by thanking my advisor, Professor Krister Svanberg, for allthe good advice, insightful comments and shared mathematical wisdom. Workingwith you has been a privilege.

I also wish to thank past and present members of the steering group for theproject. Mats Danielsson, without your hard work with applications for funding,and your talks about the merits of pursuing a doctoral degree, this thesis wouldnot have been. Christer Olsson, I am convinced that there are few persons whoare more knowledgeable of industrially applied fatigue and reliability. You are asource of inspiration when it comes to presenting deep knowledge in a pedagogicalmanner. Mårten Olsson, my co-advisor, meetings with you were always fun andthought-inspiring. Fredrik Reuterswärd, Joakim Rodebäck, Ola Sandström, who,acting as managers at Scania, have given me time and space to do research. MikaelThellner, your role not only as a project leader but as a mentor and guide in theacademic world has been much appreciated.

I wish to thank Krister Svanberg, Christer Olsson, Mikael Thellner, FredrikReuterswärd and Mårten Olsson, for the time you spent with proofreading thisthesis, and giving feedback and constructive critique.

The research presented in this thesis was jointly funded by Scania and by theSwedish Research Council, through the project grant no. 2010-4172, and I herebyexpress my gratitude for the opportunities that this funding has provided.

I wish to say thanks to all colleagues, both at the base engine department ofScania, and at the Division of Optimization and Systems Theory at KTH, who have

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viii ACKNOWLEDGMENTS

contributed to two positive work environments. In particular, I wish to thank myroom mates at KTH through the years, Albin Fredriksson, Rasmus Bokrantz, Lo-visa Engberg and Michelle Böck, it always brightened my day whenever our officehours coincided. Along with Johan Markdahl, Anders Möller, Hildur Æsa Odds-dóttir, Tove Odland, Axel Ringh, Göran Svensson, Johan Thunberg, and YuechengYang, you made work on courses more joyful.

Slutligen vill jag tacka min familj. Mamma, Pappa och Mia, tack för att nialltid finns där för mig. Lovisa och Märta, tack för att ni finns i mitt liv.

Stockholm, March 2015

Henrik Svärd

Table of Contents

Abstract v

Acknowledgments vii

Table of Contents ix

Introduction 11 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Design against fatigue . . . . . . . . . . . . . . . . . . . 62.2 Material data . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Local and non-local fatigue models . . . . . . . . . . . . 82.4 Critical plane criteria . . . . . . . . . . . . . . . . . . . . 10

3 Topology optimization . . . . . . . . . . . . . . . . . . . . . . . 123.1 Formulation of an optimization problem . . . . . . . . . . 133.2 The relaxation/penalization approach . . . . . . . . . . . 143.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . 163.4 Solving the optimization problem . . . . . . . . . . . . . 193.5 Stress constraints . . . . . . . . . . . . . . . . . . . . . . 203.6 Fatigue constraints . . . . . . . . . . . . . . . . . . . . . 24

4 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . 255 Suggested future work . . . . . . . . . . . . . . . . . . . . . . . 296 Summary of the appended papers . . . . . . . . . . . . . . . . . . 30Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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x TABLE OF CONTENTS

A Density filters based on the Pythagorean means 43A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

A.2.1 The relaxation and penalization (RP) approach for topol-ogy optimization . . . . . . . . . . . . . . . . . . . . . . 45

A.2.2 The role of the filter in RP approaches . . . . . . . . . . . 45A.2.3 Why suggest new filters? . . . . . . . . . . . . . . . . . . 46

A.3 Some existing filters . . . . . . . . . . . . . . . . . . . . . . . . 47A.3.1 Linear Density filter . . . . . . . . . . . . . . . . . . . . 47A.3.2 Sensitivity filter . . . . . . . . . . . . . . . . . . . . . . . 48A.3.3 Morphology-based filters . . . . . . . . . . . . . . . . . . 48A.3.4 Heaviside filters . . . . . . . . . . . . . . . . . . . . . . 49

A.4 Two filters based on the geometric mean . . . . . . . . . . . . . . 50A.5 Two filters based on the harmonic mean . . . . . . . . . . . . . . 52

A.5.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 53A.6 Convexity and concavity properties of the filters . . . . . . . . . . 56A.7 Test examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

A.7.1 Test problem 1: Compliance minimization . . . . . . . . . 58A.7.2 Test problem 2: The force inverter . . . . . . . . . . . . . 61A.7.3 Test problem 3: The L-bracket . . . . . . . . . . . . . . . 62A.7.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . 65

A.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66A.8.1 Obtained results for test problem 1 . . . . . . . . . . . . . 66A.8.2 Obtained results for test problem 2 . . . . . . . . . . . . . 66A.8.3 Obtained results for test problem 3 . . . . . . . . . . . . . 68

A.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70A.I Appendix: Proof of the convexity/concavity properties . . . . . . 74A.II Appendix: Mesh independence study for the new filters . . . . . . 78References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

B Interior value extrapolation: a new method for stress evaluation intopology optimization 83B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84B.2 Stresses in topology optimization . . . . . . . . . . . . . . . . . . 85

B.2.1 Density filtering . . . . . . . . . . . . . . . . . . . . . . 85B.2.2 Stress interpolation . . . . . . . . . . . . . . . . . . . . . 89

B.3 The problem of stress evaluation . . . . . . . . . . . . . . . . . . 92B.3.1 Estimating the normal of the surface . . . . . . . . . . . . 93

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B.3.2 Estimating the stress in the boundary element . . . . . . . 97B.3.3 Description of method . . . . . . . . . . . . . . . . . . . 98B.3.4 Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . 99

B.4 Verification of stress extrapolation . . . . . . . . . . . . . . . . . 100B.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

B.5 Optimization examples . . . . . . . . . . . . . . . . . . . . . . . 104B.5.1 Example 1: A uniaxially loaded bar . . . . . . . . . . . . 104B.5.2 Example 2: The L-bracket . . . . . . . . . . . . . . . . . 106

B.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109B.6.1 A uniaxially loaded bar . . . . . . . . . . . . . . . . . . . 109B.6.2 The L-bracket . . . . . . . . . . . . . . . . . . . . . . . . 109

B.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112B.I Appendix: Proofs of claims . . . . . . . . . . . . . . . . . . . . . 115

B.I.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . 115B.I.2 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . 116

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

C Using the weakest link model of fatigue in topology optimization 121C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121C.2 A fatigue model based on Weibull theory . . . . . . . . . . . . . . 124

C.2.1 The weakest link model . . . . . . . . . . . . . . . . . . 124C.2.2 Using weakest link in topology optimization . . . . . . . 128C.2.3 Choosing the parameters σth, σu, m and Vref . . . . . . . 129C.2.4 Stress measure . . . . . . . . . . . . . . . . . . . . . . . 133

C.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136C.3.1 Failure probability . . . . . . . . . . . . . . . . . . . . . 137C.3.2 Density filtering . . . . . . . . . . . . . . . . . . . . . . 137C.3.3 Stress interpolation . . . . . . . . . . . . . . . . . . . . . 138C.3.4 Optimization setup . . . . . . . . . . . . . . . . . . . . . 139C.3.5 Example problem 1: The MBB beam . . . . . . . . . . . 140C.3.6 Example problem 2: The cross . . . . . . . . . . . . . . . 142

C.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144C.4.1 The MBB Beam . . . . . . . . . . . . . . . . . . . . . . 144C.4.2 The cross . . . . . . . . . . . . . . . . . . . . . . . . . . 148

C.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

xii TABLE OF CONTENTS

D A branch and bound algorithm for evaluation of the Findley fatiguecriterion 159D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159D.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162D.3 The Findley criterion . . . . . . . . . . . . . . . . . . . . . . . . 163

D.3.1 Evaluating f(n) . . . . . . . . . . . . . . . . . . . . . . 164D.3.2 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

D.4 A branch and bound algorithm for evaluating the Findley stress . . 169D.4.1 A branch procedure for critical plane problems . . . . . . 169D.4.2 Determining whether a region is interesting . . . . . . . . 171D.4.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 175

D.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176D.5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

D.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178D.I Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . 180References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Introduction

1 Background

Fatigue is a structural failure mode which may be described as "the change ofproperties resulting from the application of cyclic loads" [54]. The "change ofproperties" is a general formulation meaning to cover many sorts of degradation,however, especially for metallic materials, the term fatigue failure is often reservedfor failure due to cracks which are initiated and propagate as a result of cyclicloads, eventually leading to a loss of load carrying ability. Ever since the industrialrevolution, in the early 19th century, when train axles and machine componentsstarted being subjected to thousands and millions of load cycles, fatigue failureshave been a common cause of mechanical failures, and today, fatigue failure hasbeen claimed to be the cause of more than 80 % of all structural failures [44], andis thus responsible for significant costs. Design against fatigue is therefore a keyconcern in the conception of any structure.

The risk of fatigue may often be reduced by adding material to a structure,thereby reducing the stresses. However, low weight is often a sought after propertyin a structure. There are many reasons for this; for products which are to be han-dled or worn by humans, for example tennis rackets and hand held drill machines,low weight is a property which enhances the user experience directly and is im-portant in its own right. For other products, a low weight has a positive impact onother properties, which are important to the end user. For example, if the weightof a truck is reduced, the fuel consumption of the vehicle is reduced, leading toeconomical benefits for its operator. For commercial vehicles with restrictions ongross weight or axle loads, low weight of the unloaded vehicle also means higherload capacity, increasing the profitability even further. Moreover, even if the prod-uct weight is unimportant for the end user, the price of the material may constitutea significant portion of the production cost, which may thus be lowered by find-ing a lighter design. An important task for the mechanical designer is therefore

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2 INTRODUCTION

to find the best trade off between weight and risk of fatigue. This is a task thatoften requires large amounts of experience, with products evolving over years anddecades to find an efficient design, which is exactly as heavy as needed in orderto obtain an acceptable risk of failure, but not any heavier. However, whenever acompletely new design is to be made, the design process often contains a certainamount of "trial and error", iterating between design and evaluation of the strengthof the product. The evaluation of strength has historically been conducted by la-borious and expensive physical testing, but as computing power and theoreticalmodels have improved during the 20th century, computational fatigue evaluations,for example using the Finite Element Method (FEM), have taken a larger role.However, the process is still in many cases an iterative one, with a human designerneeded to make each design change. The purpose of structural optimization is toautomate this task, by allowing an optimization algorithm to "design" the structure,or at least make a design proposal. There are at least two major benefits from this.Firstly, optimized designs hopefully perform better, with lower weight, and lowerproduction costs. Secondly, the time consumed by the entire design process maybe drastically reduced. A process for design of components utilizing structural op-timization is described in Figure 1. The first two steps involve formulating and

Figure 1: A process for designing components using structural optimization.

solving a structural optimization problem. The problem should capture importantrequirements which are applied to the component. In this thesis, focus is set mainlyon fatigue strength and weight.

The result of the optimization is often not a blueprint ready to be sent to aproduction facility, but rather a design proposal, which is converted into a maturedesign during the third step of the process, design interpretation. This step may beautomated to some degree, but is commonly performed manually, transferring theoptimization results into a CAD system.

Subsequently, all important properties of the component must be verified. Suchproperties may include sound emissions, wear properties, stiffness, and, as is fo-cused on in this thesis, fatigue strength. The properties may be verified by physicaltests, or as is increasingly common, by numerical computation. When the designhas passed all these tests, it is approved and ready for production.

The aim of the structural optimization is to create a structure that, after design

TOPOLOGY OPTIMIZATION OF FATIGUE-CONSTRAINED STRUCTURES 3

interpretation, passes the fatigue verification with minimal adjustments. To accom-plish this, the optimization problem needs to include the important aspects of thedesign, and in many cases, this means fatigue requirements.

This thesis deals with development of components which are constrained mainlyby their fatigue strength. It consists of an introduction to the field of structuraltopology optimization in general, and aspects of the field concerning stresses andfatigue in particular, and four appended papers. The aim of the introduction is togive the reader an overview of the subject, and in particular, to explain propertiesof fatigue which are of consequence to the formulation of optimization problems.The appended papers discuss methods applicable in different steps in the process ofFigure 1. The first three papers are devoted to the structural optimization, propos-ing procedures which may be used to incorporate accurate fatigue constraints intopology optimization. The fourth paper deals with the verification of an opti-mized design. This introduction to the thesis is divided into two main parts. In thefirst part, important aspect of the subject of fatigue are summarized. In the second,the field of topology optimization is reviewed, with focus on problems involvingstresses and fatigue.

2 Fatigue

Humans have known for a long time, that if a structure is subjected to a largeenough load, it fails. With the advent of the industrial revolution, machine compo-nents on for example trains could suddenly be exposed to millions of load cycles,each cycle not exceeding the critical failure load. Nevertheless, after months oryears of normal service, a component could suddenly break in the middle of nor-mal operation.

A famous and well documented example is the Versailles train accident in1842, where a fatigue failure in a locomotive axle of a train leaving Versailles forParis, carrying passengers returning from celebrations being held in honour of kingLouis Philippe I caused the death of 60 people [51]. In the 19th century, such fa-tigue failures of train axles were a common source of serious accidents. Though atfirst the failures seemed mysterious, research soon showed that loads much smallerthan those that cause an axle to break instantly, may cause cracks to initiate, prop-agate and eventually cause failure of the component, if they are repeatedly appliedmany times, see Figure 2.

One pioneering researcher was August Wöhler (1819-1914), who describedseveral of the fundamental properties of fatigue in metals. He concluded that the

4 INTRODUCTION

(a) Time varyingstress.

(b) A crack is ini-tiated.

(c) The crackpropagates.

Figure 2: A schematic view of a fatigue crack.

amplitude of the stress is the most important factor when fatigue is considered,and that the maximum stress is influential only through the fact that the higher themaximum stress is, the smaller the amplitude needed to cause fracture. By studyingthe life of components subjected to loads of different amplitudes, and recording thenumber of cycles to failure, Wöhler concluded that lowering the stress amplitudeleads to an increased component life, and that for many metals, there exists a limitload amplitude, below which the life of components is infinite. The number ofcycles at which this knee in the stress-life curve occurs is called the limit life, andfor many metals this life is in the region of 1-2 million cycles. In memory of thiswork, a curve with number of cycles on the abscissa and stress on the ordinateis known as a Wöhler curve in many parts of the world. An illustration of anidealized such curve, fitted to typical fatigue test data, is shown in Figure 3. Adetailed historical review of the field of metal fatigue is given by Schütz [45].

Fatigue cracks may be categorized by the number of cycles after which theyoccur. Cracks occurring after relatively few cycles, in the order of tens of thou-sands or less, are called Low Cycle Fatigue (LCF) cracks. Every load cycle in thisregime typically induces plastic deformation at least locally, and successful designagainst such cracks typically demands modelling of non linear material behaviour.An example of a component whose life is limited by low cycle fatigue is the cylin-der head of a combustion engine. Every start-stop cycle, with the accompanyingwarming up and cooling down, causes significant stresses due to thermal expan-sion of the material. If the cycle occurs twice a day for ten years, 7300 cycles areaccumulated, and the manufacturer will need to ensure that these do not cause a


Figure 3: An idealized Wöhler curve (blue curve), fitted to typical fatigue test data. Theidealized Wöhler curve consists of a finite life region which is represented by astraight line in log-log scale, and a horizontal line indicating the fatigue limit.Red crosses indicate failures, and black circles indicate run-outs, i.e. non-failedtest specimens after 2 million cycles.

fatigue crack propagated far enough to cause loss of function. This is an exampleof a finite life application, that is, the component is designed for a finite life.

Cracks occurring after application of a larger number of cycles, from tens ofthousands to millions, are normally denoted High Cycle Fatigue (HCF) cracks. Inthis regime, the amount of plastic deformation of the material is typically small, andelastic stress computations are often sufficient to predict the life of a component.Many components will need to withstand many millions of load cycles of a certainamplitude, typically beyond the knee in the Wöhler curve. Such components aredesigned with respect to the fatigue limit of the material, and are thus designedfor infinite life. The crankshaft of an engine is a typical example of a componentwhich is designed with respect to the fatigue limit, i.e. for infinite life. Every timea cylinder in the engine fires, the crankshaft is subjected to a stress cycle. Sincethis happens in the order of 1000 times per minute, millions of cycles are quicklyaccumulated, and thus, the stresses have to be below the fatigue limit in order for acrack not to occur.

Typically, there is significant scatter in fatigue life. Scatter arises both in thestrength of the components, and in the load they are subjected to. Sources of scat-

6 INTRODUCTION

ter in fatigue strength are for example casting defects, machining tolerances, anddifferences in the surrounding environment. Scatter in load occurs due to variancein product usage, for example, the loads on a truck are affected by parameters suchas driving style, road conditions and road topography.

The manufacturer needs to know with some certainty that the produced compo-nents will hold up to some minimum requirements, i.e. that even the worst compo-nent holds up to a certain standard. This is typically achieved using a safety factor.In finite life applications, this is typically done by demanding a mean fatigue lifeof the component to be substantially longer than the minimum requirement. In fa-tigue limit applications, this is done by maintaining a safety factor with respect tothe fatigue limit, i.e. the amplitude stress in the component, multiplied by a factor,is required to stay below the fatigue limit of the material. A typical safety factormay be in the order of 1.5 - 2.0.

In this work, linear elastic structures are considered throughout, which meansthat the presented methods are primarily developed for HCF applications.

2.1 Design against fatigue

In the early days of fatigue research, few tools were available other than time con-suming testing, to determine the life of a particular structure when subjected to aparticular load. However, soon theoretical models were developed, which enableddesigners to estimate the fatigue resistance of their designs before any prototypewas produced. The concept of stress, or force per unit area, was given a moremathematically rigorous description when Cauchy introduced what is now knownas the Cauchy stress tensor [11]

σ =

σ11 σ12 σ13

σ12 σ22 σ23

σ13 σ23 σ33

, (1)

where the components are defined as in Figure 4. For design against static failure,a typical requirement is that some measure, or effective stress, of the stress tensormay never exceed some critical, material dependent, value. Examples of effectivestresses for static failure applications are the ones proposed by von Mises [34] andTresca [58]. For design against fatigue, a similar criterion may be used, with aneffective stress that has to stay below some critical value. However, in this case,the effective stress is a measure of the stress history, or stress cycle. A number ofdifferent such effective stresses have been proposed for fatigue.


Figure 4: The components of the Cauchy stress tensor σ, [62].

A significant difference between designing against fatigue and designing againststatic failure, is the importance of stresses in notches and other stress concentra-tions. When considering static failures, engineering materials often exhibit signifi-cant yielding before final rupture, and this yielding reduces the stress raising effectof notches. Hence, for analysis with respect to static failure, it may be sufficient tostudy the "global" stresses. However, fatigue happens at lower loads, where elasticmaterial behaviour makes stress concentrations such as notches important. Thismeans that an accurate description of local stress is needed to make accurate pre-dictions of the fatigue strength. In a topology optimization setting, this means thatthe demands put on the representation of the structure are greater when consideringfatigue, than when considering static failure.

With the introduction of computers in the design process, the possibilities forfatigue design are vastly improved. The stress response of the most intricate struc-tures subjected to very advanced load histories may be computed using numericalmethods such as the Finite Element Method (FEM) [42]. The stress history at ev-ery Gaussian point of the FE-model may then be evaluated, to estimate the riskof fatigue cracks at that location. To do this, a fatigue model must be used. Thefatigue model is a mathematical model relating a stress history to a correspondingrisk of fatigue.

8 INTRODUCTION

2.2 Material data

To use a fatigue model, empirical test data must be obtained for the material inquestion. Fatigue tests are often performed on smooth specimens, where the stressalong one axis as a function of time may be described

σ(t) = σm + σa sin(t), (2)

with all other stress components equal to zero. This is an example of a stresshistory which may be characterised by a mean stress and an amplitude stress. Theamplitude stress is

σa =σmax − σmin

2, (3)

and the mean stress is

σm =σmax + σmin

2. (4)

From the results of many fatigue tests, performed according to Equation 2 withdifferent combinations of mean and amplitude stresses, a so called Haigh diagrammay be constructed, which shows a line of equal failure probability for a certainnumber of cycles, see Figure 5. The behaviour shown in the figure is typical formany materials, in that a positive mean stress is detrimental to the allowable stressamplitude.

The information contained in the Haigh diagram is then used to draw con-clusions on the failure probability caused by more complex stress histories, usingsome fatigue model.

2.3 Local and non-local fatigue models

Models for predicting HCF cracks may be divided into two categories: local andnon-local. Local models consider each material point separately, whereas non-localmodels take the stress distribution in the entire component into account. A localmodel typically consists of a definition of an effective stress measure, which sum-marizes a tensor valued stress history into a scalar value, and a material dependentlimit for this stress measure, below which failure is said not to occur. Accordingto such models, it makes no difference if only one point or the entire componentis loaded to this limit. There are many proposals for the effective stress measure,some examples include the ones proposed by Sines [50], Findley [20], Matake [32]and Dang-Van [13].


−300 −200 −100 0 100 2000

50

100

150

200

σm

[MPa]

σ a [MP

a]

Figure 5: A Haigh diagram for a sand casted aluminium, showing the line of 50 % failureprobability after 10 million stress cycles for different combinations of mean andamplitude stress in a uniaxial test.

A non-local model is any fatigue model that takes the stress distribution in acomponent into account, so that a component cannot be evaluated using only thestress history at one point at a time. One type of non-local models are models in-volving the spatial gradient of the stress [37, 35]. These models take into accountthe empirical fact that for a given stress at a material point, the presence of a largespatial stress gradient away from the surface reduces the risk of fatigue comparedto the presence of a small gradient. This may be explained by the fact that in thepresence of a large spatial stress gradient, a crack is growing into a stress fieldwhich is decreasing. It may also be explained by volume effects, that is, a largespatial gradient means that the volume of highly loaded material is small, whichmeans that the probability of a crack-inducing defect seeing high stresses is re-duced. Thus, such criteria only take into account the local stress history, and thelocal gradient of the stress, and may be said to be at the boundary between localand non-local models.

Another non-local model is the weakest link theory, originally proposed byWeibull [60]. According to this theory, the failure probability of each small volumeelement in a structure is a function of its stress history, and the failure probabilityof the entire structure may be computed by integrating over the volume, leading to

10 INTRODUCTION

the weakest link integral for computing the failure probability, pfail,

pfail = 1− exp(−∫ (

(σ(r)− σth)+

σu

)m dV

Vref). (5)

In this expression, σ(r) is a stress measure, evaluated at the coordinate r, andσth, σu, Vref and m are model parameters. The notation ()+ denotes the positivepart of the expression within the parentheses. The stress measure may be chosenamong the many proposals that exist in the literature, for example one of thosementioned above. In a finite element model, if the stresses are assumed to beconstant throughout each element, the expression for failure probability is

pfail = 1− exp(−∑j

((σj − σth)+

σu

)m vjVref

), (6)

where σj and vj are the stress and volume of element j.

2.4 Critical plane criteria

One class of fatigue models, often used in industrial applications for fatigue analy-sis, is criteria of critical plane type. To facilitate a clear description of such criteria,some basic solid mechanics concepts are first explained.

The stress at a point in a structure may be expressed as a stress tensor, whichdescribes the stresses acting in different coordinate directions. In the x-, y-, z-coordinate system, the tensor may be written as

σ =

σx σxy σxzσxy σy σyzσxz σyz σz

. (7)

For a general stress history, all of the six independent elements in the stress tensorvary with time independently of each other. The stress vector Tn acting on theplane with normal vector n passing through the point is a linear function of thestress tensor

Tn = σn. (8)

Hence, different planes passing through the point "feel" different stresses. If afatigue crack is thought of as the separation of the material in a certain plane, itmakes sense to assume that the probability of a fatigue crack starting in a specificplane is a function of the stress vector history on that plane alone.


To the designer, the exact plane in which a crack is going to start is of less in-terest. What is interesting is finding the plane with the worst loading and checkingthat it is not too high. Critical plane criteria for fatigue are based on this notion.A function f(Tn) that gives a measure on the fatigue loading is defined, and apoint is evaluated by finding the plane with the worst fatigue loading. Thus, thefollowing optimization problem needs to be solved

σeff = maxn

f(Tn(n)). (9)

Several critical plane criteria, i.e. suggestions for the function f(Tn), existin the literature. Examples of such criteria are the ones suggested by Matake [32],McDiarmid [33], Robert et al. [43], Papadopoulos [36], Dang-Van [13] and Findley[20]. This type of criteria have been found to show good agreement with test re-sults [52, 1]. Unfortunately, the objective function in the optimization problem (9)is in general not concave, and local maxima are present. This means that gradientbased optimization methods, which can be implemented and find a local maximumin just a few iterations, are not useful. It is of utmost importance to find the globalmaximum of this problem, since this corresponds to the fatigue load on the criti-cal plane, which determines the strength of the structure. To accomplish this task,researchers and software developers often resort to brute force methods such asgrid sampling, which refers to sampling the objective function for a large numberof planes, and approximating the solution to (9) with the highest value found. Al-though the samples may be distributed in an elegant manner, as proposed by Weberet al. [59], this process remains time consuming. The evaluation of a complete FE-model with millions of nodes, where thus millions of instances of (9) need to besolved, may take hours. Moreover, no convergence proofs have been presented forthis method, hence it is not known exactly to what degree of accuracy the prob-lem is solved, when a certain number of planes are evaluated. In Paper D, a newmethod for evaluating this type of criteria is proposed, which is proved to convergeto within some tolerance of the global maximum, and also, for the same accuracy,requires evaluation of fewer planes than previously proposed methods.

12 INTRODUCTION

3 Topology optimization

Topology optimization is a form of structural optimization, characterized by theability of the optimization to determine both shape, dimensions and connectivity,i.e. the topology of the optimized structure. Although optimization of cross sectionareas in truss structures, such as those studied in the 1960s by for example Dornet al. [15] qualifies as topology optimization if the cross sectional variables are al-lowed to be zero, topology optimization of continuum structures was first studiedin the 1980s by Bendsøe and Kikuchi [2] and Bendsøe [3]. This thesis focusesentirely on such problems. A topology optimization problem of this type is definedby a design volume, with a set of loads and boundary conditions, an objective func-tion, and constraints. An example is shown in Figure 6a, where a design domain(the gray rectangle), with a fixed boundary condition on the left and a force actingon the right side is shown. The ultimate aim of the optimization is to solve the prob-lem of finding a subset of the design volume, where material should be placed, insuch a way that constraints are fulfilled, and the objective function minimized, seeFigure 6b for an example. The infinite dimensional nature of this problem makes

(a) A topology optimization designvolume with boundary condi-tions and constraints.

(b) A possible solution.

Figure 6: A topology optimization design domain, and an example of a solution to theproblem of maximizing stiffness under a volume constraint.

it very hard to solve directly, and what is actually solved is a discretized version ofthe problem, where the design volume has been divided into elements, see Figure7. The elements often serve two purposes, both as a discretization for solving thedisplacement problem using the Finite Element Method, and as a discretization for


representing the optimized structure, where each element is assigned a design vari-able, whose value determines the presence or absence of material in that part of thedesign domain. In the early papers on topology optimization, the objective function

(a) A discretized topology optimiza-tion design volume.

(b) A possible solution.

Figure 7: A discretized optimization design volume, and an example of a solution to theproblem of maximizing stiffness under a volume constraint.

was compliance, or elastic energy. Minimizing compliance is equivalent to maxi-mizing stiffness, and this formulation tends to give reasonable, efficient structures,even in applications where stiffness is not the primary goal of the structure. Thisobjective function also has the benefit of an easily computable gradient, which hascontributed to its popularity. Later development has shown many other formula-tions, for example maximization of eigenfrequencies [30], synthesis of compliantmechanisms [47], and design of wave guides [27], to name a few. For a more thor-ough review of the field of topology optimization, the reader is referred to the bookby Bendsøe and Sigmund [4].

In this section, the basics of topology optimization according to the so calledground structure approach are summarized, and the state of the field of stresses intopology optimization is described. To demonstrate different concepts, exampletopology problems will be used, based on the design space and load shown inFigure 6.

3.1 Formulation of an optimization problem

Let the design domain be discretized with n finite elements. The discrete topologyoptimization problem then consists of deciding which of the elements should be

14 INTRODUCTION

filled with material, and which should not. Let x be a vector of design variablesx1, x2, ..., xn, with the interpretation

xj =

{1 element j is filled with material0 element j is not filled with material

(10)

Arguably the most studied topology optimization formulation is that of compli-ance minimization under volume constraint. For this choice of objective functionand constraint, a discrete optimization problem may be formulated as

minx∈{0,1}n,u∈Rd

fTu

subject to:n∑j=1

xjvj ≤ V

(∑j

xjKj)u = f

(11)

Here, u is the nodal displacement vector, f is the vector of applied nodal forces,vj is the volume of element j and Kj is the stiffness matrix of element j. Themultiplication of Kj with xj ensures that only elements with xj = 1 contribute tothe structural stiffness.

3.2 The relaxation/penalization approach

Although solving the integer formulation of topology optimization is possible forcertain small problems, see for example the works of Werme [61, 56, 57], solutiontechniques for integer programs are generally incapable of handling the large num-bers of variables present in applied topology optimization problems. A methodthat is almost completely dominating the field of topology optimization today, isthe relaxation/penalization approach. Relaxation refers to the relaxation of the in-tegrality constraints on the design variables, thus letting them take on intermediatevalues between 0 and 1. The design variables are often referred to as element den-sities. The relaxation renders a nonlinear optimization problem with continuousvariables, a class of problems for which a variety of large scale solvers exist. Ifthe solution to such a relaxed problem was integer, the original problem would besolved. However, as will be shown, this is not always the case. The problem of min-imizing compliance under a volume constraint, with relaxed integrality constraints,may be formulated formulated as in Equation 11, with x ∈ {0, 1}n replaced with


x ∈ (0, 1)n. This problem is actually convex, and thus solvable to global optimal-ity using any suitable nonlinear optimization algorithm. However, unfortunately,the drawback of the relaxation is visible in Figure 8a, where the solution of theproblem depicted in Figure 6a using the relaxed formulation is shown. As seen inthe figure, in the optimized solution, many design variables take on intermediatevalues, displayed as gray pixels. Apparently, the globally optimal structure useshigh stiffness material at the top and lower part of the structure, and lower stiffnessmaterial in its center, in what is commonly known as a sandwich structure [63].However, in most cases, this kind of structure is far too costly to manufacture, anda discrete design, consisting only of material and void, is desired. This motivates a

(a) Solution of the relaxed problem (b) Solution of the relaxed and pe-nalized problem

Figure 8: Solutions to the example problem, with and without penalization.

penalization of intermediate values. A very popular method to accomplish this is tointroduce a non linear mapping of element modulus of elasticityEj = Ej(xj). Themapping is chosen in a way which reduces the stiffness to weight ratio of elementswith intermediate design variable values, thus making them structurally inefficient.Two examples of much used such penalization functions are the immensely popu-lar SIMP (Solid Isotropic Micro structure with Penalization) proposed by Bendsøe[3], presented in Equation 12,

Ej(xj) = Emin + xpj (E0 − Emin), (12)

and RAMP (Rational Approximation of Material Properties), proposed by Stolpeand Svanberg [53], presented in Equation 13.

Ej(xj) = Emin +xj

1 + q(1− xj)(E0 − Emin), (13)

16 INTRODUCTION

The two parameters p and q serve the same purpose, namely controlling the amountof penalization, with p = 1 or q = 0 corresponding to no penalization, and highervalues reducing the stiffness to weight ratio for intermediate density elements, thusincreasing the penalization. Typical values used in topology optimization are p = 3and q = 6. The parameterEmin in Equations 12 and 13 is a small artificial stiffnessassigned to elements with zero density in order to keep the global stiffness matrixpositive definite.

With an assumption of element wise constant modulus of elasticity, the elementstiffness matrix Kj is linear inEj , and the formulation of the relaxed and penalizedexample problem is thus

minx∈(0,1)n

fTu(x)

subject to:n∑j=1

xjvj ≤ V

with u(x) = K(x)−1f = (∑

Kj(xj))−1f

(14)

The penalization unfortunately makes the problem non convex, hence any optimumthat is found may be a local one. A solution of the example problem depictedin Figure 6a using formulation (14) with SIMP penalization is shown in Figure8b. Clearly, the penalization works as intended, forcing the optimization into astructure consisting of almost no elements of intermediate density. However, anew structural feature has emerged: checkerboards.

3.3 Regularization

It is well known that without any measures taken against it, topology optimiza-tion problems of the type (14) are prone to mesh dependency, i.e. solutions thatvary qualitatively with a change of mesh discretization, and unwanted structuralfeatures, such as checkerboards. Figure 9 shows solutions to the example prob-lem using formulation (14), using different discretizations. As can be seen, allof the solutions exhibit regions where alternating black and white elements forma checkerboard pattern. Moreover, the solutions do not seem to converge withrefined discretization, but smaller and smaller structural features appear as the dis-cretization is made finer.

It has been shown that the infinite dimensional problem is not well posed, andin general does not have a solution. Given a structure, adding more holes andcreating a more complex micro structure, while keeping the total structural volume


(a) 80 by 80 elements (b) 120 by 120 elements (c) 160 by 160 elements

Figure 9: A mesh convergence study of the example problem.

constant, is in general beneficial [28, 46]. In other words, the set of solutionsis not closed. In the discretized problems, this property manifests itself in meshdependent solutions, where the structural complexity is increasing with refinementof the discretization.

The checkerboards are due to a modelling error that is particularly pronouncedwhen the structure is represented using 4-node bilinear finite elements. As Díazand Sigmund [14] show, the computed stiffness of the checkerboard pattern is toostiff, which makes the optimization favour this design.

Several so called regularization techniques exist for alleviating these problems,which all seek to set a limit to the structural complexity. One approach is to imple-ment direct constraints, for example on the total perimeter of the structure, see forexample Haber et al. [24] and Petersson [40], or the variation of the design variablebetween adjacent elements, as is done by Petersson and Sigmund [41]. Another ap-proach is to restrict the solutions to meaningful structures, using a so called densityfilter, as described by Bruns and Tortorelli [9] and Bourdin [5]. When a density fil-ter is introduced, the design variables x are separated from the element densities,and related through a filter function, as x̃ = F (x), where x̃ are the element den-sities, or physical densities. The filter function F is chosen in a way that restrictsthe attainable structures to meaningful ones. For example, checkerboard patternsshould not be in the range of F . By good choices of the filter function, also otherstructural properties, such as minimum member size, may be enforced. A benefitof the density filter is that no new constraints have to be added to the optimizationproblem, the restriction is handled through the parametrization of the structure.

A much used example of such a filter is the linear density filter [9, 5], wherethe filter function F is linear, so that

x̃j =∑i

wijxi. (15)

18 INTRODUCTION

The filter weights wij are non zero in a neighbourhood of the element, typicallywithin a radius called the filter radius. Since this type of filter smooths out thestructure, any features smaller than the filter radius are not attainable, and thus, aminimum member size is enforced. A drawback of this particular filter, however,is that sharp structural boundaries are not attainable. In some applications, suchas wave transmitters, or, as shown in Paper B of this thesis, when consideringstresses, it is important to have sharp boundaries, and for such cases, other densityfilters have been proposed, for example by Sigmund [48] and Guest et al. [23]. Theproblem formulation for the example problem when using a filter is

minx∈(0,1)n

fTu(x)

subject to:n∑j=1

x̃jvj ≤ V

with u(x) = K(x̃)−1f = (∑

Kj(x̃j))−1f

and x̃ = F (x)

(16)

The solution to the example problem with F according to Equation 15, with afilter radius of 0.0625 times the width of the design space, is shown in Figure 10a.Structural details smaller than this are thus not attainable. A side effect of thisparticular filter is that all structural edges are smoothed out, and not sharp. InFigure 10b a solution to the same problem is shown, obtained using the non linearHarmonic Erode filter presented in Paper A. Using this filter more clear structuralboundaries are attainable.

(a) Linear density filter (b) Harmonic erode filter

Figure 10: Solutions to the example problem using two different density filters.


3.4 Solving the optimization problem

When a topology optimization problem has been formulated, the analyst is in gen-eral left with a non linear optimization problem in a large number of variables.A positive aspect of the problems is that gradients of objective function and con-straints in general may be obtained rather easily, using the so called adjoint method.A number of different optimization methods may be employed to solve topologyoptimization problems. For some problem formulations, for example when mini-mizing compliance subject to a volume constraint, simple so called Optimality Cri-teria methods may be devised to solve the problem, and such methods were usedin many early papers, see for example Bendsøe and Kikuchi [2]. Interior pointmethods [21] have been used, for example by Burger and Stainko [10], as well asSequential Linear Programming (SLP) methods, as is done by Sigmund in [47].A method that has come to be used in many topology optimization applications,both academic and commercial, is the Method of Moving Asymptotes, MMA, ofSvanberg [55]. In short, the method is based on creating separable, convex subproblems which are known to have a solution, and solving a sequence of such subproblems.

Two distinct strategies exist for solving topology optimization problems: thenested approach, and SAND (Simultaneous Analysis and Design). Using the nestedapproach, the equilibrium equations are solved, outside of the optimization, asu = K(x)−1f to eliminate the displacement variables from the problem:

minx

g0(u(x),x)

subject to: gi(u(x),x) ≤ 0, i = 1...m(17)

The concept of including the displacement variables in the optimization problem,popularly denoted SAND, has been proposed by Maar and Schulz [31]. The fol-lowing problem is considered

minx,u

g0(u,x)

subject to: gi(u,x) ≤ 0, i = 1...m

K(x)u = f

, (18)

i.e. the nodal displacements are variables in the optimization problem, and theequilibrium equations are included in the problem as equality constraints. Usingthe nested approach has the advantage of reducing the size of the optimizationproblem significantly, since the number of degrees of freedom is generally signif-icantly larger than the number of elements in an FE-model. Also, this approach

20 INTRODUCTION

enables the use of standard finite element solvers for solving the equilibrium equa-tions. The formulation with nodal displacements as optimization variables, on theother hand, may save computational work by not demanding a complete solution ofthe equilibrium equations in every optimization iteration. In this work, the nestedformulation is used throughout.

It is worth mentioning that although the density approach to topology opti-mization, which is described in this work, is presently the most commonly usedmethod, several other approaches exist. Some other topology optimization ap-proaches worth mentioning are for example level set, topological derivative andphase field methods. For an overview of methods for topology optimization, thereader is referred to the review paper by Sigmund and Maute [49].

3.5 Stress constraints

Up to this point, the fundamentals of topology optimization have been presented,and the problem of minimizing compliance under volume constraint has been usedto illustrate various concepts such as penalization and regularization. Althoughstiffness is not an unimportant property, the key property of a structure is often itsdurability, or in other words its resistance to breaking when subjected to an externalload. This is closely related to the stresses in the structure, and it is commonengineering practice to keep the stresses in a structure below some suitable materiallimit, for example the yield stress or the fatigue limit. From an engineers point ofview, a problem formulation that would be attractive to solve is

minx∈(0,1)n

n∑j=1

x̃jvj

subject to: σj(x) ≤ σ̂ j : xj > 0

(19)

In (19), the volume of the structure is minimized subject to n constraints on theelemental stresses, σj . Such constraints are known as local constraints.

Stress constraints have been studied in continuum topology optimization sincethe 1990s, when Duysinx and Bendsøe [16] and Duysinx [17] showed that solv-ing topology optimization with local, i.e. element wise, constraints on the vonMises stress is possible, but noted the fact that the problem size arising from themany constraints call for development of improved algorithms for solving large-scale problems. Duysinx and Bendsøe also discuss the interpretation of stresses inelements with intermediate densities. If a SIMP material law is used, the "macro-


scopic" stress components may be computed as

σij = x̃pEijklεkl. (20)

where Eijkl is the elasticity tensor of the material, εkl is the macroscopic straintensor, and p is the penalization parameter of the SIMP model used. However, thisstress is not a suitable candidate for stress constraints in topology optimization. Forexample, for any given density distribution, multiplying all densities with a factorα < 1 would result in strains which are increased by a factor of 1/αp. However,the factor x̃p in (20) would be reduced in proportion, and thus this weakening ofthe structure would have no effect on the stresses.

Instead of constraining the macroscopic stresses for intermediate density el-ements, Duysinx and Bendsøe [16] proposed considering the stresses that wouldoccur in the members of a porous micro structure of the same average density,which leads to stresses that do not vanish as the density variable tends to zero.This leads to a discontinuity of the stress constraint, and may lead to situationswhere the optimizer is unable to reduce the density of some element to zero, eventhough that would be optimal. To avoid the discontinuity of the stress constraint,the ε-relaxation proposed by Cheng and Guo [12] may be adopted.

Another proposal is modifying the stresses, not based on physical considera-tions, but to obtain attractive properties from an optimization point of view, that is,stresses should be correctly calculated at x = 1, and should tend to zero as x ap-proaches zero. This may be accomplished by multiplying the "macroscopic" stresswith a function with the desired properties. An example is the form

σij =x̃p

x̃qEijklεkl, (21)

where q is chosen smaller than p. This method was used by Bruggi [6], denotedthe qp-approach, and later by Le et al. [29], formulated slightly differently. Al-though not shown to be consistent with the micro structural stresses, the methodhas been shown to open the degenerate regions of the design space similarly tothe ε-relaxation, meaning that gradient based optimizers may be used to solve theproblems.

Local stress constraints have been implemented by a number of researchers,see for example Pereira et al, París et al., and Bruggi and Duysinx [39, 38, 7,8]. However, many researchers have focused on more computationally attractiveaggregated stress constraints.

22 INTRODUCTION

Aggregated stress constraints

Local stress constraints give good control over the stresses in the structure, but atthe expense of a larger optimization problem, where both the sensitivity computa-tion and the solution of the optimization problem become time consuming for largeproblems. To get around this fact, many researchers propose using so called aggre-gated constraints, which circumvent this problem by replacing the local constraintsby an equivalent or approximately equivalent global constraint. One example of anaggregation function is the p-norm of stresses, which approximates the maximumstress. The p-norm σp is defined as

σmax ≤ σp =

∑j

σpj

1/p

. (22)

Note that the parameter p in this context is not the same as the penalization parame-ter of a SIMP material law. An underestimation of the maximum stress is obtainedby dividing by the number of elements, as

σmax ≥ σp =

1

N

∑j

σpj

1/p

. (23)

Both aggregation functions converge to the maximum stress as p tends to infinity.however, for reasonable values of p, the approximation made by the p-norm maybe quite poor, as can be seen in Figure 11, where the worst case behaviour of thep-norm is shown. Large values of pmake the optimization problem ill conditioned,and most authors report using values less than 12. Duysinx and Sigmund [18] usethe p-norm of stresses to formulate stress constrained problems requiring signifi-cantly less computational effort compared to local constraints, and the p-norm hassince been used by among others Duysinx et al. [19], who compared local andaggregated constraints, Le et al. [29], and later Holmberg et al. [25], who investi-gated effects of different strategies for dividing the structure in so called clusters,for which separate aggregated stress constraints were applied. A typical formula-


Figure 11: The worst case behaviour of the p-norm, as a function of p, for different numbersof elements N.

tion when using the p-norm in a stress constrained problem is

minx∈(0,1)n

n∑j=1

x̃jvj

subject to:

∑j

σj(x)p

1/p

≤ σ̂

(24)

Since the level of the p-norm constraint function does not directly relate to themaximum stress in the model, Le et al. [29] propose an adaptive right hand side,which adjusts the constraint level to constrain the maximum stress in the model.It should be noted, though, that even if the constraint level is adjusted so that themaximum stress level is at the desired stress limit, the aggregated constraint is notequivalent to local constraints, unless p→∞.

It is often easier to solve the problem where objective function and constraint

24 INTRODUCTION

of (24) have traded places,

minx∈(0,1)n

∑j

σpj

1/p

subject to:n∑j=1

x̃jvj ≤ V

(25)

This problem possesses the beneficial property of always having a feasible solution.This formulation is used to solve the example problem, with p in the p-norm equalto 12, and the result is shown in Figure 13a.

3.6 Fatigue constraints

Fatigue is caused by stresses, and thus techniques for stress minimization or stressconstraints are useful for fatigue applications. The main difference is in the choiceof stress measure. The stress measure in fatigue applications is not just a conden-sation of a single stress tensor into a scalar, but a condensation of the history ofa stress tensor. Accurate prediction of fatigue failures also requires a higher reso-lution of stresses in notches and radii than prediction of static failure. In a staticfailure, the material often yields in such stress concentrations, and they typicallydo not affect the ultimate strength to a large extent. However, when the componentis subjected to a fatigue loading, at a lower level where the material behaviour re-mains elastic, the notches act as stress raisers, which significantly increase the riskof a fatigue crack initiating. This means that greater care must be taken to ensure acorrect stress evaluation.

In a simple case, the stress as a function of the spatial coordinate r and the timet may be described as a function of time and a constant stress field, as

σ(r, t) = f(t)σ̂(r), (26)

In this case, minimizing the stress σ̂(r) minimizes both the mean stress and theamplitude stress, and is thus equivalent to minimizing the fatigue loading, if thecorrect choice of stress measure is made. However, even for just slightly moregeneral cases, such as for example when a constant stress is superimposed on thestress history (26), as

σ(r, t) = σconst(r) + f(t)σ̂(r), (27)


the situation is complicated. It is no longer obvious that minimizing σ̂(r) mini-mizes the fatigue loading at the point. In the most general case, the stress state is ageneral function of time, in which case multi axial fatigue stress measures must beemployed.

Holmberg [26] treats the case when the loading of the structure may be de-scribed by a unit load and a load spectrum, as in (26). Statistical effects such asthe size influence factor are neglected, which means that the fatigue analysis andthe topology optimization may be separated, with the fatigue analysis conductedbeforehand to determine a stress constraint level which corresponds to the desiredlife. The approach boils down to a formulation where every element is subject totwo stress constraints, one concerning fatigue, the other concerning static strength,with the element-wise constraints being approximated by aggregated ones.

4 Main contributions

The main contributions of this thesis are within three topics. These topics areelaborated on below in three subsections, one for each topic.

New restriction methods

In Paper A, alternative regularization schemes in the form of four new densityfilters for topology optimization are presented. The new filters are especially suitedfor problems involving stresses, since they are able to produce clear black and whitedesigns. The filters are based on the geometric mean, as

log(x̃i + α) =∑j

wij log(xj + α), (28)

and the harmonic mean, as

1

x̃i + α=∑j

wijxj + α

, (29)

with obvious variations obtained by replacing x with (1 − x). Numerical test ex-amples show that the new filters are competitive with regards both to discreteness,objective function value and convergence speed, compared to existing filters.

In Paper B, it is shown that using a density filter which gives soft structuraledges gives rise to errors in the computed stresses. A density filter combinationbased on the filters presented in Paper A is proposed which gives the user controlover both minimum inner radii and minimum member size.

26 INTRODUCTION

Fatigue constraints

In Paper B, a new method for stress evaluation in topology optimization is pre-sented. The method is aimed at increasing the accuracy of the computed stresses,which is especially important when considering fatigue. The jagged representationof the structural boundary introduces errors in the stress evaluation during topol-ogy optimization. The proposed method is based on the fact that stresses in theinterior of the structure are less affected by the jagged representation, and thus amore accurate stress may be computed by extrapolating results from the interior tothe structural boundary, see Figure 12. To facilitate the extrapolation technique, amethod for computing approximate structural normals is proposed. The new stressevaluation method is shown to give more accurate stress results than previous meth-ods, and also affects the optimized structures in a non trivial way.

Figure 12: Stresses evaluated at points along a line normal to the boundary (blue), for threedifferent mesh discretizations, and linear extrapolations (dashed red). Noticethat in the interior of the structure (v < 0), the different mesh discretizationsgive very similar results, while at the boundary (v = 0), there are large differ-ences between them. The extrapolations exhibit significantly less scatter at theboundary.

In Paper C, an approach for incorporating fatigue constraints in topology op-timization problems is proposed. The method circumvents the approximation of


max-operators which normally leads to a compromise between numerical effi-ciency and accuracy, and instead takes its starting point in the weakest link theorydeveloped by Weibull, which gives the failure probability of a structure as

pfail = 1− exp(−∑j

((σj − σth)+

σu

)m vjVref

), (30)

where σj is a stress measure for element j, vj is the volume of the element, and σu,σth, m and Vref are material parameters. An effective stress for fatigue loadingsis also presented, which takes mean stress effects into account. The benefits ofthis formulation, compared to approximations of the maximum stress, are several.Paired with an appropriate stress measure, the weakest link model has been shownto be good at predicting failure probability, since statistical effects of the volume ofhighly stressed material are taken into account. Only one constraint is needed forthe fatigue constraint, whereas several aggregated constraints may be needed if themaximum stress is to be accurately represented by aggregation functions. Usingthe proposed formulation, the problem of minimizing the failure probability of astructure under a volume constraint may be stated as

minx∈(0,1)n

∑j

((σeff,j(x)− σth)+

σu

)p vjvref

subject to:n∑j=1

x̃jvj ≤ V(31)

A solution to the example problem of Figure 7 using formulation (31) and theproposed fatigue effective stress is compared to one obtained by minimizing thevon Mises stress using (25) in Figure 13. A difference, compared to a minimizationof von Mises stress, is the fact that members of the structure which are subjectedto tensile stresses are reinforced, an effect of the proposed fatigue effective stress.

Efficient fatigue postprocessing

In Paper D, a method for evaluating critical plane criteria for fatigue design isproposed. The method, unlike existing methods, is proven to converge to withina user given tolerance, and is therefore useful in all applications where a highdegree of accuracy is needed. For a given tolerance, the method is also shown torequire significantly fewer evaluations of different planes, and therefore providesa reduction of execution time for the fatigue evaluation. The behaviour of the

28 INTRODUCTION

(a) Solution for minimizing p-normof mises stress.

(b) Solution for minimizing fatiguefailure probability.

Figure 13: A discretized topology optimization problem, and an example of a solution.

0 100 200 300 400 500 60010−4

10−3

10−2

10−1

100

101

102

Err

or [%

]

Mean number of iterations

Max error, new methodMean error, new methodBound for error, new methodMax error, Weber et alMean error, Weber et al

Figure 14: Errors for the two compared methods for random stress histories.

proposed method is exemplified in Figure 14, where the mean error in effectivestress evaluation is shown as a function of number of iterations for the proposedmethod and a state of the art method proposed by Weber et al. [59].


5 Suggested future work

Although recent development has brought topology optimization with respect tostresses and fatigue closer, there are still several areas of research, where improve-ment of existing methods may contribute to making such optimization more use-ful. One such area is the interpretation of optimization results. As early as 1989,Bendsøe [3] notes that research is needed for developing methods of automatedinterpretation of results from topology optimization. Such methods are neededto transfer the optimization results into Computer Aided Design (CAD) environ-ments for further processing and adaptation for production. Methods for smooth-ing a topology optimization result and importing it in CAD environments exist incommercial packages for topology optimization, however, the results are merelyimported as non editable shapes. The labour of creating proper, parametrized andeditable CAD models, a work that may consume more time than that used for theactual topology optimization, is still left to a human designer. Research aimed atdeveloping a method that truly automates these tasks, and gives the designer a de-sign proposal in the form of a fully functional, parametrized CAD model, wouldhave a huge impact, and would most likely lead to a significant increase in the us-age of topology optimization in all fields of engineering design. This would alsoenable a more fair comparison of optimization results, especially for optimizationinvolving stresses. Since formulations differ vastly, it is hardly ever possible tocompare the actual performance of two topology optimization methods based onobjective function values alone. However, an automated post processor which cre-ated usable structures would mean that the true performance of the final structurescould be compared, which would be of benefit for the development of the field.

An interesting application of topology optimization is in 3D-printing. Tech-nologies for additive manufacturing, or 3D-printing, are becoming accessible tothe broad masses as the price of the necessary technical equipment is falling, andare seeing applications in widely separated fields, from customized prosthetics toonboard production of spare parts in spacecraft. Such technologies give a userthe possibility of manufacturing products directly from a computer drawing, withvery few manufacturing constraints, since undercuts and draw directions are not anissue. This makes topology optimization a suitable candidate for an automated de-sign process, but as Gibson et al. [22] point out, there is a need for development oftopology optimization methods for taking specific properties of additive manufac-turing, such as anisotropic material properties and residual stresses, into account.Well defined methods for obtaining a smooth structure from the discretized opti-mization results also need development before a user can define a design space,

30 INTRODUCTION

loads and boundary conditions, and obtain a produced part without any other inter-action.

The papers constituting this thesis propose a number of ideas related to topol-ogy optimization in general and fatigue optimization in particular. The ideas andmethods are often presented and tested in isolation, however, there are several con-nections between them, and a natural continuation of this work would be to studythe synthesis of some of these methods. For example, the precision of the fa-tigue optimization formulation presented in Paper C hinges on correct evaluationof stresses in the optimization model. Combining this formulation with the stressevaluation method developed in Paper B, which in turn uses the density filters dis-cussed in Paper A, could be of interest. As mentioned in Section 2.3, the spatialgradient of the stress in a structure is known to have influence on the probabilityof fatigue. Using the stress evaluation method proposed in Paper B, this quantityis obtained as a side effect of the extrapolation process, and could thus be used toimprove the evaluation of the fatigue probability even further.

6 Summary of the appended papers

In this section, brief summaries of the four papers constituting the thesis are given.

Paper A

Krister Svanberg and Henrik Svärd. Density filters for topology optimization basedon the Pythagorean means. Structural and Multidisciplinary Optimization, Vol. 48,Issue 5, pp. 859 - 875, 2013.

It is a well known fact that topology optimization problems formulated using the socalled ground structure approach may suffer from mesh dependent artefacts suchas checkerboards. One way of dealing with this, which also gives some additionalcontrol of structural properties such as minimum member sizes, is the usage ofdensity filters. In this paper, four new density filters for topology optimization arepresented, and compared to existing density filters using numerical test examples.The proposed filters are based on the geometric and harmonic means, respectively,and possess properties that could be of interest in topology optimization, for ex-ample the possibility to obtain solutions which are almost completely black andwhite. The new filters are presented in detail, and convexity properties of the fil-ters are shown. A main finding of the article is that in many of the test examples,drastically different solutions are obtained using the different density filters. In


particular, many of the obtained structures are close to each other in terms of ob-jective function value, but exhibit different topologies. Although the non convexnature of the studied problems do not allow for general statements of one methodssuperiority over another, the article concludes that the proposed new filters do showpromising results in most of the examples. Moreover, it is argued that the alterationof density filter requires a relatively small amount of work, once the problem is setup. This means that a "filter tool box" could be of benefit, enabling the user to ob-tain multiple optimization results, and choosing the one which suits the particularapplication best, for example with respect to manufacturing aspects.

Paper B

Henrik Svärd. Interior value extrapolation: a new method for stress evaluation dur-ing topology optimization. Structural and Multidisciplinary Optimization, Vol. 51,Issue 3, pp. 613 - 629, 2015.

There is a significant difference between evaluation of stiffness and evaluation oflocal stresses with respect to the demands on the finite element discretization ofa structure. For an adequate representation of stiffness, a rather coarse elementmesh is sufficient, and local inaccuracies such as those arising from a jagged rep-resentation of the structure do not have a significant effect on the result. For localstresses to be computed correctly, a finer mesh is in general necessary. However,as is shown by examples in this paper, a finer discretization alone is not sufficientto obtain correct local stress results. The jagged representation of the structuralboundary affects the stress computed in the boundary elements, a phenomenonthat does not vanish with mesh refinement.

In this paper, a method for evaluating the local stresses at the boundary of astructure during topology optimization is proposed. The method is based on theobservation that the impact of the jagged structural representation vanishes in theinterior of the structure, where the computed stresses consequently are more accu-rate. Using the proposed method, the stresses at the boundary are extrapolated fromthe stresses computed in the interior of the structure. The name Interior Value Ex-trapolation, IVE, is therefore proposed. As part of the method, a way of calculatingthe structural normal for a boundary element is proposed, based on calculating thecenter of gravity of a region around the element in question. The normal calculatedusing this method is proved, under some assumptions of smoothness of the optimalstructure, to converge to the real structural normal, as the mesh is refined.

32 INTRODUCTION

For the stress evaluation using any method to be accurate, the discretizationof structural properties such as radii must not be too coarse. Also, when using thenew stress evaluation technique, the extrapolation strategy used to evaluate stressesnecessitates a minimum member size, to ensure that there is sufficient amounts ofinterior to extrapolate results from. To meet these two demands, a density filterbased on the filters proposed in Paper A is proposed, which gives the optimizedstructure both a minimum member size and a minimum feature radius.

Using a number of example geometries, it is shown that the new method gives amore accurate stress evaluation than a direct stress computation, and that the errorusing the new method appears to vanish as the discretization is refined.

It is also shown that the new stress evaluation method causes changes in thesolution to a typical stress minimization problem, and that in fact the stresses eval-uated using IVE are closer to the ones calculated in a post processed, smooth struc-ture based on the optimization results.

Paper C

Henrik Svärd. Using the weakest link model of fatigue in topology optimization.Submitted for publication

In this paper, a method for incorporating fatigue failure probability in topology op-timization is presented. The method is based on the weakest link model developedby Weibull in the 1930s. The model is based on assuming a failure probabilityof a small volume element as a function of the applied stress. Under assumptionsof statistical independence between volume elements in the structure, and the as-sumption that the entire structure fails when any volume element does, the failureprobability of the entire structure is an integral over all elements. Furthermore, if aspecific functional form is assumed for the failure probability as a function of theapplied stress, the total failure probability takes a particularly attractive form. Infact, the expression is in some cases identical to the much used p-norm of stresses,giving a physical motivation for its usage other than as an approximation of themax operator. In order to use the weakest link model in a fatigue setting, an effec-tive stress measure must be defined, i.e. a scalar measure of the stress history. Inthe paper, such an effective stress measure is defined for load histories defined bytwo alternating extreme cases.

Suggestions on how to choose the parameters in the Weibull model are given,and examples are used to investigate the consequences of different parameter choiceson the optimal solutions. It is shown that using the proposed method, both failure


probability minimization under volume constraint, and volume minimization underfailure probability constraint is possible.

Paper D

Henrik Svärd. A branch and bound algorithm for evaluation of the Findley fatiguecriterion. International Journal of Fatigue, Vol. 73, pp. 27 - 38, 2015.

In this paper, the problem of evaluating so called "critical plane" criteria for fa-tigue is considered. Evaluation of a fatigue criterion of critical plane type for a sin-gle material point involves finding the plane passing through the considered point,where some combination of the shear and normal stress is the most damaging. Themost damaging plane is the one where a function f(n) is maximized, where nis a normal direction defining a plane. The problem of finding the critical planeis therefore equivalent to the optimization problem max‖n‖=1 f(n). The functionf(n) is in general not concave, and hence, gradient based methods are not able toguarantee convergence to a global maximum. The solution of such a problem mustbe repeated for every point which is to be evaluated, typically every node in a finiteelement model, which means that millions of instances of this optimization prob-lem need to be solved. It is therefore of great importance that an efficient methodis used to solve the problems.

In this paper, a method is developed for evaluation of the Findley critical planecriterion, using a branch and bound technique to find the critical plane. Contraryto previously existing methods for the solution of the critical plane problem, whichmainly rely on sampling of a large number of points, the proposed method main-tains a bound on the worst case error, and the algorithm thus gives the user a possi-bility to specify a tolerance of the maximum allowable error in the fatigue criterionevaluation.

It is shown that for a given tolerance on the error, the proposed algorithm needsfewer function evaluations on average, compared to methods previously describedin the literature.

Contributions by co-authors

Paper A was written jointly with Krister Svanberg, who originally suggested thefilters that came to be named "Harmonic Erode" and "Geometric Erode". He alsocontributed with the proofs of the convexity and concavity properties of the filters.The numerical implementation and test examples, as well as the the composition

34 INTRODUCTION

of the manuscript, was mainly conducted by Henrik Svärd. In the writing of theremaining papers, Krister Svanberg acted as a supervisor, and contributed withideas and directions for the work.

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