analysis and optimization of laminated composite shell structures

Institute of Mechanical Engineering

Aalborg University, Denmark.

Special Report No. 54

Analysis and Optimization of

Laminated Composite Shell Structures

Ph.D. Thesis

by

Jan Stegmann

Institute of Mechanical Engineering, Aalborg UniversityPontoppidanstræde 101, DK-9220 Aalborg East, Denmark

e-mail: [email protected]

Copyright c© 2004, 2005 Jan Stegmann

This report, or parts of it, may be reproduced without the permission of theauthor, provided that due reference is given. Questions and comments are mostwelcome and may be directed to the author, preferably by e-mail.

Typeset in LATEX and printed in Aalborg, May 2005.

ISBN 87-89206-94-0

iii

Preface

This thesis has been submitted to the Faculty of Technology and Science atAalborg University in partial fulfillment of the requirements for the Ph.D. degree inMechanical Engineering. The underlying work has been carried out at the Institutefor Mechanical Engineering, Aalborg University during the period from July 2001to August 2004. Based on a preprint version of this thesis a public defencetook place on November 19, 2004, with Professor Kai-Uwe Bletzinger (TechnicalUniversity of Munich), Professor Martin P. Bendsøe (Technical University ofDenmark) and Professor Niels Olhoff (Aalborg University) acting as opponents.Incorporated into this final version of the thesis are minor corrections suggestedby the three opponents.

The project has been supervised by Associate Professor, Ph.D. Erik Lund towhom I express my sincere gratitude for his competent guidance, endless patience,support and friendship. I also wish to thank my colleague and friend, AssistantProfessor, Ph.D. Henrik Møller, for many invaluable discussions over a hot copof coffee and, not the least, for thorough proof reading of this manuscript.Furthermore, I wish to thank my friends and fellow Ph.D. students Jens Chr.Rauhe and Lars R. Jensen for working with me on our joint Master’s thesis in2001, which helped me get a running start for this work. I would also like to thankmy colleagues at the Institute of Mechanical Engineering for creating a pleasantand inspiring atmosphere.

I am indebted to Professor Krister Svanberg from the Royal Institute of Technologyin Stockholm, Sweden for providing me with the source code for his excellent op-timizers MMA and GCMMA and to Professor Ole Sigmund, Technical Universityof Denmark for fruitful discussions on multiphase topology optimization.

Finally I want to thank my family – my wife Ditte for encouraging me and makingthe family work around me in times of much work, and my children Maja andEmilie for bringing me joy and always lifting my spirits after a long days work.Their love and support is invaluable.

Jan StegmannAalborg, May 2005.

v

Abstract

The objective of the present work is to develop finite element based optimizationtechniques for laminated composite shell structures. The platform of implementa-tion is the finite element based analysis and design tool MUST (MUltidisciplinarySynthesis Tool) and a number of features have been added and updated. Thisincludes an updated implementation of finite elements for shell analysis, tools forinvestigation of nonlinear effects in multilayered topology optimization and a novelframework called Discrete Material Optimization (DMO) for solving the materiallayout and orientation problem.

A necessary tool for optimization is robust finite elements and consequently,the finite element library in MUST is extended with a new three-node elementand an updated four-node element. These are designated MITC3 and MITC4,respectively, since they use Mixed Interpolation of Tensorial Components to avoidproblems with shear locking. The SHELLn family of standard isoparametric shellfinite elements in MUST has also been updated for improved performance. Allelements have laminate and geometrically nonlinear capabilities and tests showthat the performance and computational efficiency are very good.

Geometrically nonlinear effects are investigated to determine if these should betaken into account when designing for maximum stiffness of laminated compositestructures using structural optimization. Facilities for nonlinear topology opti-mization of multilayered shell structures is implemented using a Newton-Raphsonscheme for the analysis, the adjoint variable method for sensitivity analysis andthe MMA optimizer for solving the optimization problem. The SIMP methodis used for layer-wise stiffness scaling to allow material to be added/removed inspecific layers. Several examples illustrate the effect of the nonlinearities on theoptimal topologies and, depending on the problem, the increase in performance issignificant.

Existing methods for solving for optimal material orientation and maximumstiffness inherently suffer from problems with local optima, which inspired thedevelopment of Discrete Material Optimization (DMO), which is a novel approachfor simultaneous solution for material distribution and orientation. The DMOmethod uses an element level parametrization in a weighted sum formulationthat allows the optimizer to choose a single material from a set of pre-definedmaterials by pushing the weights to 0 and 1. The success of the method is thereforedependent on the optimizers ability to push the weights to 0 and 1 and severalweighting schemes are implemented. Numerical examples indicate that the methodis indeed able to solve the combined material distribution and orientation problem.Furthermore, an industry related design problem of a wind turbine blade main sparis solved and the obtained results are very encouraging. The DMO method thusshows promising potential for application to problems of industrial relevance andno problems with local optima could be identified in the tested examples.

vii

Abstrakt

Formålet med nærværende arbejde er at udvikle finite element baserede op-timeringsteknikker til laminerede kompositte skalkonstruktioner. Implementer-ingsplatformen er det finite element baserede analyse- og optimeringsværktøjMUST (MUltidisciplinary Synthesis Tool), og en række funktioner er blevettilføjet og opdateret. Dette inkluderer en opdateret implementering af elementertil analyse af skaller, værktøjer til undersøgelse af ikke-lineære effekter i multi-lags topologioptimering samt en ny metode kaldet Diskret Materiale Optimering(DMO), der kan løse med hensyn til optimal fordeling og orientering af materialer.

Robuste elementer er et nødvendigt værktøj i optimering, og finite elementbiblioteket i MUST er derfor blevet udvidet med et nyt tre-knuders element samtet opdateret fire-knuders element. Disse benævnes henholdsvis MITC3 og MITC4,idet de bruger Mixed Interpolation of Tensorial Components til at undgå problemermed shear locking. For at opnå bedre ydelse er SHELLn familien af standardisoparametriske skalelementer i MUST også blevet opdateret. Alle elementer kanhåndtere laminater og geometriske ikke-lineariteter, og test viser at elementernesydelse er god.

Geometriske ikke-lineære effekter undersøges med henblik på at afgøre, om dissebør medtages ved design af laminerede konstruktioner for maksimal stivhed medstrukturel optimering. Funktioner til ikke-lineær topologioptimering af multi-lagsskalstrukturer implementeres ved brug af Newton-Raphson metoden til analyse,adjoint variabel metoden til sensitivitetsanalyse og MMA optimizeren til at løseoptimeringsproblemet. SIMP metoden bruges til lagvis skalering af stivheden,hvilket giver mulighed for at tilføje/fjerne materiale i specifikke lag. Flereeksempler illustrerer effekten af ikke-lineariteterne på den optimale topologi og,afhængig af problemet, kan forbedringen af designets ydelse være signifikant.

Eksisterende metoder til løsning af optimal materialeorientering lider underproblemer med lokale optima, hvilket inspirerede til udviklingen af DiskretMateriale Optimering (DMO), der er en ny tilgang til samtidig løsning foroptimal materialefordeling og -orientering. DMO metoden anvender en vægtetsum formulering til at lave en parametrisering på elementniveau, der tilladeroptimizeren at vælge et enkelt materiale fra et sæt af pre-definerede materialer,ved at skubbe vægtene mod 0 og 1. Metodens succes afhænger således afoptimizerens evne til at skubbe vægtene til 0 og 1, og flere formuleringer afvægtene er implementeret. Numeriske eksempler viser, at metoden er i standtil at løse det kombinerede fordelings- og orienteringsproblem. Endvidere løseset industrirelevant designproblem med en hovedbjælke fra en vindmøllevinge, ogresultaterne er meget lovende. DMO viser således potentiale til anvendelse påproblemer i industrien, og der kunne ikke identificeres problemer med lokale optimai de kørte eksempler.

ix

Publications

Parts of this work has been published.

Publications in refereed journals

• Stegmann J, Lund E (2005): Nonlinear topology optimization of layered shellstructures. Structural and Multidisciplinary Optimization, 29(5), pp. 349–360

• Stegmann J, Lund E (2005): Discrete material optimization of general compositeshell structures. International Journal for Numerical Methods in Engineering,62(14), pp. 2009–2027.

• Lund E, Stegmann J (2005): On structural optimization of composite shellstructures using a discrete constitutive parameterization. Wind Energy, 8(1), pp.109–124.

Publications in proceedings

• Lund E, Stegmann J (2004): On Structural Optimization of Composite ShellStructures Using a Discrete Constitutive Parameterization. In: The Science of

making Torque from Wind, (ed. G.A.M. van Kuik), 19-21 April 2004, DUWind,Delft University of Technology, pp. 556–567.

• Stegmann J, Lund E (2003): Optimizing General Shell Structures Using aDiscrete Constitutive Parameterization. In: American Society for Composites 18th

Technical Conference, ASC 18, Gainesville, FL, US, 19-22 October 2003, pp. 1–10.

• Stegmann J, Lund E (2003): Discrete Fiber Angle Optimization of General ShellStructures using a Multi-Phase Material Analogy. In: Fifth World Congress on

Structural and Multidisciplinary Optimization, WCSMO 5 (ed. C. Cinquini et al),Venice, Italy, 19-23 May 2003, pp. 1–6.

• Stegmann J, Lund E (2002): Nonlinear topology optimization of laminated shells.In: 15th Nordic Seminar on Computational Mechanics, NSCM 15 (ed. E. Lund etal), Aalborg, Denmark, 18-19 October 2002, pp. 215–218.

• Stegmann J, Lund E (2002): Topology Optimization of Multi-Layered ShellStructures Undergoing Large Displacements, In: Fifth World Congress on Compu-

tational Mechanics, WCCM V (ed. H.A. Mang et al), Vienna, Austria, 7-12 July2002, pp. 1–10.

Publications partially based on Master’s Thesis

• Stegmann J, Jensen RL, Rauhe JM, Lund E (2001): Shell Element for Geomet-rically Non-linear Analysis of Composite Laminates and Sandwich Structures, In:14th Nordic Seminar on Computational Mechanics, NSCM14 (ed. L. Beldie et al),Lund, Sweden, 19-20 October 2001, pp. 83–86.

• Stegmann J, Jensen RL, Rauhe JM, Lund E (2001): Finite Element Analysis ofLaminated Composite Shells Undergoing Large Displacements, In: 2nd Max Planck

Workshop on Structural Optimization (ed. M.P. Bendsøe et al), Nyborg, Denmark,12-13 October 2001, pp. 65–68.

Contents

1 Introduction 1

1.1 Structural design optimization . . . . . . . . . . . . . . . . . . . . 1

1.2 Background of work . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Finite element analysis of laminated composites . . . . . . . 4

1.2.2 Topology optimization . . . . . . . . . . . . . . . . . . . . . 5

1.2.3 Optimization with orthotropic materials . . . . . . . . . . . 6

1.3 Objectives of work . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 The MUltidisciplinary Synthesis Tool – MUST . . . . . . . 7

1.3.2 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Analysis and optimization 11

2.1 Analyzing the design . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Solving the equations . . . . . . . . . . . . . . . . . . . . . 13

2.2 Improving the design . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Design sensitivity analysis . . . . . . . . . . . . . . . . . . . 15

2.2.2 The optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Multilayered shell finite elements 19

3.1 Geometry, kinematics and material . . . . . . . . . . . . . . . . . . 20

3.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.3 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

xii Contents

3.2 Laminate description . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 Numerical integration . . . . . . . . . . . . . . . . . . . . . 29

3.3 Unlocking – Assumed Natural Strain . . . . . . . . . . . . . . . . . 30

3.3.1 Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.2 Linear MITC elements . . . . . . . . . . . . . . . . . . . . . 31

3.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.1 The MITC elements . . . . . . . . . . . . . . . . . . . . . . 34

3.4.2 The SHELLn elements . . . . . . . . . . . . . . . . . . . . . 35

3.5 Numerical verification . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5.1 Patch testing . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5.2 Nonlinear comparative test . . . . . . . . . . . . . . . . . . 37

3.5.3 General remarks . . . . . . . . . . . . . . . . . . . . . . . . 38

3.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . 40

4 Nonlinear topology optimization 41

4.1 Design parametrization . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Optimization schemes . . . . . . . . . . . . . . . . . . . . . 42

4.1.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Objective function sensitivities . . . . . . . . . . . . . . . . . . . . 44

4.2.1 Adjoint sensitivity analysis . . . . . . . . . . . . . . . . . . 45

4.2.2 Multiple load cases . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3.1 Simply supported 3-layer square plate . . . . . . . . . . . . 49

4.3.2 Hinged 4-layer spherical cap – single load case . . . . . . . 52

4.3.3 Hinged 4-layer spherical cap – multiple load cases . . . . . . 57

4.3.4 Corner hinged 5-layer cylindrical shell . . . . . . . . . . . . 60


5 Discrete material optimization 65

5.1 Orientation optimization with orthotropic materials . . . . . . . . 66

Contents xiii

5.2 The discrete material optimization method . . . . . . . . . . . . . 69

5.2.1 The methodology . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3 Element level parametrization . . . . . . . . . . . . . . . . . . . . . 71

5.3.1 DMO scheme 1 . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.2 DMO scheme 2 . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.3 DMO scheme 3 . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.4 DMO scheme 4 . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3.5 DMO scheme 5 . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3.6 DMO schemes 6 and 7 . . . . . . . . . . . . . . . . . . . . . 78

5.3.7 Multi layered structures . . . . . . . . . . . . . . . . . . . . 78

5.3.8 Patch design variables . . . . . . . . . . . . . . . . . . . . . 79

5.4 The optimization problem . . . . . . . . . . . . . . . . . . . . . . . 80

5.4.1 Design sensitivity analysis . . . . . . . . . . . . . . . . . . . 80

5.4.2 DMO convergence . . . . . . . . . . . . . . . . . . . . . . . 81

5.4.3 Explicit penalization . . . . . . . . . . . . . . . . . . . . . . 81

5.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.5.1 Cantilever beam with distributed top load . . . . . . . . . . 82

5.5.2 Beam subjected to four-point bending . . . . . . . . . . . . 83

5.5.3 Hinged 8-layer spherical cap . . . . . . . . . . . . . . . . . . 84

5.5.4 Wind turbine blade main spar . . . . . . . . . . . . . . . . 88


6 Conclusions 103

Bibliography 107

Index 114

xiv Contents

1

Introduction

The term optimal is widely used for “the best” but in the present contextit is necessary to define it more rigourously. Here, optimal denotes the

best design available given the performance criteria and restrictions defined bythe engineer. Obtaining such a solution is called design optimization, meaningsystematic improvement of an initial design by selection of better and betterdesign parameters. This implies the iterative nature of design optimization wherethe design is continuously analyzed, evaluated and improved until no furtherimprovement can be made and the design is optimal. In its basic form thisprocedure is far from new and, indeed, it is probably one of the oldest disciplinesin engineering and, to be philosophical, in human existence. Optimization canbe said to have occurred throughout human history although the process hasbeen characterized by small and often painstaking steps towards the optimum. Inmodern history this task has been left largely to engineers who through knowledgeand skill have managed to find still better solutions to known problems on aheuristic “trial and error” basis. It is therefore not surprising that solutionsobtained with modern day optimization methods often resemble well known andwell tried solutions from engineering history. One example is frame structures,which were widely used in the 19th century and the first part of the 20thcentury for steel bridges. Two typical examples of such structures are shown inFig. 1.1 together with optimal solutions obtained using modern structural design

optimization techniques for similar loading and support conditions.

1.1 Structural design optimization

The optimal structural design will always constitute the best compromise betweena number of contradictory demands and wishes for the structure. Take the exampleof a commercial airliner. Starting from a Boing 747 we might want to increase thenumber of passengers it can carry and its maximum speed and at the same timereduce operation cost and weight. However, increasing the number of passengerswill increase weight and reduce speed and increasing speed will increase operationcost. Furthermore, the size of the plane is limited by the airport (length of therunway, height of the gates etc.), the weight is restricted by the capacity of the

2 1.1. Structural design optimization

Figure 1.1: Railroad suspension bridge design (left) and railroad frame bridge design

(right). The black/white structures have been obtained using topology optimization with

similar boundary conditions (from Bendsøe and Sigmund (2003), courtesy Ole Sigmund).

engines and the weight cannot be reduced to a point where the fuselage mightloose stiffness and break. So, finding the best solution available is a far fromsimple task and becomes virtually impossible to do “by hand” when the number ofparameters and restrictions is high. Consequently, the development of computersplay an important role in design optimization but at the same time poses a numberof challenges for the engineer who must a priori decide by what measure a designis good and also, by which bounds the design is limited. In design optimizationthe measure of “goodness” is called the objective function since the objective of theoptimization is to “increase goodness” (or “reduce badness”, which amounts to thesame thing) and the limiting factors are called constraints. Common to both is thatthey must be quantities that can be computed and evaluated as a number. Theconceptual difference is that the constraints pose nonnegotiable boundaries calledconstraint bounds on the design while the objective must simply be improved to apoint where further improvement cannot be made within the constraint bounds.A constraint is said to be active when it is imposing its bound on the design, i.e.when the value of the constraint is equal to one of the constraint bounds.

In order to evaluate the objective and constraints the design must be described interms of a set of well defined parameters that govern geometry, material properties,densities, etc. This is called design parametrization and constitutes choosing anumber of characteristic parameters called the design variables, which are theparameters we wish to change. The goal of the optimization is then to find the

Chapter 1. Introduction 3

combination of design variables that yields the best design while observing theconstraints. Performing the optimization is characterized by three distinct steps.First, the performance of the design is evaluated by analyzing it with the currentvalues of the design variables. In structural optimization this is most commonlyachieved using the finite element method. Second, the sensitivity of the design tochanges in the design variables is evaluated for all design variables - this is calleddesign sensitivity analysis and the sensitivities are the gradients of the objectiveand constraints. Third, the sensitivity information is used to update the designvariables in a way that improves the objective. This is most commonly done usingmathematical programming techniques.

To state the procedure outlined above more rigourously we proceed by definingthe different quantities as follows. We want to minimize the objective, f , which isa function of the vector of design variables, a, i.e. f = f(a). The design variablescannot attain any value but must stay between the limits amin and amax. Atthe same time we want the design to obey some physical constraints, g, whichmust remain below the constraint bounds, G. Finally, the design must of coursefulfill the physical laws governing the problem at hand (Newton’s laws, laws ofthermodynamics etc.)1. Now, the problem may be stated in mathematical termsas:

Objective : mina

f(a)

Subject to : g(a) ≤ G

amin ≤ a ≤ amax

Physical laws

(1.1)

The problem in (1.1) is solved iteratively by gradually changing the designvariables, a, according to the gradients computed in the sensitivity analysis untila lower value of the objective, f , cannot be found. This methodology is referredto as gradient based design optimization and is generic to the three commonlyused classes of methods for doing structural design optimization: topology, shapeand size. To distinguish between the three let us take the example of the 2Dstructure in Fig. 1.2. Topology optimization (left) can be used to figure out whereto distribute a limited amount of material and as such, it can introduce internalholes in the structure. This usually provides a coarse outline of the structure soshape optimization (middle) can be used to refine the boundaries. Finally, sizeoptimization (right) can be used to find the optimal thickness distribution overthe structure, indicated in the edge view in Fig. 1.2. The applicability of thethree methods is far greater than indicated here, but the example illustrates thefundamental differences.

In the following the present study will be mapped out in more detail with referenceto related studies.

1In the present study we employ the Nested ANalysis and Design (NAND) approach in whichthe equilibrium equations will be assumed to be satisfied prior to solving the optimization problemitself.

4 1.2. Background of work

Figure 1.2: Side and edge view of some 2D structure subjected to topology optimization

(left), shape optimization (middle) and size optimization (right).

1.2 Background of work

Over the last two decades the strive for lighter and stronger structures has resultedin an increasing use of composite materials. In particular, fiber reinforced polymers(FRP) have gained an ever increasing popularity due to their very high strengthto weight ratio. In structural applications fiber reinforced polymers are usuallystacked in a number of layers, each consisting of strong fibers bonded together by aresin, to form a laminate. The fibers may be uniformly oriented in typically one ortwo directions or they may be oriented in no particular ordered fashion. The bestuse of the material is achieved when ordering the fibers in specific directions toobtain high stiffness in the loading directions and lower stiffness in other directions.Exploiting this “directionality” of the material is at the core of efficient design withlaminated composites. Furthermore, to obtain an optimal design the engineer mustchoose where to put material and which materials to use – both in general and inindividual layers. However, proper choice of material layout, materials, stackingsequence and fiber orientation is a far from simple task since laminates can oftenconsist of as many as 500 or more different layers.

This brings forward the need for efficient and reliable numerical design tools– in particular when dealing with large scale structures involving complicatedgeometries, multiple layers, multiple materials and multiple load cases. Efforts fordeveloping such tools are already well under way and this work naturally drawson results from previous studies in the fields of both finite element analysis andgradient based optimization. With the vast amount of literature available on thesetopics it is well beyond the scope of this brief review to give an exhaustive accountof the works preceding this. In the following particular emphasis will therefore beplaced on topics directly related to the present work.

1.2.1 Finite element analysis of laminated composites

The use of composite materials, especially in the automotive and aerospaceindustries, has naturally fueled the effort for developing finite element methodssuitable for such materials. Laminated composite shell structures are usedextensively in these industries and have therefore received an increasing amount ofattention over the last decade, see Noor et al. (1996), MacNeal (1998), Yang et al.(2000) and Mackerle (2002). This progress has been supported by the developmentof robust and reliable shell finite elements, which do not suffer from deficiencies


such as locking and hour-glass modes usually associated with shell finite elements.The finite element method has, of course, also benefitted tremendously from theexponential growth in computer power, which has made possible the analysis ofrealistic and industry relevant models (Bathe et al., 1997; Noor, 1999).

Several classes of finite elements have been developed and the choice dependson the level of detail required for design purposes. The simplest and mostpopular class combines an Equivalent Single Layer (ESL) laminate descriptionwith finite elements based on First order Shear Deformation Theory (FSDT)to obtain highly efficient and reliable elements for global response analysis. Inthis group the most popular choice are linear elements, rather than higher-orderelements, as it compensates for the increased computational cost of computing thelaminate stiffness. To avoid locking problems when using lower-order elements theAssumed Natural Strain (ANS) technique developed by e.g. Hughes and Tezduyar(1981), MacNeal (1982), Dvorkin and Bathe (1984) and Bletzinger et al. (2000) isemployed, resulting in very robust elements. This is also the strategy chosen in thiswork (as described in Chapter 3) and other examples from the literature includeBarut et al. (2000), Alfano et al. (2001) and Wagner and Gruttmann (2002). TheANS elements have also appeared in major commercial codes such as ANSYS,ADINA and MSC.NASTRAN/MARC while more advanced methods such as layer-wise elements e.g. Reddy (1993), Brank and Carrera (2000) and To and Liu (2001)or solid shell elements e.g. Klinkel et al. (1999) and Sze et al. (2002), accountingfor varying degrees of local behavior, are still reserved for research codes. This ismainly due to their higher complexity and computational cost. For an extensivereview of layer-wise methods in finite element applications see Ochoa and Reddy(1992), Carrera (2003) and Reddy (2004).

1.2.2 Topology optimization

Topology optimization was introduced some 30 years ago and was an importantcontribution to structural optimization in that it provided engineers with theability to optimize not only the shape of existing topologies but also the topologyitself. Since then the field has been the subject of extensive research and isprobably the most active field in optimization at present. The basis for topologyoptimization as it is today was laid out by Bendsøe and Kikuchi (1988) who usedthe homogenization technique and Bendsøe (1989) who introduced the SIMP2

methodology, which was derived independently and extensively implementedby Zhou and Rozvany (1991) and Rozvany et al. (1992). Ever since, numerousextensions have been made to the method – both in terms of capabilities and itsrange of applicability to industrial problems. For extensive reviews of the methodthe reader is referred to Eschenauer and Olhoff (2001), Bendsøe and Sigmund(2003) or Mackerle (2003).

The extension of topology optimization to shells (e.g. Maute and Ramm, 1997) has

2Solid Isotropic Material with Penalization

6 1.2. Background of work

not received the same amount of attention as other fields in topology optimization.This is partly due to the fact that the essence of “classical” topology optimizationis the ability to introduce holes in a structure, thereby reducing material usage andin turn reducing weight. However, the ability to introduce through-the-thicknessholes in shell structures has little practical relevance since holes deteriorate themembrane load carrying ability of the shell. Furthermore, for the majority ofengineering shell structures (fuselages, wings, ship hulls, turbine blades, pressurevessels etc.) it is not viable to introduce holes since the boundary is usuallyprescribed by other factors (e.g. aerodynamical considerations). Thus, topologyoptimization of shell structures only has real engineering applications with the useof multilayered shell finite elements. In that context topology optimization canbe used to “add or remove” material in specific layers rather than through-the-thickness. This opens the door to stiffener and core layout design, which has beentreated for plates by e.g. Diaz et al. (1995) and Krog and Olhoff (1999) and forshells by e.g. Lee et al. (2000), Belblidia et al. (2001) and Belblidia and Bulman(2002).

The above mentioned works are all concerned with linear problems. However,nonlinearities play an important role in the failure of large composite structures,such as wind turbine blades, and should therefore be considered as well. Someof the earliest work involving geometrical nonlinearities in stiffness design ofcontinuum structures was that of Jog (1996) and Yuge et al. (1999). Recentdevelopments of importance to the present work include Buhl et al. (2000),Gea and Luo (2001) and Bruns et al. (2002) who used topology optimization on2D structures with geometrical nonlinearities. To the best of our knowledge nowork prior to this addresses the influence of nonlinearities on the stiffening topologyof laminated composite shell structures as treated in Chapter 4.

One of the exiting new developments in topology optimization is the extensionbeyond two phases (solid and void) to include multiple phases. This work hasbeen pioneered by Sigmund and co-workers, e.g. Sigmund and Torquato (1997) orSigmund (2001), who used multiphase topology optimization for material designand design of 2D continua as well as compliant mechanisms. The same idea wasused recently by Wang and Wang (2004) in a level-set framework for solving similarproblems. These ideas lay out the ground for Chapter 5 and will be discussed indetail there.

1.2.3 Optimization with orthotropic materials

Topology optimization can solve the material distribution problem but the fiberorientation has at present not been solved by any of the optimization branchesmentioned above. For this another branch of optimization has emerged, dedicatedto finding the optimal orientation layout for orthotropic materials (such as fiberreinforced polymers). This work has been established in large part by Pedersen(Pedersen, 1989, 1991) and the key aspects of the method are summarized inPedersen (2004), which also provides a number of examples. Other authors


have contributed as well, e.g. Luo and Gea (1998) who used a very similarapproach to that of Pedersen for plates, and Thomsen and Olhoff (1990) who usedoptimality criteria combined with mathematical programming to solve for fiberorientation and density in discs and plates. Other approaches have been taken bye.g. Miki and Sugiyama (1993) and Foldager et al. (1998) who used laminationparameters to overcome the inherent difficulties with local minima in this typeof problems. The prevailing method seems to be that of Bruyneel and Fleury(2002) and Moita et al. (2000) who use customized mathematical programmingtechniques and such methods have been implemented in the commercial softwarepackages BOSS QUATTRO from SAMTECH and OPTISTRUCT from ALTAIR.In the present work the methods above are not extended further but used asreference in Chapter 5.

1.3 Objectives of work

The general objective of this work is to develop finite element based optimizationtechniques for laminated composite shell structures. Furthermore, these methodsshould be applicable to practical problems of engineering interest in Danishindustry. The key aspects chosen for investigation in this work are:

• Robust and efficient analysis methods for laminated composites

• Solution of the material distribution problem

• Solution of the fiber angle optimization problem

• Investigation of large displacement effects on optimal design

The last point springs from our collaboration with the Danish wind turbineindustry who wants to develop still longer and lighter wind turbine blades, whichare subject to very large displacements under running conditions. In the context ofsuch large laminated composite structures it has been chosen to focus on stiffnessmaximization (compliance minimization) as the optimization objective.

1.3.1 The MUltidisciplinary Synthesis Tool – MUST

The MUltidisciplinary Synthesis Tool (MUST) is a finite element based analysisand optimization code developed by the Computer Aided Engineering DesignGroup at the Institute of Mechanical Engineering, Aalborg University (MUST,2004). The system has been developed in Fortran 90/95 by Associate ProfessorErik Lund and co-workers over the last six years and the development is set tocontinue in the future. Some of the major contributors are Henrik Møller (Møller,2002) and Lars Jakobsen (Jakobsen, 2002) who have both been associated with theresearch project “Interdisciplinary Analysis and Design Optimization of Systemswith Fluid-Structure Interaction” that spawned MUST.

8 1.3. Objectives of work

MUST

Input module(ANSYS, COSMOS, ODESSY)

Finite element library(Beam, Solid/fluid 2D/3D, shell)

Sensitivity analysis modules(Finite difference, analytical)

Optimizer library(MMA, simplex)

Optimization modules(Shape, size, topology, fiber)

Database module(Result generation)

Postprocessor(FEPlot)

Solver library(Linear, nonlinear)

Analysis modules(Solid, fluid, thermal, FSI)

Figure 1.3: Major components of the MUltidisciplinary Synthesis Tool (MUST). The

gray components are those changed in the present work.

The major reason for choosing MUST as platform of implementation is that theneed for developing a finite element framework from scratch is circumvented.Furthermore, the use of MUST allows the implemented methods to be used bycolleagues and students – an opportunity that has already been exploited inseveral graduate studies. This supports the philosophy behind MUST which isto support both research and education by providing understandable programcode and allowing easy implementation of new features.

MUST is a stand-alone application but relies on external software for meshingand general preprocessing. It reads a modified input file from ANSYS, COSMOSor ODESSY3, solves the problem, and generates a database for visualization inFEPlot, which is an in-house postprocessor continuously developed by Erik Lundand Henrik Møller. The major features of the MUST system are depicted inFig. 1.3 where the parts affected by this work are marked by gray.

1.3.2 Outline of thesis

The thesis is organized in four main chapters.

Chapter 2 introduces the basic concepts of structural analysis and optimization.This includes brief treatments of governing equations, equation solving, sensitivityanalysis and the optimizer.

Chapter 3 is dedicated to finite element analysis of laminated composite shells

3The Optimum DESign SYstem. In many ways a predecessor to MUST, today maintainedand used largely by Associate Professors Erik Lund and John Rasmussen.


and describes the implemented shell elements in MUST in terms of assumptions,implementation and performance.

Chapter 4 treats the influence of large displacements (geometrical nonlinearities)on multilayered shell topology optimization problems, and a number of benchmarkexamples demonstrate the difference in optimal topologies between linear andnonlinear solutions.

Chapter 5 is devoted to material layout and fiber angle optimization and introducesthe concept of discrete material optimization (DMO), which is tested for various2D and 3D examples.

Chapter 6 summarizes the conclusions drawn from the present work.

10 1.3. Objectives of work

2

Analysis and optimization

This chapter lays out the analysis and optimization tools necessaryfor solving the generic optimization problem stated in (1.1). Solving the

optimization problem is a process involving several steps as illustrated in Fig. 2.1.Starting from an initial design (with defined design variables) a design analysis isperformed (solution of the physical problem) and subsequently, analysis and designimprovement is performed consecutively to gradually obtain the final, optimaldesign.

The optimization process is thus a chain of events, each leading (hopefully) towardsthe optimal design. As usual, the chain is only as strong as its weakest link and so,a considerable amount of time has in this work been invested in implementationof reliable finite element technology. The methodology for doing so is introducedin Section 2.1 and later in Chapter 3, the particular implemented elements will bedescribed in greater detail. Another important aspect of solving the optimizationproblem is the design improvement step, Fig. 2.1. In this work a shortcut has

Design analysis(finite element)

Sensitivity

analysis

Improve design(optimizer)

New design

variables

Optimize?Preprocessing

(Initial design)

Yes

Postprocessing(Final design)

No

Figure 2.1: Flow chart of the solution process for the generic optimization problem in

(1.1). The gray boxes indicate topics treated in this chapter.

12 2.1. Analyzing the design

been taken in this step since an “off the shelf” optimizer has been implemented inMUST, only requiring an appropriate interface to be programmed. A thoroughtreatment of the theory behind optimizers will be left the established literaturebut the fundamentals will be presented in Section 2.2 with special emphasis onthe applicability of such methods in this work.

2.1 Analyzing the design

The problem statement so far has been generic for any class of problems butin the following, focus will be on structural problems. The governing equationsfor the physical problem are Newton’s laws of motion and for the finite elementmethod these are recast as energy conservation equations. This is standard andfor the sake of brevity a derivation of the governing equations will not be givenhere, as it may be found in numerous textbooks such as Cook et al. (1989),Zienkiewicz and Taylor (1991), Bathe (1996), Bonet and Wood (1997), Hughes(2000) or Belytschko et al. (2000).

In this work both linear and nonlinear problems are considered and as theformulation of the latter encompasses the former, emphasis will be on derivingthe nonlinear expressions. The starting point is the governing equations for thestatic structural problem, which are stated as an axiom:

∫

V

sT δε dV

︸︷︷︸Internal work

−

∫

V

pbδu dV +

∫

A

psδu dA

︸︷︷︸External work

= 0 (2.1)

Here δu ≡ δui is a displacement increment and s ≡ sij is the second Piola-Kirchoff stress, which is work conjugate with the Green-Lagrange strain increment,δε ≡ δεij . The external forces are divided into body forces, pb, and surface forces,ps. All quantities in (2.1) are tensors but have been expressed using matrixnotation, which is convenient for deriving the element matrices. In that contextthe strain and stress vectors are defined as ε = {ε11, ε22, ε33, 2ε12, 2ε23, 2ε13}

T ands = {s11, s22, s33, s12, s23, s13}

T , respectively.

In finite element analysis the governing equations (2.1) are recast in vector formby introducing the strain-displacement matrix, B, which is defined from ε = Bu

where u indicates nodal values. Alternatively, the strain-displacement matrix canbe expressed from the variation of the strain δε = ∂ε

∂uδu as:

B =∂ε

∂u(2.2)

The form in (2.2) is convenient for deriving the strain-displacement matrix aswill be shown in Section 3.4. Introducing B and rewriting the strain variation in(2.1) to a displacement variation, an internal element nodal force vector, r, can be

Chapter 2. Analysis and optimization 13

derived from the internal work term as:

r =

∫

V

BT s dV (2.3)

This must be balanced by the external element nodal force vector, p, which canbe derived in the standard way by computing equivalent nodal forces from thedistributed loads pb and ps in the external work term. Thus, the residual vector1,R, is defined over all elements, Ne, as:

R(u) =Ne∑

k=1

(rk − pk

)(2.4)

For the system to be in static equilibrium the internal and external forces mustbalance each other, i.e.

R(u) = 0 (2.5)

which represents the governing system of equations for linear and nonlinear staticproblems. Thus, the generic statement “physical laws” in (1.1) can be replaced byR = 0 to form the basic form of the optimization problem solved in this work:

Objective : mina

f(a)


amin ≤ a ≤ amax

R(u,a) = 0

(2.6)

in which u is a fixed point indicating that R(u,a) = 0 has been solved prior tosolving the optimization problem, which constitutes a Nested ANalysis and Design(NAND) approach.

2.1.1 Solving the equations

Obtaining the stationary solution to the physical problem (2.5) requires aniterative methodology since the internal force vector, r, is a function of thedisplacements due to the nonlinear terms in the Green-Lagrange strain tensor(3.6). We employ Newton-type solvers, which use the following linearization ofthe governing equations, (2.4):

R(u + ∆u) ' R(u) +∂R(u)

∂u∆u = 0 (2.7)

To solve the system in (2.7) the tangent stiffness matrix, KT , is defined as:

KT =∂R(u)

∂u(2.8)

1This notation is somewhat inconsistent since capital letters usually denote matrices but thisis standard in the literature

14 2.2. Improving the design

and used to find the displacement increment ∆u by solving the linear system:

KT ∆u = −R(u) (2.9)

In MUST (2.7) is solved continuously until the ratio ‖R‖/‖p‖ is below somespecified value, typically around 10−8 − 10−6. At present, MUST encompasses anumber of nonlinear solvers due to Møller (2002), including a full Newton-Raphsonsolver, a modified Newton-Raphson solver and a quasi-Newton BFGS solver allof which can be used with or without line search. Line search gives better globalconvergence properties than full N-R and modified N-R directly and has been usedwith full N-R when solving for a single equilibrium point. For tracking equilibriumpaths an arc-length solver has been used with modified N-R for the sub-iterations.

Derivation of the tangent stiffness matrix is an essential step in implementingfinite elements and to proceed we assume design independent loads and use thedefinition of the internal nodal force vector (2.3) to write (2.8) as:

KT =Ne∑

k=1

∫

V

∂(BTk sk)

∂uk

dV

=

Ne∑

k=1

∫

V

(∂BT

k

∂uk

sk + BTk

∂sk

∂uk

)dV

(2.10)

If the problem under consideration is geometrically linear the need for (2.4)–(2.10)can be circumvented and the static problem simplified. Using that s = CBu alinear stiffness matrix can be defined from the internal work term in (2.1) as:

K =

∫

V

BT CB dV (2.11)

whereby the static equilibrium for the linear problem becomes:

Ku = p (2.12)

which is an algebraic linear system of equations that can be solved directly foru using any linear solver. In MUST we employ the direct sparse solver from theCompaq eXtented Math Library (CXML), which has proven extremely efficientand far superior the direct profile solver used previously. Furthermore, a numberof iterative linear solvers are available but these have not been employed in thepresent work.

The equations (2.3) and (2.10) above form the basis for proceeding with the finiteelement formulation and implementation as described in Chapter 3.

2.2 Improving the design

Having successfully analyzed the design the next step towards the optimal designis the design improvement phase as depicted in Fig. 2.1. To improve the design it


must first be established how the performance of the design changes with changesin the design variables. This is achieved through a design sensitivity analysis(DSA) in which the gradients of the characteristic properties of the optimizationproblem (2.6) are found.

2.2.1 Design sensitivity analysis

Sensitivity analysis is an important part of the gradient based optimization methodand particularly, efficient and accurate computation of gradients is essential for thesuccess of the method. In this work we strictly use analytical sensitivities, whichcan be formulated explicitly and implemented in a general way.

In this work the objective of the optimization is to maximize stiffness, as mentionedin Section 1.3, which can be recast to a minimization problem by introducing thecompliance, C(a):

C(a) = pT u(a) (2.13)

which now becomes the objective, i.e. f(a) = C(a). However, this does notin any way imply that the methods developed later are restricted to complianceminimization. We will assume that the external load is independent of the designvariables and write the compliance sensitivity for the i’th design variable as:

dC(a)

dai

= pT du(a)

dai

(2.14)

This gradient of the objective will indicate whether a change in a particular designvariable will increase or decrease the performance of the design. In the same waythe constraint gradients, dg(a)/da, indicate how the design variables should bechanged to keep the design within the constraint bounds. In the following weassume that a design sensitivity analysis has already been performed. Detailsconcerning computation of the gradients for the specific optimization problemsconsidered in this work will be treated more extensively in Chapters 4 and 5.

Solving the problem of changing the design variables based on the gradientinformation such that the performance is improved (minimization of f) whileobserving the constraints falls to an optimizer.

2.2.2 The optimizer

Several options are available for the choice of optimizer but the most popular instructural topology optimization is the family of convex approximation methodssuch as CONvex LINearization (CONLIN) (Fleury and Braibant, 1986) or theMethod of Moving Asymptotes (MMA) (Svanberg, 1987). Both these methodssolve the optimization problem by generating convex approximations as illustratedin Fig. 2.2 and solving the approximated problem using a dual formulation.Since their introduction these optimizers have proven very efficient in numerousapplications in various fields of structural optimization. The convex approximation


a

a1

2

g a G( ) £

Original design space

Convex approximationg a G( ) £~

aopt

a~opt

f ( )a

Figure 2.2: Design space and convex approximation.

methods are particularly efficient for problems involving many design variables andfew constraints, which makes them ideally suited for topology optimization.

In this study the Method of Moving Asymptotes has been implemented inMUST using FORTRAN77 source code kindly made available by Professor KristerSvanberg, Royal Institute of Technology, Stockholm, Sweden.

MMA solves the optimization problem by creating a convex monotonic approxi-mation from a first order Taylor series expansion around the design point, a. Theapproximation is made in mixed variables of linear and reciprocal terms, i.e. a

and 1/a. This provides a set of approximation functions for both objective, f(a),and constraints, g(a), and thus, the approximated optimization problem to solveis stated from (2.6) as:

Objective : mina

f(a)


amin ≤ a ≤ amax

R(u,a) = 0

(2.15)

The solution of (2.6) is then achieved by successive solution of increasingly good(hopefully) approximations (2.15) until convergence is reached. The quality ofthe approximation is controlled using the lower and upper asymptotes, L andU , respectively as shown in Fig. 2.3. Depending on the sign of the gradient inthe design point, df(a∗)/da, either L or U (never both) is active, as indicatedin Fig. 2.3. The asymptotes are moved from iteration to iteration based oninformation from the previous two iterations and, as the optimization progresses,the asymptotes will move closer together.

The strategy for moving the asymptotes is a key issue for the success of MMAand several updated schemes have been suggested to improve its convergence


f

a~opt

L

Movelimit

f ( )a~

aopt

U

f

a

a~opt

L

Movelimit

f ( )af ( )a~

aopt

Ua

f ( )a

(a) (b)a* a*

Figure 2.3: Convex MMA approximation to a fictitious design space. In (a) the slope

in a∗ is negative and L is active, in (b) the slope is positive and U is active.

properties, see e.g. Bruyneel et al. (2002) for a review. One of these extendedschemes is the Globally Convergent MMA (GCMMA) algorithm, which has alsobeen implemented in MUST from FORTRAN77 source code provided by ProfessorKrister Svanberg. GCMMA uses a non-monotonic approximation to achieve betterconvergence. However, the term “global” is somewhat misleading as GCMMA doesnot guarantee convergence to the global optimum solution but just convergenceto a stationary point in the approximated problem from any starting point. TheGCMMA algorithm has been implemented but not used to any great extent in thiswork since it tends to converge slower than MMA in terms of number of iterationsused and thus computational time.

Besides controlling the approximation the asymptotes also function as move limits,which means that the move limits will also gradually tighten. However, early in theoptimization the asymptotes may be too far apart to provide practical boundariesso an additional move limit strategy should be employed. In the present study astationary move limit of typically 2–5% has been used to stabilize the iterations inthe beginning. This has proven a reliable approach and in general, the performanceof the MMA optimizer in this work has been very satisfactory. We have successfullysolved stiffness design problems of more than 740000 design variables and MMAused only about 4 seconds per iteration to solve the approximated problem.

Now, the basic aspects of the analysis and optimization process have been discussedand in the following the particular element technology implemented will be treatedin detail.

3

Multilayered shell finite elements

The formulation of shell finite elements for laminated composite struc-tures requires insight into shell kinematics, composite laminate behavior and

finite element theory since assumptions from these disciplines will be built into theelements. However, a comprehensive theoretical treatment of these subjects willbe left to the established literature such as Flugge (1990) and Kraus (1967) forshell theory, Jones (1998) or Reddy (2004) for laminated composite structures andBathe (1996) or Hughes (2000) for finite element theory and technology. Instead,focus will be on specific topics of interest to the present thesis.

The procedure adopted for implementing isoparametric shell elements is thedegenerated solid approach (Ahmad et al., 1970), which has two major advantages.First, implementation is straightforward since the procedure is similar to that of3D isoparametric elements and second, the method allows us to use general 3Dconstitutive laws.

The framework for shell analysis in MUST was originally developed and imple-mented in 2000/2001 by Lars R. Jensen, Jens M. Rauhe and Jan Stegmann andis documented in the joint masters thesis Jensen et al. (2001). During the courseof elaborating the present work the original routines have been reimplemented forhigher computational efficiency and also to accommodate extra features as requiredfor the implemented optimization procedures. Furthermore, a number of errorshave been corrected in the element routines and a number of features for geometrichandling, pre- and postprocessing have been added. It is inevitable, however, thatsome figures and topics presented in the following will resemble those found inJensen et al. (2001)1.

The chapter is organized as follows. Section 3.1 provides an outline of theshell element technology implemented and in Section 3.2 this is extended tomultilayered structures by introducing the laminate description and the associatednumerical integration scheme. Then in Section 3.3 the problem of shear locking is

1With the presently available electronic version this is actually the other way around sincethe original printed masters thesis was revised by the author with a number of key new figuresand extended explanations for the elaboration of Stegmann and Lund (2002).

20 3.1. Geometry, kinematics and material

addressed and the solution used in MUST is described. Finally, aspects of elementimplementation is discussed in Section 3.4 and in Section 3.5 the performance ofthe elements in MUST is demonstrated.

3.1 Geometry, kinematics and material

One of the major challenges when dealing with shell elements is keeping track ofthe geometry and thus obtaining an unambiguous description of kinematics andmaterial orientation. To do this two essential coordinate systems are introduced,the node director system in Section 3.1.2 and the material coordinate system inSection 3.1.3. In turn, an equivalent number of mappings are introduced tokeep track of quantities in the different coordinate systems, which is essentialfor the finite element formulation. In the following these aspects will bebriefly discussed with particular reference to the implemented shell elements inMUST. For a rigorous treatment of the individual subjects please refer to e.g.Sokolnikoff (1956) or Heinbockel (2001) for tensors, Chapelle and Bathe (2003) orBonet and Wood (1997) for kinematics and e.g. Hughes (2000) for general shellelement implementation and Jensen et al. (2001) for additional details about theelements implemented in MUST.

3.1.1 Geometry

It is common practice when dealing with shells to consider the geometry of thestructure as a surface instead of a volume. This is justified because shell structuresare thin compared to the overall size of the structure. The characteristic entitychosen for describing the shell, the reference surface, is most commonly eitherthe geometric top, bottom or middle surface of the physical structure. Thisgeometric representation is still 3D but can be made 2D by introducing curvilinearcoordinates for the surface, Fig. 3.1. In that context it proves convenient to operatewith a covariant vector base as local reference frame for describing the shell.

The covariant base vectors are defined in any point in terms of the position vector,x = {x, y, z}, and the local curvilinear coordinates, (r, s, t) ≡ ri, as:

gi =∂x

∂ri

=

{∂x

∂ri

;∂y

∂ri

;∂z

∂ri

}(3.1)

The “in-plane” covariant base vectors g1 and g2 are tangents of the coordinatecurves r and s, respectively, and span the tangent plane of each point on thesurface. The third covariant vector, g3, is tangent to t and is generally not normalto the surface and furthermore, none of the covariant vectors will in general bemutually orthogonal. This is of no consequence since the vectors are still linearlyindependent and thus may serve as coordinate base for the geometric description.In practice, however, a third vector is defined normal to the shell surface to providea meaningful way of expressing the shell thickness (as will be shown shortly). This

Chapter 3. Multilayered shell finite elements 21

g3

g1

g2

xy

z

r

s

x

Figure 3.1: Reference surface (usually the top, bottom or middle of the physical

structure) of some doubly curved shell with global Cartesian reference (x, y, z) and

covariant vectors (g1,g2,g3)

vector will automatically be linearly independent of both g1 and g2 since theyspan the surface tangent plane.

In mathematical terms the shell geometry is well-defined from these definitionsprovided that the whole shell surface is prescribed in curvilinear coordinates.In finite element analysis this description is achieved by braking down the shellsurface into a number of smaller surfaces, each described in terms of discretenodal coordinates and Lagrange shape functions, N . Consequently, the localcoordinates, ri, no longer exist globally but only in a local space within eachelement. Furthermore, the local coordinates are bounded due to the Lagrangianpolynomials such that ri ∈ [−1; 1] – this is usually called the natural space of theelement. With these definitions any point, x, on the shell may be written as:

x = N(ri)x (3.2)

with N being the shape functions for the particular element containing x andx being a vector of nodal coordinates of all nodes, a, in the element, i.e. x ={· · · |xa, ya, za| · · · }T . The particular formulation of (3.2) depends on the orderand type of elements chosen, which may vary across the shell geometry.

The use of simple shape functions and (3.2) simplifies matters considerablycompared to having complicated mathematical expressions for the entire geometry.The covariant base vectors require no additional work since the components in(3.1) are identical to the components of the Jacobian matrix, J = [gT

1 ,gT3 ,gT

3 ]T ,which can be easily determined as the shape functions are known. Defining thekinematics is therefore straightforward.


(a) (b)

1

2

34

r

st

1

2

34

r

st

5

6

78

Figure 3.2: Degeneration of eight-node solid element (a) into four-node shell element

(b). The nodal vectors are the node directors and the shaded surface is the reference

plane. Deleted nodes (ghost nodes) are marked by a “◦”.

3.1.2 Kinematics

In general terms the displacement, u, for any 3D element may be expressed,analogues to (3.2), as u = Nu where u is the vector of nodal displacements,i.e. u = {· · · |ua

x, uay, ua

z | · · · }T . The kinematics of a shell element is closely related

to this expression, the difference being that assumptions regarding the structuralbehavior of the shell will be built into the kinematic description. To do so thedegenerated solid approach is applied in the following for a four-node elementwithout any loss of generality.

The starting point is an eight-node volume element and the result is a four-nodeshell element, which is geometrically reduced to a surface, here taken to be thegeometric midsurface of the solid element (the shaded area shown in Fig. 3.2).This is the reference surface of the shell element, so named since it will be usedas geometric and kinematic reference for the element. In Fig. 3.2 the referencesurface is shown to be identical to the midsurface but it may be displaced up ordown within the element volume to serve specific modeling needs. As mentioned,the mostly used alternatives, which are both supported in MUST, are the geometrictop and bottom of the solid element. The choice of reference surface has no bearingon the kinematics and can be handled efficiently in the element implementation.

The degeneration procedure shown in Fig. 3.2 basically involves replacing theoriginal nodes with nodes lying on the reference surface of the shell. In doingso the element can no longer deform in the transverse direction. At the sametime a vector is introduced in each of the new nodes, pointing from the deletedbottom node towards the deleted top node (called ghost nodes). The new nodalvectors in Fig. 3.2 are called node directors and effectively link points oppositethe mid-surface. The node director is allowed to rotate about the node but itcannot stretch. Together these properties make up two of the Reissner-Mindlinassumptions, namely zero transverse strain (ε33 ≡ 0) and “normals remain straight


a

b

b

-a

v

v

1

2

v3

v3*

Figure 3.3: Rotation of node director, v3, under deformation to deformed state, v∗

3 .

but not necessarily normal”. This also implies that the transverse shear strains willbe continuous across the thickness, which is important in relation to multilayeredstructures as discussed in Section 3.2. The last assumption of zero transverse stress(s33 ≡ 0) will be enforced through the constitutive relation as shown later.

The node directors are very important in that they are used to define the nodaldisplacements and rotations. This is done by setting up, in each node, a Cartesianbase called the director coordinate system having the node director as z-axis. Thenode director will be denoted v3 and defined as:

v3 =g1 × g2

‖g1 × g2‖(3.3)

This might seem inconsistent with Fig. 3.2 since the ghost nodes need not lieon a surface normal but in practice (3.3) is the only viable approach since thecoordinates of the ghost nodes are non-existent in the model. However, using(3.3) directly can lead to discontinuities in the displacements since the surfacenormal may change from element to element depending on the mesh generator andelement type. This will in particular be the case when using four-node elements tomodel curved geometries. Consequently, a simple algorithm that averages adjacentnode directors for all elements is applied in MUST to ensure continuity. The twoadditional base vectors in the director coordinate system are defined from v3 andthe auxiliary vector a as:

v1 =a × v3

‖a × v3‖; v2 = v3 × v1 (3.4)

where a = j if j · v3 6= 0 and a = k if j · v3 = 0. Here, the vectors j and k

are global unit vectors along the y- and z-axis, respectively. The key role of thedirector coordinate system is to define the local rotations, α and β, of the nodedirector, v3. This introduces two additional degrees of freedom per node andis consistent with Reissner-Mindlin shell theory. These rotations are defined asshown in Fig. 3.3.


As indicated in Fig. 3.3 a rotation, α, of v3 about v1 causes a linear displacementof magnitude − sin α in the direction of v2. In the same way the director vectortip is displaced in the direction of v1 a distance sin β. Assuming small rotationsthis amounts to a local linear displacement of ua = {βa,−αa, 0} which can betransformed to a global displacement using a simple tensor transformation rule (seee.g. Sokolnikoff (1956)). This results in a global displacement expressed in termsof the local rotations as ua = βava

1 − αava2 . This expression is essential in shell

analysis since it describes the displacement of any point in the thickness direction,relative to the reference surface. Now, the total displacement of any point in theshell may be written in terms of the in-plane part of the shape functions, i.e.N(r, s, 0), and the local rotations as:

u =A∑

a=1

Na(r, s, 0){u +

t

2h(βv1 − αv2

)}a

(3.5)

where a is the nodal number, A is the number of nodes in the element and his the shell thickness. Note that all quantities must be evaluated at node a.The expression in (3.5) completely defines the kinematics of the shell element interms of five degrees of freedom, (ui, α, β), while observing the Reissner-Mindlinassumptions. The expression (3.5) may be expressed in terms of a modifiedshape function operator, N, defined as N = N(N(r, s, 0), t) such that u = Nu

where u is defined as a vector containing all the nodal degrees of freedom, i.e.u = {· · · |ua

x, uay, ua

z , αa, βa| · · · }T . This rather tedious notation will be abandonedand from this point and on, N and u will always imply “modified shell quantities”as defined by (3.5).

The strain components follow naturally from (3.5) and may be expressed in termsof the covariant base vectors as:

εij =1

2

(tgi · tgj − 0gi · 0gj

)

=1

2

(∂u

∂xj

· gi +∂u

∂xi

· gj +∂u

∂xj

·∂u

∂xi

) (3.6)

where εij indicates local strain since (3.6) is stated in the covariant base vectors.The subscripts t and 0 indicate deformed and initial state, respectively, i.e.

tgi = gi(x + u). The strain may also be obtained from terms of the deformationgradient, F = 0∇tx, which is perhaps a more direct approach:

εij =1

2

(FT F − I

)(3.7)

This can be efficiently evaluated in terms of the displacement gradients, which arereadily available in a finite element framework such that F = 0∇tu + I. Whichof (3.6) and (3.7) is most convenient depends on the type of finite element beingimplemented.


3.1.3 Material

As briefly mentioned above the third of the Reissner-Mindlin assumptions (s33 = 0)is enforced through the linear constitutive relation s = Cε, where s is the secondPiola-Kirchoff stress. This will be addressed in the following.

First, we define the constitute matrix of an orthotropic material, which is the mostgeneral case for the present work:

C =

C11 C12 C13 0 0 0C22 C23 0 0 0

C33 0 0 0C44 0 0

Sym. C55 0C66

(3.8)

Enforcing s33 = 0 simply involves deleting the third row and column in (3.8) andthe most general form of C is then:

C =

C11 C12 0 0 0 0

C22 0 0 0 00 0 0 0

C44 0 0Sym. C55 0

C66

(3.9)

where the coefficients are given in terms of the engineering constants as:

C11 =E1

1 − ν12ν21C22 =

E2

1 − ν12ν21C12 =

ν21E1

1 − ν12ν21

C44 = G12 C55 = γG23 C66 = γG13

(3.10)

where the modified terms C11, C12 and C12 can be derived from the generalexpressions by setting the transverse Poisson ratios equal to zero (see e.g. Reddy(2004)). In (3.10) the factor γ is the shear correction factor taken to be 5/6 for bothmulti- and single-layered structures. This is somewhat crude but the predictivecapabilities of the formulation are quite good so no further steps have been takentowards improving this along the lines of e.g. Pai (1995) or Auricchio and Sacco(1999). For sandwich structures, however, MUST supports an alternative schemein which γ is set to 1.0 for the core layer while the transverse shear stiffnessesare set to zero for the skin layers. This corresponds to assuming that transverseshear is supported only by the sandwich core and provides very good predictivecapabilities for sandwich structures (Jensen et al., 2001).

From (3.10) it is apparent that only in-plane normal material properties and out-of-plane shearing properties are taken into account, which is consistent with the shellassumptions. Each of the expressions above are stated in the orthotropic principal


a

bv3

a

v3b

Fiber direction

c

v3c

Figure 3.4: Conceptual difference between director vector (dotted line) and material

coordinate system (solid lines), which changes through the thickness.

directions, which are of no particular interest. Therefore the need arises for acoordinate system that uniquely defines the orientation of the orthotropic materialat any point in any layer of the element. This coordinate system is very fittinglycalled the material coordinate system and the major difference between this andthe director coordinate system is that the material coordinate system changesfrom layer to layer as illustrated in Fig. 3.4 for a curved nine-node element. Forflat (three- and four-node) elements the coincidence of all normal vectors couldbe exploited to bypass evaluation of some of these coordinate systems but asthe implemented element routines are generic the same approach is used for allelements.

The material coordinate system is spanned by the base vectors mi and for theconstitutive properties to be meaningful the system must span the tangent planein any given point. As such it is natural to define m3 to be normal as:

m3 =g1 × g2

‖g1 × g2‖(3.11)

which is the same definition as used for the director vector, v3, the differencebeing that (3.11) is updated in all Gauss-points and layers (Fig. 3.4). The in-plane material base vectors may in principle point in any direction as long as theyare orthogonal to m3 but for practical modeling purposes it proves convenient tointroduce a projection of a defined (or default) global material system onto theelement plane. This procedure has been adopted from the commercial softwarepackages ANSYS and COSMOS and can be stated as follows.

A global material system is defined in terms of three vectors, di, which are eithertaken to be the global Cartesian base vectors (default) or, if defined, read fromuser input. These vectors, di, will be projected onto the element plane as follows.If the angle between d1 and m3 is smaller than 45◦, d1 is projected as m1. Elsed2 is projected as m1. The procedure is outlined in Algorithm 3.1.

The definition in Algorithm 3.1 provides a material coordinate system that by


Algorithm 3.1: Pseudo code for setting up material coordinate system

Get covariant base vectors g1 and g2

Compute shell normal, m3, from (3.11)if (Global material coordinate system defined) then

Read defined system into vectors, d1, d2 and d3

else . DefaultUse global Cartesian, i.e. d1 = i, d2 = j and d3 = k

end if

if(abs(d1 · m3) < 45◦

)then . Project using either d1 or d2

Compute a = d1 × m3

else

Compute a = d2 × m3

end if

Compute m1 = m3 × a

Compute m2 = m3 × m1

default is oriented identically for all elements and has an “intuitive feel” whenhandling materials in modeling situations. The user also has the option of definingglobal material coordinate systems, which allows the user to control the materialdirection explicitly on various parts of the model. The procedure outlined aboveprovides a well-defined framework for handling orthotropic materials using a singleangle, θ. This angle is defined relative to the local, projected material coordinatesystem in any point, as shown in Fig. 3.5.

The constitutive behavior can now be described by (3.9) and an in-plane rotation ofC such that C = TT (θ)CT(θ) where T(θ) is a standard transformation matrix (seee.g. Cook et al. (1989)). Depending on the element formulation the constitutivematrix must be transformed once more since the material coordinate system isusually not the preferred frame for setting up the element stiffness matrix. Inthe present work the element stiffness is either expressed in the global Cartesianframe (i, j,k) or the covariant frame (g1,g2,g3) depending on what providesfewest element level computations. The transformations employed are statedin Cook et al. (1989) for transformation to (i, j,k) and in Heinbockel (2001) fortransformation to (g1,g2,g3).

3.2 Laminate description

The topics discussed in the previous section are generic for single- and multilayeredstructures but provide no framework for handling multilayered elements. Thiswill be addressed in the following by introducing the Equivalent Single Layer(ESL) laminate description, which has been adopted in this work. In an ESLdescription the layers of the laminate are assumed to be perfectly bonded togetherand thus, displacements will be continuous across the thickness. Due to thekinematic assumptions this implies that in-plane strains (ε11, ε22 and ε12) are

28 3.2. Laminate description

m1

m2

m3

q 1

2

Figure 3.5: Orthotropic principal directions (1, 2) with reference θ to the material

coordinate system base vectors mi.

continuous across the thickness as well and furthermore, that transverse strains(ε13 and ε23) are constant through the thickness. Consequently, a single node inthe thickness direction is sufficient for describing the kinematics of the laminateand consequently, all quantities derived in the previous sections apply directlyto ESL shell models. The downside of using ESL is that interlaminar effectssuch as delamination are difficult to predict due to the absence of normal stressand strain components in the thickness direction. This can be remedied to someextent by introducing stress recovery as discussed by e.g. Cho and Choi (2001)in which the equilibrium equations are integrated a posteriori to obtain a betterresult for the stress components. Another approach is to introduce a modifiedkinematic assumption, allowing for piecewise continuous displacements. This isknown as the zig-zag approach and has become increasingly popular over the lastyears, see e.g. Carrera (2003) for a review. However, the unmodified laminatedescription employed here is still the most widely used since it provides a goodapproximation of the structural stiffness and is computationally less expensive.Both these properties are desirable in the present context and thus, the methodchosen provides a sufficient and well-established basis for doing global analysis andoptimization.

The notation associated with laminates is shown in Fig. 3.6 where the number oflayers is designated N l and the layer thickness is hl. Each layer is also associatedwith a material, Cl, as well as an angle, θl, for orthotropic materials. When mixingelement- and layer-wise quantities superscripts e and l will be used, respectively,to relate each quantity.

Each layer, l, in the laminate of Fig. 3.6 is described by the constitutive relation:

sl = Cl(θl)εl (3.12)

and consequently, the stresses will be layer-wise continuous since strains arecontinuous and Cl changes from layer to layer. The constitutive behavior ofthe entire laminate is obtained by integration through the thickness. In ClassicalLaminate Theory (CLT) this is achieved explicitly by forming the extension matrix,


Layer 1Layer 2Layer 3

Layer N -1

Layer N

h

hl

l

l

Figure 3.6: Laminated composite shell structure.

A, the bending-extension matrix, B, and the bending matrix, D, which togetherdescribe the laminate behavior (see e.g. Jones (1998)). In the present studynumerical through the thickness integration has been employed as a more generalscheme for obtaining the response of any laminate while maintaining the 3Dcontinuum mechanical formulation. This is also the prevailing approach in theliterature and commercial finite element software packages.

3.2.1 Numerical integration

Evaluation of the integrals in (2.8) is done using full Gauss quadrature and forlaminated structures this can be extended to a layer-wise integration scheme.From laminate theory it is well-known that the laminate behavior is dependenton the thickness coordinate to a power of three, i.e. we may write tentativelyC = C(t, t2, t3). Using a two-point Gauss quadrature in the thickness direction istherefore sufficient as long as the constitutive relation is linear, which will alwaysbe the case in this study. The integration is made layer-wise by introducing anadditional mapping such that the thickness coordinate, t, is expressed from thelayer thickness coordinates, tl, as:

t = −1 +2

h

N l∑

i=1

(hi − hl(1 − tl)

)(3.13)

where tl = ± 1√3

for two-point quadrature. Now, the global tangent stiffness matrix

becomes:

KT =Ne∑

k=1

N l∑

q=1

∫∫∫ (∂BT

q

∂uq

sq + BTq

∂sq

∂uq

)hl

hdrdsdtl

k

(3.14)

30 3.3. Unlocking – Assumed Natural Strain

where the ratio hl/h arises from (3.13) since dt = (hl/h)dtl. Similarly, the internalforce vector is obtained as:

r =Ne∑

k=1

N l∑

q=1

∫∫∫ (BT

q sq

)hl

hdrdsdtl

k

(3.15)

In the following the layer-wise form of volume integration in (3.14) and (3.15) willbe implied whenever stating the stiffness matrix or internal force vector in integralform.

Using (3.14) effectively subdivides each element into N l sub-elements in thethickness direction and so, the computational cost will increase linearly with thenumber of layers. For large models with many layers this can be an impediment,particularly for higher-order elements. Alternatively, explicit thickness integrationcan be used by making simplifying assumptions regarding the thickness variation ofthe inverse Jacobian, which allows the element matrices to be decomposed and inte-grated analytically. This is discussed for the AG-method of Zienkiewicz and Taylor(1991) by Kumar and Palaninathan (1997, 1999) who used it for geometricallylinear and nonlinear structures, respectively. Such methods are not widely usedand cannot be implemented in the existing element routines in MUST without firstmaking considerable changes. Consequently, no steps have been taken towardsimplementing explicit integration but this could be considered in the future toobtain higher computational efficiency.

3.3 Unlocking – Assumed Natural Strain

An important aspect of shell finite elements is the problem of locking, whichhas historically haunted shell elements. The problem arises in elements deriveddirectly from the kinematics presented above, i.e. by inserting the displacementinterpolation (3.5) in the strain definition (3.6) to obtain the strain-displacementmatrix. This approach results in deficiencies in the element formulation in thatthe element will exhibit overly stiff behavior in some situations – the element meshis said to lock.

3.3.1 Locking

Locking is usually characterized as either shear locking, membrane locking, volumelocking or thickness locking and the different types are encountered depending onthe element type and conditions of the model. This brief discussion is limitedto the elements in MUST and an exhaustive account is left the literature, e.g.Cook et al. (1989) or Bischoff (2004).

In short, locking arises due to an elements inability to properly represent thedeformation it is supposed to model. Such problems are more profound in someelement types than others and as a rule of thumb the problem decreases as the


element order gets higher. Indeed, the SHELL16 element in MUST has provenvery robust and only exhibits locking behavior under extreme conditions suchas when elements get highly curved and distorted, which can cause membranelocking. The SHELL16 element can therefore serve as reference for other elementsalthough it is not efficient for optimization problems in general due to its highcomputational cost. The SHELL9 element in MUST is less expensive and hasbeen used extensively with good results. However, the element may exhibit shearlocking when subjected to non-constant bending moments and can also suffer frommembrane locking when curved. The difficulty in using the SHELL9 and SHELL16elements lies in the fact that the occurrence of locking is hard to predict and canbe difficult to detect unless it is severe. The only viable approach with theseelements is therefore to recognize the problem and perform comparative solutionsto validate the results. For this and other reasons (Section 3.5) the linear MITCelements remain the choice of preference.

3.3.2 Linear MITC elements

The problem of locking is particularly profound in linear three- and four-nodeelements, which are known to suffer from shear locking when the elementthickness becomes small compared to the element edge length. This is due toparasitic transverse shear strains, which result in an gross overestimation of theelement stiffness. Still, the linear elements remain very popular due to their lowcomputational cost and consequently, considerable effort has gone into improvingthese elements and eliminating the locking behavior. Historically, this was firstachieved using a reduced order of integration in the Gauss quadrature but this givesrise to problems with spurious energy modes. In recent years, the preferred curefor locking has instead been the Assumed Natural Strain (ANS) methods, whichaim at fixing the deficiencies in the element formulation by introducing modified(assumed) strain expressions in the natural element (r, s, t)-space. Elementsderived using assumed strain expressions will be called stabilized elements anddenoted as MITCn while elements derived directly from the kinematics will bereferred to as non-stabilized elements and denoted SHELLn (n indicates thenumber of element nodes).

In the present work the ANS method introduced by Dvorkin and Bathe (1984)as Mixed Interpolation of Tensorial Components (MITC) has been appliedsuccessfully to four- and three-node elements, effectively eliminating problemswith shear locking. The basic idea is to use a new set of points, p = 1 . . . Np,for strain evaluation instead of the Gauss points, which would be used in non-stabilized elements. These new points are called tying points and are chosen suchthat the strains evaluated in these points are free from parasitic strains. Havingobtained a set of “correct” strains these are then interpolated across the element,thus eliminating problems with parasitic strains altogether. The strains in thetying points, (rp, sp, tp), are evaluated directly from the strain definition in (3.6),i.e. εij |p ≡ εij(rp, sp, tp). Introducing a set of interpolation functions for each

32 3.3. Unlocking – Assumed Natural Strain

r

s

A

C

BD

1 2

4 3

s = -1

r = 1r = -1

s = 1

1 2

3

BD

C

A

r

s

Figure 3.7: Three- and four-node elements with MITC tying points (A–D) used for

transverse shear strain evaluation.

strain component, N ijp , the assumed strains (AS) are expressed as:

εASij =

Np∑

p=1

N ijp (r, s)εij |p (3.16)

where Np is the number of tying points, p, for the ij’th strain component. TheMITC interpolation functions, N ij

p , must naturally fulfill the relation:

N ijp |q = δkq, q = 1 . . . Np (3.17)

so that the p’th interpolation function assumes the value 1 in the p’th tying pointand the value 0 in all other tying points. It is therefore natural to choose Lagrangepolynomials of an order appropriate to the number of tying point, i.e. first orderLagrange polynomials for two tying points etc.

The two expressions (3.16) and (3.17) above are general for all MITC elements,which include linear, quadratic and cubic elements. The location and number ofthe tying points will vary depending on the order of the element but the basic idea isthe same (see Bucalem and Bathe (1993) for details). For the three- and four-nodeMITC elements there are four tying points, A–D, used for transverse shear strainevaluation (two tying points for each strain component). Both transverse shearstrains are interpolated linearly but ε13 is interpolated along s between points Aand C and ε23 is interpolated along r between points B and D. The four points arelocated at the element mid-sides, i.e. A = (0,−1, 0), B = (1, 0, 0), C = (0, 1, 0),D = (−1, 0, 0), as shown in Fig. 3.7. The reason for choosing these locations for thetying points is that the element mid-sides are the only points in which the straindefinition (3.6) is able to evaluate zero transverse shear strain in pure bending.This is because the mid-sides are the only points in which the deformation figureis correct, i.e. the normal remains normal whereby the transverse shear strains arezero, Fig. 3.8. However, the choice of tying point for the MITC3 element involvesa problem since the transverse strains become dependent on the node numbering.


(a) (b)

Figure 3.8: Linear shell element subjected to constant bending moment (a) and

corresponding actual deformed shape (b) with correct deformed shape (dotted line).

This problem is also encountered in other formulations, e.g. Bletzinger et al.(2000), but recently Lee and Bathe (2004) proposed a MITC3 element with analternative interpolation, which renders the element insensitive to node numbering.This approach has not been adopted yet but tests have revealed that the resultsfrom the MITC3 element in MUST vary by no more than 3% when permutatingthe node numbering2. This is satisfactory considering that the MITC3 element ismainly complementary to the MITC4 element in mixed meshes and as such rarelyused exclusively.

The assumed transverse shear strains for both the three- and four-node elementsare now found by evaluating the tying point strains, εij |p, and using (3.16) forNp = 2 with first order Lagrange polynomials:

ε13 =1

2(1 − s)ε13|A +

1

2(1 + s)ε13|C

ε23 =1

2(1 − r)ε23|D +

1

2(1 + r)ε23|B

(3.18)

which provides linearly varying transverse shear stresses in the r, s-plane andconstant stresses in the t-direction. The assumed strains (3.18) are used fordetermining the strain-displacement matrix, and as such the stiffness matrix, andallow the elements to show proper behavior for thin elements.

3.4 Implementation

The shell element library in MUST originally consisted of the MITC4, SHELL3,SHELL4, SHELL6 and SHELL9 elements with linear and nonlinear capabilities.These were implemented by Jensen et al. (2001) and later, a linear MITC3 elementwas added and tested by Moser (2002) and Poulsen et al. (2003) adapted MUSTto accommodate a SHELL16 element and performed extensive testing. In thoseimplementations the invariant properties of the energy terms in (2.1) were notexploited and consequently, a number of essentially redundant transformationswere used. Removing these transformations and formulating the stiffness matrixin the covariant base vectors instead of the global Cartesian system lowered thecomputational cost of forming the global tangent stiffness matrix by approximately

2Tests performed by Moser (2002) in connection with the original implementation of the linearMITC3 element.

34 3.4. Implementation

15% and rendered the elements very efficient when comparing to commercial codes.However, a number of trivial errors existed in the original implementation of thetangent stiffness matrix of the MITC4 elements. This did not cause noticeableproblems for moderate nonlinear analysis but was quite clear when comparing thestiffness matrix to a central finite difference approximation. The implementationstrategy was (and still is) to formulate the tangent stiffness matrix analytically,which results in some rather involved expressions and it is natural to suspectinsufficient differentiation in the derivation of the linearized terms or other errorsin the derivations. Unfortunately, even after spending considerable time on theproblem, a proper solution has not been found. Instead, the MITC3 and MITC4element routines have been completely re-implemented and furthermore, theSHELLn element routines have been optimized for higher computational efficiency.

3.4.1 The MITC elements

As in the original implementation we proceed from (2.10) by splitting the straininto a linear and nonlinear part, i.e. ε = e + η, whereby the strain-displacementmatrix is also split such that B = Be + Bη. So, using that C = ∂s/∂ε, (2.8) maybe written as:

KT =Ne∑

k=1

∫

V

(∂(Bη)T

k

∂uk

sk + (Be + Bη)Tk Ck(Be + Bη)k

)dV (3.19)

The idea in the modified implementation is to create a general framework forcomputing (3.19) using different routines depending on the element. In otherwords it is a modular approach in which routines are needed for determining Be,Bη and (∂BT

η /∂u)s. The benefit of taking this approach is that these routinesmay be developed independently using the mathematical software MAPLE, whichcan manipulate expressions symbolically and subsequently generate FORTRAN77code suitable for MUST. In turn, this should eliminate problems with trivial errorsin the derivations as well as insufficient differentiation. It turned out, however, thatMAPLE 8 suffers from a number of shortcomings in the translation to FORTRANand would in fact introduce sign errors as well as fail to apply the sufficient amountof parentheses. This can be circumvented by carefully reviewing the code andtaking control “manually” of the derivation. Doing so, the method was eventuallyapplied with great success and efficiency to the MITC elements.

The MAPLE routines start from the manually derived strain expression, whichare obtained by inserting (3.5) in (3.6). The linear strain-displacement matrix,Be, is then obtained as:

Be =∂e

∂u=

1

2

∂

∂u

(gT

i

∂N

∂rj

u + gTj

∂N

∂ri

u

)

=1

2

(gT

i

∂N

∂rj

+ gTj

∂N

∂ri

) (3.20)


where i and j are set as appropriate for the given strain component (row in Be),i.e. i = j = 1 for e1 = e11 and so forth. This leads to the usual definition of thelinear strain-displacement matrix, i.e. δe = Beδu and from the strain definition(3.6) it follows that e = Beu. It is implied in (3.20) that the covariant vectors arecomputed in the appropriate node depending on the degree of freedom being usedfrom u. The nonlinear contribution is derived as:

Bη =∂η

∂u=

1

2

∂

∂u

(uT ∂NT

∂rj

∂N

∂ri

u

)

= uT ∂NT

∂rj

∂N

∂ri

(3.21)

and thus it follows that δη = Bηδu and from the strain definition in (3.6) thatη = 1

2Bηu. Note that the expressions in (3.20) and (3.21) should be considered asa conceptual aid rather than a model for implementation. Instead, all componentsof the B-matrices are derived explicitly row by row as shown in Algorithm 3.2 inpseudo code. Once the nonlinear strain-displacement matrix is known the secondterm of the tangent stiffness matrix in (3.19) can be formed directly while thefirst term, usually called the initial stress stiffness matrix and denoted Kσ, can bedetermined as shown in Algorithm 3.3. Due to symmetry it is only necessary todetermine the upper or lower triangle of the matrix, which saves quite a numberof lines in the code and improves the computational efficiency. In both Algorithm3.2 and 3.3 the variable ndof is the number of degrees of freedom for the element,which is the number of nodes multiplied by the number of degrees of freedom pernode, e.g. ndof = 4 × 5 = 20 for four-node elements.

The code generated by MAPLE is efficient but the practical down-side is that itconsists of many lines of code, rendering debugging somewhat difficult. Also, theelement routines are rather cumbersome to work with in the editor but still, froma computational and modeling point of view the result is very satisfactory. Toobtain a more editor-friendly implementation the element routines could be storedin a pre-compiled library, but this has not been implemented.

3.4.2 The SHELLn elements

The SHELL3, SHELL4, SHELL6, SHELL9 and SHELL16 elements are imple-mented using the so-called AG-method suggested by Zienkiewicz in which thenonlinear element matrices are expressed in terms of three auxiliary matrices A,G and H such that Bη = AG and Kσ =

∫GT HG dV (see Zienkiewicz and Taylor

(1991) for details). The current implementation of the AG-method in MUST isnot computationally efficient but very convenient since the element routines can begeneralized for any number of nodes (as opposed to the MAPLE routines, whichmust be derived for each individual element). Therefore, the AG-method is stillused in the current implementation although the element routines have been codeoptimized resulting in about 10% performance increase compared to the originalroutines implemented by Jensen et al. (2001).

36 3.5. Numerical verification

Algorithm 3.2: Pseudo code for deriving Be and Bη in MAPLE

Compute e

Compute η

for i = 1 to 6 do

for j = 1 to ndof do

Compute (Bij)e = ∂ei/∂uj

Compute (Bij)η = ∂ηi/∂uj

end for

end for

Algorithm 3.3: Pseudo code for deriving (∂BTη /∂u)s in MAPLE

Compute BTη

Compute s

for i = 1 to ndof do

for j = 1 to i do

Compute (Kij)σ =∑6

k=1(∂(BTik)η/∂uj)sk

end for

end for

3.5 Numerical verification

The numerical verification of the shell elements consists of two parts: patch testingand general verification by result comparison. The elements chosen for evaluationare the MITC3, MITC4 and SHELL9 elements, which are the three elements usedmost frequently for analysis and optimization with MUST. Extensive testing hasalso been done to determine if the MITC elements suffer from shear locking whenthe element thickness becomes small compared to the element edge length. Thesetests have all shown that this is not the case but for the sake of brevity no suchresults are presented here.

3.5.1 Patch testing

Patch testing is done in order to verify that the elements exhibit properconvergence properties by checking element consistency and stability as discussedby e.g. Razzaque (1986) and Felippa (2003). For the element types implementedin MUST, which are fully integrated and employ standard shape functions, theserequirements are expected to be met and consequently, the patch test can beconstrued as a check for correct implementation.

As patch test strategy for choice of mesh and boundary conditions we adopt theapproach of Dvorkin and Bathe (1984) and Lee and Bathe (2004) for the MITC4and MITC3 elements, respectively. The patch test is successfully passed if theelement can display constant stress when subjected to appropriate boundaryconditions in a non-regular mesh. If the element passes it implies that the


2.000E+005

2.000E+005

2.000E+005

2.000E+005

2.000E+005

2.000E+005

Name: MITC3 patch test s13-

2.000E+005

2.000E+005

2.000E+005

2.000E+005

2.000E+005

2.000E+005

Name: MITC4 patch test s11-

Sxx Sxz

Figure 3.9: Randomly selected patch test results for MITC4 elements (left) and MITC3

elements (right). The same uniform stress results are obtained for all other components.

kinematics are correct since stresses are computed from strains, which are in turncomputed from displacements. As expected both MITC elements pass all patchtests for the five stress components s11, s22, s12, s23 and s13. Not all results arereproduced here but the results for s11 and s13 obtained with the MITC4 andMITC3 elements, respectively, are shown in Fig. 3.9.

The same results have been obtained for the SHELLn family of elements, but thesehave been left out for brevity.

3.5.2 Nonlinear comparative test

To demonstrate the nonlinear performance of the implemented elements in MUSTthey are tested in a single- and multilayer configuration and the results arecompared to those obtained with commercial software packages. The problems aresolved using the arc-length method. The geometry chosen for the tests is shownin Fig. 3.10 for the multilayered case. The cap spans a base of 1000 × 1000 [mm]and the apex rises 100 mm above the base (detailed geometric information isavailable in Section 4.3). For the multilayered case the total thickness of the shellis 20 mm distributed as 1/1/16/1/1 [mm] with the thin sheets being glass/epoxycomposite with Ex = 36 GPa, Ey = Ez = 6.4 GPa, Gxy = Gyz = Gxz = 2.4GPa, νxy = νxz = 0.27 and νyz = 0.33 while the thick middle layer is taken tobe isotropic polymeric foam with E = 125 MPa and ν = 0.3. For the single-layercase the total thickness is 25 mm of aluminum with E = 70 GPa and ν = 0.3.

For the single-layer case three reference solutions have been obtained with ANSYSand ADINA. The ANSYS SHELL181 element is a four-node element formulatedwith MITC as the MITC4 element in MUST and is therefore a natural choice.Furthermore, two quadratic elements have been used – the SHELL93 element inANSYS and the MITC9 element in ADINA, which is widely recognized as one ofthe most robust elements available. As can be seen from Fig. 3.11 the results fromMUST and the commercial codes correlate very well.

38 3.5. Numerical verification

Glass/epoxy composite (45°)

Polymeric foam

Glass/epoxy composite (-45°)

Glass/epoxy composite (-45°)

Glass/epoxy composite (45°)

Figure 3.10: Geometry of spherical cap used for performance comparison.

For the multilayer test three reference solutions have been obtained using ANSYS,ADINA and COSMOS. The element of choice was again the SHELL181 elementin ANSYS but unfortunately the element was unable to converge. ANSYShas recognized the problem and their Research and Development department iscurrently investigating the cause of this problem. Instead the SHELL4L elementin COSMOS has been used as well as quadratic elements in ADINA and ANSYS.The results are compared in Fig. 3.12 and again, the correlation is very good.

Equally good correlation has been found with solutions obtained using thequadratic TYPE75 element in MSC.MARC3 and the MITC4 element in ADINA,but these have been left out for the sake of brevity.

3.5.3 General remarks

During the course of this work a large number of other examples have been tested,both by the author and students working on MUST, and the performance of theshell elements in MUST has consistently been found to be very good. Furthermore,with the modifications made in this work to the element routines the computationalefficiency of MUST for the tested cases is actually higher than or equal to thatachieved with commercial codes. In the multilayered example above MUSTuses approximately 4 minutes with the MITC4 element for generating the entireequilibrium path, only matched by COSMOS, while ADINA uses approximately10 minutes. For the same example MUST uses 14 minutes with the SHELL9element while ANSYS solves the same problem in 18 minutes using the SHELL91(eight-node) element and again, ADINA takes last place by using 25 minutes withthe MITC9 element4.

3Thanks to my colleague, Ph.D. student Lars Christian Terndrup Overgaard, who actuallypushed the buttons in PATRAN.

4The poorer performance of ADINA is probably due to its force-control solver, which seemsto be less efficient than the arc-length solvers employed by the other packages.


0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

Normalized center point displacement

Lo

ad

facto

r,

MUST (MITC4)

MUST (MITC3)

MUST (SHELL9)

ANSYS (SHELL181)

ANSYS (SHELL93)

ADINA (MITC9)k

Figure 3.11: Nonlinear performance of MUST elements compared to commercial codes,

single-layer test with isotropic material. All results are normalized to MUST (MITC4)

results (solid line).

k

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0


Lo

ad

facto

r,

MUST (MITC4)

MUST (MITC3)

MUST (SHELL9)

ANSYS (SHELL91)

COSMOS (SHELL4L)

ADINA (MITC9)

Figure 3.12: Nonlinear performance of MUST elements compared to commercial codes,

multilayer test with iso- and orthotropic materials. All results are normalized to MUST

(MITC4) results (solid line).

40 3.6. Summary and conclusions

These comparisons are interesting in that computational efficiency is very im-portant for the efficiency of the optimization where multiple analyses are runsuccessively. The results render the MITC3 and MITC4 elements particularlyinteresting since they have demonstrated to offer computational efficiency whilemaintaining good predictive capabilities.

Another interesting observation regarding element choice is that higher-orderelements are often overkill in optimization since the benefit of using fewer elementsto obtain the solution contradicts the need for many elements to accuratelyparametrize the design. This is especially true in topology optimization wherea fine mesh is needed to represent the boundaries of the optimal design. Indeed,when reviewing the literature, linear elements are found to dominate the arena inthis field, which just emphasizes the fact that the MITC3 and MITC4 elementsrepresent an important tool.

3.6 Summary and conclusions

This chapter provided a birds eye view on the theory supporting the shellfinite elements in MUST. The elements are formulated using the assumptionsfrom First order Shear Deformation Theory (FSDT) and composite laminatebehavior is encompassed using an Equivalent Single Layer (ESL) description. Thepreferred elements are the stabilized MITC3 and MITC4 elements, which useMixed Interpolation of Tensorial Components (MITC) to avoid problems withshear locking. These elements have been implemented using a modular approachbased on FORTRAN code generated directly from MAPLE, which has resultedin computationally efficient elements. A family of non-stabilized elements hasalso been implemented and of these particularly the SHELL9 element has provenefficient. The MITC3, MITC4 and SHELL9 elements were tested and showed goodpredictive capabilities as well as high computational efficiency.

4

Nonlinear topology optimization

This chapter shifts focus from analysis to optimization by addressingthe topic of material distribution in multilayered structures. Specifically, the

multilayer shell formulation developed in Chapter 3 is deployed in a topologyoptimization framework. This allows material to be added/removed in individuallayers, simulating stiffening of the shell structure. The purpose of the chapter isto investigate the influence of nonlinear effects from large displacements on theoptimal topology in order to determine whether nonlinearities can be disregardedwhen designing laminated structures using structural optimization. Additionally,we want to get experience with nonlinear optimization of laminated compositeshell structures on “familiar territory” before proceeding to other optimizationfields. The bulk of the topics covered in this chapter are also documented inStegmann and Lund (2005b).

The testing ground for this investigation is topology optimization of multilayeredstructures, which is fundamentally identical to “classical” topology optimizationof single-layered structures. The difference is that instead of adding/removingmaterial over the entire thickness we add/remove material in specific layers. Thiswill be explained in greater detail in Section 4.1.

The remaining part of the chapter is organized as follows. In Section 4.1 the designparametrization is discussed and the optimization problem is briefly defined. InSection 4.2 the evaluation of sensitivities is explained and finally, four numericalexamples for plates and shells are presented in Section 4.3.

4.1 Design parametrization

For multilayered topology optimization we distinct between voided and solid layersof the laminate as shown in Fig. 4.1. The solid layers are those that remainunchanged during optimization and the voided layers are those in which material isadded/removed. The strategy for adding/removing material is the SIMP method,see e.g. Bendsøe (1989) or Bendsøe and Sigmund (2003) for details.

In this methodology the material stiffness is scaled by introducing a density

42 4.1. Design parametrization

Solid

Solid

Voided

Voided

Voided

Solid

(b) (c)(a) (d)

Voided

Solid 1

Solid 2

Solid 1

Voided

Voided

Figure 4.1: Typical lamina sequences for plate/shell topology optimization: (a)

“classical” topology optimization, (b) symmetric lay-up with voided core, (c) symmetric

lay-up used with voided outer layers and (d) generic scheme.

parameter, ρe, in the constitutive relation as:

sl = ρeClεl (4.1)

This is standard in topology optimization and the only difference for multilayeredstructures is that the scaling is employed on specific layers (as indicated bysuperscript l in (4.1)) instead of on the entire element. The solid layers arenot scaled and the stiffness of these is calculated using the standard constitutiverelation, i.e. s = Cε. As in topology optimization the element densities, ρe,constitute the design variables and only a single design variable is assigned to anyelement. It should therefore be noted that even though several voided layers maybe specified, all voided layers within the same element will be scaled in the sameway.

Choosing in which layers to apply the scaling in order to perform various types ofoptimization is the next step and several optimization schemes are available.

4.1.1 Optimization schemes

In principle there are no restrictions on how the scaling should be appliedbut a few simple schemes have become standard in shell and plate topologyoptimization of laminated structures. The available options are shown inFig. 4.1 but in the literature schemes (a) and (c) constitute the most employedones, see e.g. Soto and Diaz (1993), Lee et al. (2000), Belblidia et al. (2001) orBelblidia and Bulman (2002).

The single-layer scheme in Fig. 4.1(a) represents “classical” topology optimizationand allows formation of through-the-thickness holes. Using this involves numericalproblems since the stiffness matrix becomes singular when the element stiffnesstends to zero. Remedies for this problem have been treated by e.g. Buhl et al.(2000) and Bruns and Tortorelli (2001) who introduced various “numerical tricks”to bypass the problem. As the topic of interest has not been this type of

Chapter 4. Nonlinear topology optimization 43

K 0=

K 0¹

K 0¹

d

d¹ K 0¹

y

x

y

x

Figure 4.2: Laminate optimized with scheme (b) to zero stiffness (K = 0) of the core,

which is not equivalent to removing the layer.

optimization no such measures have been implemented, and scheme (a) has notbeen utilized.

The scheme in Fig. 4.1(b) seems interesting since core layout design is ofpractical importance in many applications involving stiffening with sandwich cores.However, the scheme lags in that the “removal” of core material by scaling ρe tozero (or near-zero) does not eliminate its stiffness contribution entirely since thepresence of the solid layer in the numerical integration (3.13) still contributes tothe total stiffness of the laminate. This can be illustrated by considering themoment of inertia, Izz, of the structure illustrated in Fig. 4.2. Scaling the stiffnessto zero eliminates the contribution to Izz from the core layer (white) but due tothe parallel axis theorem there will still be a contribution to the total moment ofinertia, proportional to d2. Consequently, scheme (b) does not provide a viablemethod for solving the core layout problem and should be replaced by some updatescheme, which removes/adds both the stiffness of layers and their contribution to(3.13). There is no immediate solution to this problem and no solution has beensought out in this work.

The two remaining schemes, however, have both been employed in the presentstudy. The rib stiffener scheme in Fig. 4.1(c) allows formation of symmetric ribson both sides of a fixed surface, which has many practical applications and sufferfrom none of the deficiencies outlined above. Scheme (c) has been used by e.g.Lee et al. (2000) and will be used in the first numerical example in Section 4.3.The general scheme in Fig. 4.1(d) illustrates the point that any lay-up is treatablein the present formulation and indeed, two of the three numerical examples in (4.3)employ an asymmetric lay-up to form ribs on one side of a shell structure. However,care must be taken to obtain a physically sound model when using scheme (d)and furthermore, introduction of internal voided layers should be avoided unlessspecifically required by the physical model being optimized.

4.1.2 Problem formulation

Having established a design parametrization the optimization problem may nowbe stated. As previously mentioned the objective function is compliance and

44 4.2. Objective function sensitivities

introducing a volume constraint, Vc, the problem is stated as:

Objective : minρ

C(ρ) = pT u(ρ)

Subject to : V ≤ Vc

0 < ρmin ≤ ρe ≤ 1

R(u,ρ) = 0

(4.2)

The constraint acts on the effective volume, V , defined as the sum of the scaledvolume of all voided layers, Nvl, in all elements, Ne, i.e.:

V =Ne∑

e=1

Nvl∑

n=1

ρeVne (4.3)

The volume constraint is included to ensure that the optimizer does not increasethe stiffness (reduce the compliance) by pushing all design variables to 1, whichwould be a rather useless result. To reduce the occurrence of intermediate densitiesthe SIMP methodology is applied to the stiffness scaling whereby (4.1) is actuallycomputed for each layer as:

sl = ρpeC

lεl (4.4)

where the SIMP power p ≥ 1 penalizes densities between 0 and 1 by makingthem uneconomical. This helps push the optimal topology towards a black/white(0/1) design with no or few intermediate values of ρe as described by e.g.Bendsøe and Sigmund (2003). The power is often chosen statically to p ≥ 3 butin this work an incremental scheme is also used in which the power, p, is increasedfrom a starting value to a final value in steps of some chosen size. In this way thepenalization is increased as the design converges, which in our experience resultsin more distinct topologies. As the power increases, however, the term ρp

e tends tozero quite rapidly, which may cause numerical problems. To avoid this problementirely, a conservative lower bound of ρmin = 1 ·10−3 has been used for all designvariables, as indicated in (4.2).

Solution of the problem in (4.2) is done as described in Section 2.2 using analyticaldesign sensitivities and the Method of Moving Asymptotes (MMA). This approachhas proven efficient in other works on nonlinear topology optimization (see e.g.Bruns et al. (2002); Buhl et al. (2000)).

4.2 Objective function sensitivities

In determining the sensitivities of the objective function we assume designindependent loads and write for a particular design variable, ρe:

dC

dρe

= pT du

dρe

(4.5)


The sensitivities du/dρe are not readily available and hence, it is not possible todetermine the sensitivities of the objective function directly from (4.5). To circum-vent this problem a few paths are available: (1) finite difference approximation,(2) direct differentiation method and (3) the adjoint variable method. For thefinite difference to be accurate a central finite difference approximation should beused but this involves two additional analyses and is therefore highly impracticalfrom a computational efficiency point of view. The direct differentiation methodis only favorable if the number of constraints is larger than the number of designvariables, which is not the case here. The adjoint variable method is fairly simpleto implement and has proven efficient for large numbers of design variables andhas therefore been chosen for this work. The procedure for forming an adjointproblem (see e.g. Haug et al., 1986) is described in the following.

4.2.1 Adjoint sensitivity analysis

To establish an expression for the displacement derivative, du/dρe, we use thechain rule to obtain the derivative of the residual, R(u(ρ),ρ) = 0, defined in(2.4):

dR

dρe

=∂R

∂u

du

dρe

+∂R

∂ρe

= 0 (4.6)

Assuming that the structural problem has been solved, i.e. R ≡ 0, (4.6) may berewritten to yield:

du

dρe

= −

[∂R

∂u

]−1∂R

∂ρe

(4.7)

where ∂R/∂u is the global tangent stiffness matrix defined in (2.8). Inserting theabove into (4.5) eliminates the unknown derivative du/dρe:

dC

dρe

= −pT

[∂R

∂u

]−1∂R

∂ρe

(4.8)

In principle (4.8) is sufficient for performing the sensitivity analysis. However,doing so requires finding the inverse of the global tangent stiffness matrix andas such, that would be a very impractical approach. Alternatively, the followingequation could be solved for each design variable:

KT

du

dρe

= −∂R

∂ρe

(4.9)

and the compliance sensitivities can then be obtained by substituting this solutioninto (4.5) for all design variables. This constitutes the direct differentiationapproach but instead we proceed from (4.8) by defining the adjoint variable vector,λ, as follows:

λT = −pT

[∂R

∂u

]−1

(4.10)

46 4.2. Objective function sensitivities

Finding the adjoint variables is very simple and fast since it just requires solvingthe linear system:

KT λ = −p (4.11)

in which the factored stiffness matrix is readily available from solving the analysisand the force vector, p, is known. Now, having solved for λ the objective functionsensitivities may be found directly as:

dC

dρe

= λT ∂R

∂ρe

(4.12)

Finding the derivative of the residual in (4.12) with respect to ρe can be doneby finding the derivative of the internal force vector defined in (2.4) sincedisplacements have been assumed to be independent of the design variables, ρ.The internal element force vector is stated again for reference:

r =

∫

V

BT s dV (4.13)

Finding derivative of (4.13) effectively only involves finding the derivative of thestress vector since the strain-displacement matrix, B, is not dependent on thedesign variables. Thus, the residual derivative can be found by computing thederivative of (4.4) as:

∂sl

∂ρe

= p ρ(p−1)e Clεl (4.14)

And so, the vector product in (4.12) only yields contributions for the given element,e, which renders the sensitivity calculation quite efficient.

It is important to note that the analysis must be solved accurately in order toobtain accurate sensitivities from (4.12) since one of the basic assumptions isR = 0. This is because of the NAND approach taken in this work where theanalysis is solved prior to computing the sensitivities. In our experience the errortolerance on the residual in the nonlinear solution should be no more than around10−10 to 10−8 on the final iteration to avoid problems with numerical noise.

4.2.2 Multiple load cases

When solving the nonlinear topology optimization problem for a number of distinctloads it is likely that the individual designs, although optimal at a single load, arenot optimal at other loads. To obtain a “good compromise” that works well atboth light, medium and heavy loading we solve the problem for the same loadcase but for different load magnitudes. The objective, C(ρ), from (4.2) is thenexpressed as a weighted sum:

C(ρ) =Nq∑

q=1

wq pTq uq(ρ) (4.15)


0

200

400

600

800

1000

0 20 40 60 80 100

p

u

a

b

c

e

d

Figure 4.3: Fictitious load-displacement space with selected load points a = (1, 10),

b = (15, 200), c = (25, 350), d = (80, 550) and e = (100, 1000).

where q is the load case number and wq is a weight factor for load case q. Theweights can be chosen arbitrarily but the natural approach would be to choosethem such that the contribution from each load case is approximately equal. Thisensures that no single load is favored over others and should therefore yield thedesign that performs best at all loads. In the present work two strategies havebeen tested:

I : wq =‖p1‖

‖pq‖II : wq =

(p)T1 u1

(p)Tq uq

(4.16)

where (p)Tq uq in scheme II is the estimated compliance at load number q.

Obtaining these estimates either requires qualified guessing or additional analysis.The approach used most frequently in this work is to run an initial nonlinearsolution using the arc-length solver to obtain the response over the entire load-displacement space. This initial analysis, although not optimized, can then beused to estimate the compliance at different load levels. This requires quite a bitof manual processing and so, scheme I is by far the easiest to handle and could bepreferred for this reason. However, using scheme I can lead to favoring of the highload cases since the displacements in this scheme (4.16) are disregarded resultingin a factor of 10 − 1000 on the weight. To illustrate the difference between thetwo schemes a fictitious load-displacement space has been created and five distinctloads, a–e, have been selected, Fig. 4.3.

Now, using (4.16) the weights for the compliance in (4.15) can be found fromFig. 4.3 by inserting. The results are listed in Table 4.1. As indicated, thedifference between the two schemes is significant but for large numbers of q thiscan be leveled out to some extent by choosing more loads in the lower part of thespectrum. In general, however, it is more “fool proof” to choose scheme II sinceit is hard to predict how the weighting affects the results. There is no “right”approach to choosing the weighting scheme and a bit of tweaking of the weightingfactors must probably be endured before obtaining satisfactory results.

48 4.3. Numerical examples

Table 4.1: Weights computed with scheme I and II in (4.16) for fictitious load-

displacement space in Fig. 4.3. Values are rounded off to six digits.

Scheme I Scheme IIWeight: wi wiCi wi wiCi

Load a 1.00000 10.0000 1.00000 10.0000Load b 0.05000 150.000 0.00333 10.0000Load c 0.02857 250.000 0.00114 10.0000Load d 0.01818 800.000 0.00023 10.0000Load e 0.01000 1000.00 0.00010 10.0000

The weighting factors naturally enter into the sensitivity expression as wellwhereby (4.12) for multiple load cases is written as:

dC

dρe

=Nq∑

q=1

wq λTq

∂Rq

∂ρe

(4.17)

The major disadvantage of solving the multiple load case problem is the increasedcomputational time and memory consumption caused by the need to store multipletangent stiffness matrices. The latter issue is not a generic problem but exists in thecurrent implementation in MUST. For solving problems of practical engineeringinterest this currently poses a severe limitation on MUST as memory usage mayeasily approach the 4 GB limit on standard PC systems. This will, however, beremedied in a later version of the software.

Following a general discussion four examples will now be presented to demonstratethe capabilities and limitations of the methodology used.

4.3 Numerical examples

When presenting the results, load-displacement plots showing the evolution of thetopology with increasing load factor, κ, will be used. These curves are the result ofmultiple nonlinear optimizations, solved at various loads, and thus do not representthe load-displacement curve from a single nonlinear analysis. Consequently, eachmarker “◦” (in Figs. 4.5, 4.9 and 4.16) represents the end result of a nonlinearoptimization at that particular load (a dotted line marks the response predictedwith linear theory.). The load-displacement curve in these plots therefore marksthe boundary for optimal designs, i.e. all non-optimal designs must lie beneaththat curve.

The optimal boundary curve may in practice be obtained in a number of ways. Oneis to start all analyses from the same conditions, e.g. evenly distributed material


(uniform gray) and u = 0. This strategy may lead to local optimum solutionsso at each load, alternatives should be sought out to see if a better solution canbe found. This constitutes solving for the same load several times with differentinitial conditions, e.g. restarting from the previous or subsequent load with statevariables alone or both state variables and topology (this is illustrated in Fig. 4.11).As such, the generation of the optimality boundary in the load-displacement spacemay require a substantial amount of work. Even then, it is not guaranteed thatno better solution can be found at a particular load. This could be the casefor problems exhibiting non monotonic behavior in the load-displacement space,e.g. snap-through behavior, since the Newton-type solvers are not well suitedfor these types of problems. To provide a better understanding of the nonlinearbehavior over the entire load spectrum, the arc-length method has been used onsome optimal designs as illustrated in Figs. 4.12 and 4.14.

The first example is a plate example, which serves as a validation example for thepresent formulation since the linear topology solution is well known. The secondand third examples are for a doubly-curved shell structure and are included todemonstrate the effects of the nonlinearities on a more general structure for singleand multiple load cases, respectively. Finally, a multilayered, curved panel oforthotropic lamina is included.

Note that in the chapter the examples have been solved exclusively using theSHELL9 element since the MITC4 element had not been fully reimplementedwhen these results were generated. Consequently, checker-boarding does notpose a problem but a sensitivity filter has still been employed to ensure mesh-independency (Bendsøe and Sigmund, 2003).

4.3.1 Simply supported 3-layer square plate

The square plate supported at the four edges and subjected to a central pointload is a well-described problem in topology optimization. It has been includedmainly to validate the current formulation by comparing linear results to thoseobtained by others. For the sake of comparison the model data have been adoptedfrom Lee et al. (2000). The dimensions of the plate are 2000 × 2000 [mm] with atotal thickness of 150 mm distributed over three layers as 37.5/75.0/37.5 [mm] allconsisting of isotropic material with E = 21 MPa and ν = 0.30. The skin layersare voided and the core layer is maintained constant (scheme (c) in Fig. 4.1).

A quarter of the plate is modeled using a 20× 20 mesh of SHELL9 elements. Themodel is simply supported on two edges and has symmetry boundary conditionson the opposing edges. This approach is valid in this example since the platedoes not exhibit snap-through and therefore has well-defined nonlinear behavior.The SIMP power is initially 3.0 and is increased by 1.0 every 10 iterations untilit reaches 6.0. This causes some minor fluctuations in the objective function asshown in Fig. 4.4 but is without consequence to the results. The volume constraintis Vc = 0.5.


0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0 10 20 30 40 50Iterations

No

rmal

ized

ob

ject

ive

3

4

5

6

Po

wer

Figure 4.4: History of objective function (left axis, continuous curve) and SIMP power

(right axis, step curve) for linear optimization of square plate example. Load factor is

0.027.

Figure 4.5 shows the evolution of the topology with increasing load factor, κ, andtwo distinct shifts in topology can be observed (the gray areas). As expectedthe topologies at small loads are the same for the linear and nonlinear cases, andcorrespond well to those obtained by e.g. Lee et al. (2000).

With an increasing load two effects can be noted in Fig. 4.5 due to increasingstiffening from in-plane stresses; 1) the four corner reinforcements become widerand extend further into the plate and 2) the center reinforcement is reduced in sizeand instead ribs extend towards the boundaries until at maximum load, the cornerand center reinforcements become a single reinforcement. This example bothvalidates the formulation and gives a first indication that the nonlinear effects areof increasing importance as the load increases. This corresponds well to the usualprogressive nature of nonlinear effects. At light loading the optimal topologiesare identical, see Fig. 4.5, but show increasing deviation as the load increases.However, the increase in performance in very small as indicated in Fig. 4.6, wherethe topology obtained with a linear solution (κ = 0.001) has been compared to thetopology obtained with a nonlinear solution at maximum load (κ = 1.000). This isachieved by performing a nonlinear analysis of the optimal topologies and tracingthe equilibrium path over the entire load spectrum using an arc-length solver.

As indicated in Fig. 4.6 the linear solution performs slightly better than thenonlinear solution in the lower part of the spectrum and vice versa as the loadincreases. This is to be expected but the difference in performance is onlyapproximately 0.45%, which amounts to nothing for practical purposes. This wasnot particularly encouraging so we continue to investigate the same phenomenafor shell structures.

Chapter

4.

Nonlin

ear

topolo

gy

optim

izatio

n51

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1


Lo

adfa

cto

r,k

Figure 4.5: Topological evolution of square plate with increasing center point load. The dotted line is the linear load-displacement

curve, the solid line is the equivalent nonlinear response. Both curves are for the optimized structure.


0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0


Lo

adfa

cto

r,

Boundary of optimal designs

= 0.001

= 1.000

k

k

k

Figure 4.6: Nonlinear response of optimized topologies for nonlinear optimization of

plate example. The curves represent the response of the optimal topologies obtained for

κ = 0.001 and κ = 1.000, respectively. The solid line indicates the boundary of optimal

designs as obtained from Fig. 4.5.

4.3.2 Hinged 4-layer spherical cap – single load case

The doubly-curved shell has been chosen because its behavior is fundamentallydifferent from that of the plate. Instead of stiffening with increasing load thespherical cap becomes more compliant and eventually snaps – a behavior oftenencountered in shell structures. The base of the shell structure is spanned by a1000 × 1000 [mm] square and the center point rises 100 mm above the base. Thefour edges are prescribed by parabolas and the surface is generated by draggingone parabola along an identical parabola. In mathematical terms the surface maybe described as:

z(x, y) = h −2h

l2(x2 + y2

)(4.18)

where h is the height of the center point and l is the edge-length, here 100 [mm]and 1000 [mm], respectively. The thickness of the shell is 24 mm divided over fourlayers as 1/20/1/2 [mm] where the voided layer is the 2 mm skin at the centerof curvature side. The three skins consist of aluminum with E = 70 GPa andν = 0.3 and the core is an isotropic foam with E = 125 MPa and ν = 0.3. The fullgeometry is shown in Fig. 4.7 with actual thickness and distribution of layers. Inthis configuration the stiffening constitutes a relatively small portion of the totalload carrying capacity.


Solid: Aluminum

Solid: Foam

Solid: Aluminum

Voided: Aluminum

Figure 4.7: Geometry of spherical cap example with actual thickness and distribution

of layers. The model is subjected to a center point load and hinged along the four edges.

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0 10 20 30 40 50Iterations

No

rmal

ized

ob

ject

ive

3

4

5

6

Po

wer

Nonlinear,

Linear

= 1k


(right axis, step curve) for nonlinear optimization of spherical cap example. Normalized

to initial compliance.

The full shell geometry is modeled using a 40× 40 mesh of SHELL9 elements andthe model is hinged (ui = 0) on the four edge curves. The volume constraint isVc = 0.5 and the SIMP power is increased from 3.0 to 6.0 in steps of 1.0 over thefirst 30 iterations as shown in Fig. 4.8. As indicated in Fig. 4.8 the optimal designonly has a moderately improved performance over the initial design (uniform grey),which is due to the moderate thickness of the stiffening layer as mentioned above.

Figure 4.9 shows the evolution of the topology with increasing load factor.

54

4.3

.N

um

ericalex

am

ples

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0


Lo

adfa

cto

r,k

Figure 4.9: Topological evolution of spherical cap with increasing center point load. The dotted line is the linear load-displacement

curve, the solid line is the equivalent nonlinear response. Both curves are for the optimized structure. The gray areas mark the

transitions between topologies.


(a) (b) (c)

Figure 4.10: Actual deformed shapes of the three basic topologies of a quarter of the

spherical cap geometry: (a) moderately nonlinear, κ = 0.10, (b) intermediate topology,

κ = 0.37, (c) final configuration, κ = 1.00.

As the load increases the optimal stiffening topology changes five times as markedby the gray zones in Fig. 4.9. The topology changes over these zones due tochanges in the load carrying mechanisms of the structure as shown for the threemajor topologies in Fig. 4.10. The first transition (from (a) to (b) in Fig. 4.10)occurs because material is needed to reinforce the corners, which are subjected tolocal bending. At the same time the center area is highly affected by the load andmust be reinforced as well. The change (from (b) to (c) in Fig. 4.10) is caused byincreasing membrane effects due to the near-flat surfaces extending from the loadtowards the hinged boundaries and along the symmetry planes.

It is interesting to note the major jump in topology that occurs when the structuresnaps through (the horizontal gray area in Fig. 4.9). The sudden change intopology can be seen clearly when starting an optimization from a high loadfactor (the post-snap regime, κ > 0.35) and solving for a smaller load factor.This is illustrated in Fig. 4.11 where the state variables obtained at κ = 0.36 havebeen reused as initial guess for solving for κ = 0.35. As can be seen, the optimizereffectively pushes the design to the pre-snap configuration between iteration 19 and20. This also indicates that the curve in Fig. 4.9 may be discontinuous betweenthese two points or, at least, have near-zero slope. This makes good sense sincethe pre-snap configuration is by far more attractive in terms of compliance thanthe post-snap configuration.

The increased performance of the nonlinear design over the linear solution ismodest since the stiffening layer only contributes moderately to the total stiffness.However, the structure exhibits noticeably different nonlinear response for thedesign obtained with the linear method compared to some of those obtained withthe nonlinear method. Consider the three load-displacement curves in Fig. 4.12,generated using an arc-length solver for small, medium and heavy load (κ = 0.01,κ = 0.42 and κ = 1.00, respectively). As expected all three curves stay belowthe boundary of optimal designs (obtained from Fig. 4.9) but the response of thethree designs is not the same over the entire load spectrum. At low load factorsthe curves are almost indistinguishable but as the load increases the differenttopologies show their strengths and weaknesses.


0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0 25 50 75 100 125 150Iterations

No

rmal

ized

ob

ject

ive

3

4

5

6

Po

wer


(right axis, step curve) for nonlinear optimization of spherical cap example, restarted from

state variables at κ = 0.36 and solved for κ = 0.35.

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0


Lo

adfa

cto

r,

Boundary of optimal designs

= 0.01

= 0.42

= 1.00

k

k

k

k

Figure 4.12: Nonlinear response of optimized topologies for nonlinear optimization

of spherical cap example. The curves represent the response of the optimal topologies

obtained for κ = 0.01, κ = 0.42 and κ = 1.00, respectively. The solid line indicates the

boundary of optimal designs as obtained from Fig. 4.9.


Solid: Aluminum

Solid: Foam

Solid: Aluminum

Voided: Aluminum

Figure 4.13: Geometry of spherical cap (multiple load cases) example with actual

thickness and distribution of layers. The model is subjected to a center point load and

hinged along the four edges.

In Fig. 4.12 it is noteworthy that the result for κ = 0.42, although not optimal inall configurations, has a superior ability to sustain load during snap-through. Thisis another interesting aspect of utilizing nonlinear optimization for such problems.

As demonstrated, the spherical cap example shows significant influence of nonlin-ear effects, particularly for very large displacements. The changes in topology canbe explained by the change in load carrying properties, which is not accounted forwith small displacement analysis. The characteristic property of the topologies forlarge displacements is that they seem to be more “global” since the deformation inthese cases involve the majority of the structure. The result is a simpler and morecontinuous reinforcement pattern, which is desirable from a manufacturing pointof view since that type of pattern will be much easier to manufacture than thediscontinuous reenforcement pattern suggested by the optimal topology obtainedwith a linear solution. This will be discussed further in the last example.

4.3.3 Hinged 4-layer spherical cap – multiple load cases

This example is identical to the previous one in geometry and materials but thethickness distribution is now 1/10/1/4 [mm], where the 4 mm layer is voided asshown in Fig. 4.13. With a thickness share of 25% the stiffening layer constitutesthe majority of the total structural stiffness and consequently, the distributionof material in the stiffening layer plays a decisive role for the load carryingmechanisms of the structure. This reduces the robustness of the present method inthe sense that individual topologies will potentially exhibit very different nonlinearbehavior over the load spectrum. This is not an acceptable behavior for engineeringstructures and choosing a single topology is thus not a viable approach. Instead,we solve the problem for 5 and 10 different loads simultaneously, as described inSection 4.2.2.


The multiple load case optimizations are solved at κ values of 0.01, 0.17, 0.33,0.67, 1.0 (5 load cases) and 0.03, 0.10, 0.17, 0.33, 0.40, 0.47, 0.53, 0.60, 0.67, 0.83(10 load cases). For the 10 load case optimization the weights are chosen as ascaling of the load (scheme I in (4.16)) while the 5 load case optimization is scaledusing the estimated compliance values at each load (scheme II in (4.16)). Theestimated values are obtained by running an initial analysis of the non-optimizedstructure. Consequently, the weighting is not entirely uniform as indicated inTable 4.1, page 48, but uniform to an order of magnitude, which is sufficient toavoid severe favoring of individual load cases. The choice of load factors, κ, is theresult of an iterative process and is used primarily to seek out different topologies,testing their nonlinear response, and deciding which ones perform satisfactorily.This procedure is somewhat “unscientific” and illustrates the tweaking involved inusing the weighted sum formulation.

Using the approach described above should result in the topology (or topologies)that constitutes a “good compromise” between the very different topologies ob-tained at distinct loads, given the criteria defined in the weighted sum formulation.In Fig. 4.14 the obtained topologies for 5 and 10 load cases are shown together withreference topologies obtained using a linear and nonlinear solution, respectively.An arc-length solver has been used to determine the nonlinear response of both thetopology obtained from a linear solution, the topology obtained from a nonlinearsolution at κ = 1 and the topologies obtained with 5 and 10 load cases.

Note that the curves in Fig. 4.14 represent transverse center point displacementin the thickness direction. The step in the curves for the nonlinear solution(κ = 1.00) and the 5 load case solution indicates that the geometry is deformingwith components perpendicular to the transverse direction, which is a well-knownbehavior of such structures. The reason that this is observed here and not inthe previous example is the reduced thickness, which reduces the stiffness of theshell in a way that allows it to snap in other directions before actually snappingthrough.

Considering the graphical representation of the optimal topology obtained for 10load cases (Fig. 4.14) it is apparent that it represents a compromise between thelinear and nonlinear topologies. Furthermore, the nonlinear response of the 10 loadcase design represents a good compromise at all loads. This is a very promisingresult, which indicates that the weighted sum formulation may indeed be used togenerate structures that perform well at very different load levels.

Chapter

4.

Nonlin

ear

topolo

gy

optim

izatio

n59

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0


Lo

adfa

cto

r,k

5 LC 10 LCLinear k = 1.00

Figure 4.14: Nonlinear response of optimized topologies for nonlinear optimization with multiple loads of spherical cap example. The

curves represent the response of the optimal topologies obtained with the linear solution and the nonlinear solution for κ = 1.00 as

well as the results of solving with 5 and 10 load cases (LC).


Voided: Glass/epoxy biax (+45°/-45°)

Solid: Foam

Solid: Glass/epoxy UD (0°)

Voided: Glass/epoxy biax (+45°/-45°)

Solid: Glass/epoxy UD (0°)

Figure 4.15: Geometry of cylindrical shell example shown with actual thickness and

distribution of layers. Fiber angles are taken with respect to the straight edges of the

shell. The model is subjected to a center point load and hinged at the four corners.

4.3.4 Corner hinged 5-layer cylindrical shell

This example demonstrates the use of orthotropic materials in the presentformulation. The geometry is a cylindrical shell segment as shown in Fig. 4.15where all edge lengths are 1000 mm and the curved edge spans θ = 1/0.86 radof a circle with radius 860 mm. The shell rises 141.3 mm above the base, whichmeasures 1000 × 944.6 [mm]. This is a standard geometry for shell examplesand has been used by numerous authors. The lay-up of the shell is a 5-layersymmetric laminate [±45/0/0/0/±45] with a total thickness of 14 mm distributedas 1/1/10/1/1 [mm]. The four skin layers consist of an orthotropic glass fiberreinforced epoxy composite with the following material properties: Ex = 36 GPa,Ey = Ez = 6.4 GPa, Gxy = Gyz = Gxz = 2.4 GPa, νxy = νxz = 0.27 andνyz = 0.33. The core is taken to be the same isotropic foam as used above withE = 125 MPa and ν = 0.3. For the ±45 biax layer we use averaged materialproperties, which discards the bending-twist coupling that would occur if a +45layer was stacked upon a −45 layer. The properties used are Ex = Ey = 6.6GPa, Ez = 5.8 GPa, Gxy = 5.2 GPa, Gyz = Gxz = 2.2 GPa, νxy = 0.50 andνyz = νxz = 0.30.

The shell is modeled using a mesh of 50 × 50 SHELL9 elements and the modelis supported in three directions (ui = 0) at the four corner nodes. This rendersthe model very flexible and snap-through occurs at a relatively low load factor asshown in Fig. 4.16, which also shows the change in topology with increasing load.

In Fig. 4.16 the cross-shaped topology obtained for large displacements coincidewith the primary material directions of the biax reinforcement layer and as suchcorrespond well to the anticipated design.

Chapter

4.

Nonlin

ear

topolo

gy

optim

izatio

n61

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0Normalized center point displacement

Lo

adfa

cto

r,k

Figure 4.16: Topological evolution of cylindrical shell with increasing center point load. The dotted line is the linear load-displacement

curve, the solid line is the equivalent nonlinear response. Both curves are for the optimized structure. The gray areas mark the

transitions between topologies.


Figure 4.17: Optimal stiffening topology for cylindrical shell, solved for 10 load factors

(0.01, 0.04, 0.08, 0.12, 0.20, 0.40, 0.56, 0.64, 0.80, 1.00).

As observed in the previous example the large displacement topologies arecontinuous over the geometry and consequently have a better manufacturability.This is particularly the case for laminated composite structures since fibres aremost often laid out in mats covering larger areas. Making the local patternsobtained with a linear solution involves a number of problems.

First, manufacturing is more difficult because several smaller and geometricallymore complex patches of fibre mat must be cut out and handled. Second, localizedreinforcement would lead to an increased number of local stress concentrationsand incidently, an increased number of locations from which a potential crackmight spring. These aspects are very important when designing reenforcements forlaminated composite structures and should be taken into account when choosingthe optimal topology to go from.

As in the previous example the problem has also been solved for multiple loadfactors to obtain a compromise design that will perform well throughout the loadspectrum. The 10 load factors considered are 0.01, 0.04, 0.08, 0.12, 0.20, 0.40,0.56, 0.64, 0.80, 1.00, which yields the design shown in Fig. 4.17 using scheme Ifrom (4.16). Again, the multiple load factor design is a compromise between thedifferent topologies, which is evident when comparing Fig. 4.16 to Fig. 4.17.


In this chapter the effect of including the full Green-Lagrange strain measurein the topology optimization formulation for layered shell structures has beeninvestigated. The static problem is solved iteratively using the Newton-Raphsonmethod and the minimum compliance optimization problem is solved using theadjoint variable method and the Method of Moving Asymptotes (MMA). Thedesign parametrization allows multiple voided layers in each element to be scaledby a single design variable while any additional layers remain fixed (solid). Thiscircumvents problems of near-zero terms in the stiffness matrix and renders the


problem solvable without the need for any additional numerical tricks. Multipleload cases are handled in a weighted sum formulation and allows us to solve for“best compromise” designs in a wide load spectrum.

Four numerical examples demonstrate the capabilities of the method and showthe effects of the nonlinear terms. The performance of the nonlinear designs overthe linear solutions is better in all examples (lower compliance) but the differencein compliance may be relatively small. This is the case in the first spherical capexample where the optimal distribution of an isotropic reinforcement material isfound for a single and relatively thin layer. If the thickness of the reinforcementlayer is increased substantially the difference in compliance also increases asdemonstrated in the second spherical cap example. Here, a major difference inthe nonlinear response for topologies obtained at low and high loading can alsobe observed. Furthermore, the example shows the “best compromise” responseobtained when solving for 10 different load factors simultaneously. This indicatesthat a multiple load case approach should be taken for structures operating at bothlow and high loads. Finally, a cylindrical shell example shows the reinforcementpattern of two opposite orthotropic biax layers. The result correspond well to theanticipated result in that the reinforcement coincides with the principal materialdirections of the biax.

In all the examples a significant change in topology is observed for increasing loads.Also, the results are consistent with the progressive nature of nonlinear effects inthat a gradual increase of loads yields a gradual change in topology (starting fromthe linear and nonlinear topologies being identical and tending towards increasingdeviation of the nonlinear topologies). A common characteristic of the largedisplacement topologies is that they are “global” in nature where the linear resultstend to involve local reinforcements. This is to be expected since an increasingportion of the geometry becomes affected when the load increases. An interestingobservation is that the manufacturability of the large displacement results seems tobe higher since they are continuous over the geometry. For laminated compositesthis is essential since application of smaller reinforcement patches would be a highlyimpractical approach.

5

Discrete material optimization

This chapter is dedicated to presenting a novel approach to materiallayout and fiber angle optimization, called Discrete Material Optimization

(DMO). For laminated composites the choice of material and orientation canbe difficult but it is an important topic since efficient design with laminatedcomposites is dependent on the engineer being able to exploit the directionality ofthe material. This has historically been associated with great difficulty but it isbelieved that DMO can be used to solve the general material optimization problemefficiently for laminated composite shell structures. This objective is three-foldsince it requires simultaneous solution for material, orientation and stacking. Aswill be shown, this can be encompassed in the same parametrization and solvedwithin a finite element framework. The main points presented in this chapter arealso published in Stegmann and Lund (2005a).

Displacements will in this chapter be assumed small and material behavior linearbut the proposed parametrization could easily be used for nonlinear problems aswell. For the linear case the element stiffness matrix, Ke, is defined as in (2.11)but stated here again for reference:

Ke =

∫

V

BT CB dV (5.1)

which is summed over all elements and used for solving the linear static equilibriumequations:

Ku = p (5.2)

Again, the compliance, C, is used as objective function and may for the linear casebe stated as:

C = uT p = uT Ku = 2U (5.3)

where U is the total strain energy. These equations form the basis for implementingthe proposed parametrization and will be addressed in more detail in Section 5.2.

The rest of the chapter is organized as follows. Section 5.1 discusses orientationaloptimization methods to place DMO in a context, Section 5.2 introduces theDMO parametrization, Section 5.3 takes a closer look at the element level

66 5.1. Orientation optimization with orthotropic materials

q1

qm

qNe

Figure 5.1: Illustration of the classical concept of orientation optimization in a finite

element framework. The arrows represent the 1st principal material direction (denoted

by θm) of the orthotropic material.

parametrization and Section 5.4 states the optimization problem. In Section 5.5 anumber of 2D and 3D examples are presented to illustrate the capabilities of theDMO method.

5.1 Orientation optimization with orthotropic materials

The classical approach to solve for optimal orientation of orthotropic materials andminimum compliance has been to use the local orientation (denoted by local fiberangles, θ) as design variables. In a finite element framework this can be depicted asshown in Fig. 5.1, where each arrow represents the 1st principal material directionand is uniquely defined in each element (or layer) by the angle, θm, relative tosome fixed frame. The design variables are then the continuous parameters, θm,and thus the method is often referred to as Continuous Fiber Angle Optimization(CFAO). The optimization problem may be stated as:

Objective : minθ

C(θ) = pT u

Subject to : Ku = p

θmin ≤ θ ≤ θmax

(5.4)

where θmin and θmax contain the lower and upper bounds on the design variables,respectively (typically −90◦ and +90◦). The design sensitivities (compliancesensitivities) can be obtained as the derivative of (5.3) and when assuming designindependent loads, the following expression may be derived (see e.g. Pedersen(1991) or Masur (1970)).

dC

dθm

= −uT dK

dθm

u = −(uem)T ∂Ke

m

∂θm

uem (5.5)

This expression is valid if θm only pertains to a single element. Later we willintroduce so-called patch variables which affect several elements and in such cases

Chapter 5. Discrete material optimization 67

x

y

y

x

1

(a) (b)

P

P

PP

q

2q

2

5

10

Figure 5.2: Example geometry and boundary conditions (a) and optimum solution (b).

the sensitivity dC/dθm must be evaluated as a sum over all elements affectedby the variable. The expression in (5.5) is common to both CFAO and topologyoptimization and is in practice quite efficient to evaluate either analytically or morecommonly for CFAO by a finite difference approximation for general geometries.

The major difficulty faced when using this type of formulation is that the globaldesign space becomes non-convex. To illustrate this the simple problem in Fig. 5.2has been constructed and solved using a continuous formulation and the Methodof Moving Asymptotes (MMA) by Svanberg (1987). The result is the design spaceshown in Fig. 5.3(a), where four extrema can be found – one global and three local(see also Table 5.1). To obtain the global optimum solution the initial guess mustbe within the non-shaded part of the design space (the safe domain) as illustratedin Fig. 5.3(b). In a simple case like this it is fairly easy to determine how to startthe optimization but for complicated geometries with multiple layers it becomesalmost impossible to state the problem a priori within the safe domain. It followsthat for generic problems the results obtained in this manner with CFAO will oftenbe sub-optimal albeit better than the initial design.

Table 5.1: Extremum values for the design space in Fig. 5.3.

Optimum: Global Local #1 Local #2 Local #3

Element 1 24.2◦ 29.3◦

−90.0◦

−90.0◦

Element 2 −41.6◦ 90.0◦ 90.0◦

−47.3◦

This is, of course, not a new realization and several methods have already beenproposed to circumvent the problem of local optimum solutions. These methodsroughly fall within one of four categories:

1. Analytical methods, which rely on the closed-form formulation of anoptimality criterion as described by Prager (1970). This constitutes a

68 5.1. Orientation optimization with orthotropic materials

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1,40

1,48

1,56

1,64

1,72

1,80

Normalized ocjective

(compliance)

Element 2

Element 1

(a) (b)

Optimum

Figure 5.3: Normalized objective function for CFAO test example. The red dots mark

the local extrema and the shaded area marks values that will lead to local optimum

solution.

limitation in the applicability of the methods but for simpler geometries themethods have been applied very successfully and often serve as benchmarksolutions for purely numerical methods. The major contributor to optimalitycriterion methods in orientation design is Pedersen (1991) who has usedthe method to optimize 2D continua, beams and plates. Luo and Gea(1998) used a similar approach to solve for optimal orientation of plate beadstiffeners.

2. Mathematical programming techniques are purely numerical and are con-cerned with tuning the optimizer itself rather than reformulating the problemor change the parametrization. Among others Bruyneel and Fleury (2002)and Moita et al. (2000) have applied such methodology with success. Still,these methods do not ensure convergence to the globally optimum solution.

3. Parametrization methods start from the realization that the problem oflocal optimum solutions is inherent in the CFAO method. The goal ofthe methods is to change the parametrization and obtain a convex designspace. Such methods include the Lamination Parameter Method introducedby Tsai and Pagano (1968) and used by e.g. Miki and Sugiyama (1993) fororientation optimization of plates and Foldager et al. (2001) for plates andcylindrical shells. The method requires closed-form analytical formulation ofthe feasible domain of the lamination parameters which has so far only beenachieved for relatively simple geometries.

4. Evolutionary techniques, attributed to Holland (1975), are fundamentallysearch methods but employ ideas derived from genetics and Darwin’s “sur-vival of the fittest” principle to more effectively select the best solution from


a population of solutions. Genetic Algorithms (GAs) are particularly usefulfor discrete problems and problems where the sensitivities are impossibleor extremely difficult to compute. If the population is sufficiently largethe method reduces the risk of obtaining a local optimum solution but stillcannot guarantee convergence to the global optimum. The major problemfaced with GAs is the amount of computational effort involved when solvingproblems with a large number of design variables, particularly if the analysisitself is computationally expensive. GAs have been successfully applied toe.g. stacking sequence optimization by Riche and Haftka (1993), Adali et al.(1995) and others.

The work-horse of engineering design in industry today is the mathematicalprogramming techniques, which have been integrated in e.g. the commercial codesBOSS QUATTRO from SAMTECH and OPTISCTRUCT from ALTAIR. Suchsoftware cannot guarantee that the obtained solution is the global optimum butis still a valuable design tool since it can provide engineers with a significantperformance increase. The goal of the present work is to provide a novel methodthat obtains a close approximation to the global optimum solution while beingapplicable to problems of industrial interest. To this end a new parametrizationis proposed.

5.2 The discrete material optimization method

The basic idea in the discrete material optimization (DMO) parametrization isessentially an extension of the ideas used in structural topology optimizationbut instead of choosing between solid and void we want to choose betweenany distinct number of materials. This methodology can be stated as: for all

elements in the structure find one distinct material from a set of pre-defined

candidate materials such that the objective function is minimized. Metaphoricallythis is like painting a picture from a palette of colors – the optimal picture isthe one having the best combination of colors over the canvas (although someartists might object to this crude portrayal of their trade). The idea of usingtopology optimization for material selection in this way was first introduced fortwo and three candidate materials (phases) as multiphase topology optimization

by Sigmund and co-workers. The first applications of the three-phase methodwas that of Sigmund and Torquato (1997) who used it for designing materialswith extreme thermal expansion properties and later by Gibiansky and Sigmund(2000) for designing materials with extreme bulk modules. The method has alsobeen extended to actuator design, Sigmund (2001), which is closely related tocompliant mechanism design, Sigmund (1997). Recently Wang and Wang (2004)introduced a level-set method based on the same ideas as an alternative to theapproach of Sigmund and co-workers. The level-set method is applied with up tofour phases to 2D structural problems and shows promising results.

Common to the parametrization used in the works stated above is that materials

70 5.2. The discrete material optimization method

are assumed to be isotropic, the elements are single layered and the maximumnumber of phases involved is four – three distinct materials and void. Furthermore,the structures considered have been either 2D continua or plates. The DMOformulation extends the scope of application by considering multiple phases and3D structures and furthermore by allowing the structures to be multilayered andof orthotropic materials. However, the method is still limited to operate on a fixeddesign domain, i.e. thicknesses and shape are not changed during optimization.

In the context of orientation optimization, different materials simply mean thesame material oriented at various angles in space (fiber angles) but in general,it might as well mean Carbon or Glass Fiber Reinforced Plastic (CFRP/GFRP),polymer foam, steel, aluminum or any other material at any orientation. As such,the proposed formulation is very versatile and can be used to optimize the materialconstitution of structures in general and composite structures in particular.

5.2.1 The methodology

As in topology optimization the parametrization of the DMO formulation isinvoked at the finite element level. The element constitutive matrix, Ce, isexpressed as a weighted sum of candidate materials, each characterized by aconstitutive matrix, Ci. In general, this may be expressed as a sum over theelement number of candidate materials, ne:

Ce =ne∑

i=1

wiCi = w1C1 + w2C2 + · · · + wneCne , 0 ≤ wi ≤ 1 (5.6)

It follows that the number of candidate materials is also the number of elementdesign variables and if Ne is the number of elements, the total number of design

variables, Ndv, for single layered structures is Ndv =∑Ne

i=1 nei . Note that

the “classical” topology optimization formulation having one design variable perelement is obtained by setting ne = 1 in (5.6).

The weights, wi, in (5.6) must have values between 0 and 1 as no matrix cancontribute more than the physical material properties and a negative contributionis physically meaningless. In this way, as in classical topology optimization,the weights on the constitutive matrices become “switches” that turn on and offstiffness contributions such that the objective is minimized and a distinct choice ofcandidate material is made. This underlines that the DMO method relies heavilyon the ability of the optimizer to push all weights to the limit values. Any elementhaving intermediate values of the weights must be regarded as undefined sincethe constitutive properties are non-physical. For the same reason any elementhaving more than a single weight of value 1 must be considered undefined as well.Consequently, the single most important requirement for the DMO method is thatevery element must have one single weight of value 1 and all other weights of value

0. Failing to comply with this essentially renders the results meaningless. It followsthat the choice of weighting functions, wi, is very important for the performance of


-90 -60 -30 0 30 60 90

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1,72-1,80

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1,16-1,24

1,08-1,16

1,00-1,08

y

x

-41.6°

(b)(a)

24.2°qi

qi

Figure 5.4: Normalized objective function for DMO test example. (a) The candidate

materials at 12 angles, θi. (b) The yellow dots mark possible combinations of candidate

materials and the large and small white dots mark optimum solutions obtained with

DMO and CFAO, respectively.

the DMO method and several formulations have been developed and evaluated (asdescribed shortly). The initial values of the design variables, xi, may in principlebe any set of numbers between 0 and 1 but in general the values should be chosensuch that the initial weighting is uniform, i.e. wi = wj for all i, j = 1 . . . ne. Thisprovides the most “fair” starting guess since no materials are favored a priori.

To illustrate the methodology for fiber angle optimization we solve the examplein Fig. 5.2 using DMO1 with the same orthotropic material oriented at 12different angles (0◦,±15◦,±30◦,±45◦,±60◦,±75◦, 90◦) as the candidate materials(Fig. 5.4(a)). The possible material constitutions for the structure are then allcombinations of the 12 candidate materials for two elements, e.g. 12 × 12 = 144combinations in all (±90◦ are identical). These are marked by yellow dots inFig. 5.4(b) where the obtained optimum solution 30◦/−45◦ is marked by thelarge white dot. This solution is the “best fit” to the global optimum solution24.2◦/−41.6◦ obtained with CFAO (see Table 5.1). The normalized compliance ofthe DMO solution is 1.0040, which is very close to the global optimum solution.

5.3 Element level parametrization

From the general form of the DMO material interpolation given in (5.6) severalinterpolation schemes can be derived. In this work seven schemes have been tested,four of which are new, two have been adapted and one has been adopted. In the

1This simple test case has been solved using DMO scheme 4, see Section 5.3.5 for details.

72 5.3. Element level parametrization

following, the seven implemented interpolation schemes will be introduced anddiscussed, based on experiences obtained through a number of tests performedduring the course of this work. However, only the main conclusion drawn will bepresented here since an elaborate account of the entire process would be excessiveand most likely disguise the main points rather than emphasize them. All of theinterpolation schemes have been implemented and tested in MUST, but it is veryeasy to implement the schemes in any existing finite element code as well, as willbe illustrated in Section 5.3.4.

5.3.1 DMO scheme 1

This adapted scheme is probably the simplest and most obvious choice ofweight functions. The idea is to extend the classical topology optimizationparametrization to multiple design variables, xi, by adding terms as:

Ce =

ne∑

i=1

(xei )

p

︸︷︷︸wi

Ci = (xe1)

pC1 +(xe2)

pC2 + · · · +(xene)pCne , 0 ≤ xi ≤ 1 (5.7)

In this formulation each design variable scales only one constitutive matrix and hasno influence on any of the other matrices. To push the design variables towards0 and 1 the SIMP method has been adopted by introducing the power, p, as apenalization of intermediate values of xi (see e.g. Bendsøe and Sigmund (2003)for details). The method in (5.7) is not very efficient as it fails to push the designvariables to the limit values for all the cases tested.

5.3.2 DMO scheme 2

The problems with scheme 1 were also realized by Sigmund and co-workers(Gibiansky and Sigmund, 2000; Sigmund and Torquato, 1997) who instead usedthe following formulation for two distinct materials (three phases), which has beenadopted in MUST:

Ce = (xe0)

p([1 − (xe

1)p]C1 + (xe

1)pC2

), 0 ≤ xi ≤ 1 (5.8)

This formulation is fundamentally different from scheme 1 in that x0 scales theentire contribution to Ce while x1 “slides” between C1 and C2. As such, theformulation encompasses simultaneous topology optimization (through x0) andmultiple material optimization (through x1). The major difference between thisformulation and (5.7) is the term (1 − xe

i ), which links a single design variable tomore than one material. The benefit of that is that adding weight in one placeautomatically reduces weight in others, thus helping to push the weights towards0 and 1. For pure material selection design (such as fiber angle optimization) thex0 variable can be left out.

As can be seen from (5.8) the SIMP methodology is still encompassed in thisformulation. However, several implementations of the power, p, have been


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0

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1

0

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1

x1x1 x1

w2 w2 w2

w1 w1 w1

Figure 5.5: Weight functions, w1 and w2, for two materials, computed with DMO

scheme 3. Top and bottom row represent w1 and w2, respectively.

suggested but here we follow the strategy suggested by Gibiansky and Sigmund(2000). The formulation in (5.8) is not very general since it can only handle twomaterials, so the scheme has been adapted to multiple materials in MUST.

5.3.3 DMO scheme 3

This scheme is a simple extension of scheme 2 that can encompass any number ofmaterials by adding terms to (5.8), e.g. for three distinct materials (four phases):

Ce = (xe0)

p([1 − (xe

1)p]C1 + (xe

1)p{

[1 − (xe2)

p]C2 + (xe2)

pC3

})

= (xe0)

p([1 − (xe

1)p]︸︷︷︸

w1

C1 + (xe1)

p[1 − (xe2)

p]︸︷︷︸w2

C2 + (xe1)

p(xe2)

p

︸︷︷︸w3

C3

)(5.9)

where the limits 0 ≤ xi ≤ 1 have been excluded for brevity. The expression in (5.9)becomes tedious to write out for larger number of variables but can be generalizedfor any number of candidate materials:

Ce = (xe0)

p

ne−1∑

i=1

[[1 − (xe

i )]p

i−1∏

j=1

(xej)

p

]

︸︷︷︸wi

Ci +ne−1∏

j=1

(xej)

p

︸︷︷︸wne

Cne (5.10)

where we have that∑

wi = 1. To accommodate fiber angle optimization thescheme can in MUST be used either with or without the topology optimizationvariable, x0. DMO scheme 3 has proven efficient for up to three phases but whenthe number of phases is greater the formulation tends to get stuck in local optima.This is particularly the case when the remaining choice is between two almost


p q= 3 = 3, p q= 3 = 9, p q= 3 = 15,

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1

x1x1 x1

w2 w2 w2

w2 w2 w2

p q= 1 = 1, p q= 1 = 3, p q= 1 = 15,

Figure 5.6: Weight function, w2, for two materials, computed with DMO scheme 3. Top

and bottom row represent w2 with q-penalization and w2 with combined pq-penalization,

respectively.

p = 1 = 3, q p q= 3 = 3,

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Figure 5.7: Sum of weight functions, w1 and w2, for two materials, computed with

DMO scheme 3.

equally good materials. This can be explained by considering a plot of the weightsfor two materials as shown in Fig. 5.5. Since the two weights add up to 1 any spotin the design space is equally good and the optimizer cannot figure out where togo since nothing forces it to choose. This behavior has been observed for elementsin pure shear where the same orthotropic material at ±45◦ is equally good. Tocounter this a modified SIMP scheme has been tried in which a second power, q, isintroduced on the weights as [1 − (xe

i )p]q and (xe

j)pq. This changes the behavior

of the weight as illustrated for the second weight, w2, in Fig. 5.6 while the firstweight, w1, is unaffected. Using the pq-power scheme the weights no longer addup to one and the middle region of the space becomes unfavorable as shown inFig. 5.7. This improves the capabilities of the method in some cases but requiressome tweaking of the power q to converge to the optimum solution. Consequently,interpolation scheme 3 in (5.10) is not generally employed.


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w1 w1 w1



5.3.4 DMO scheme 4

As an alternative to scheme 3 above the following scheme has been developed,which is essentially an extension of scheme 1 in (5.7). The difference is that eachdesign variable now affects all weights as:

Ce =ne∑

i=1

[(xe

i )p

ne∏

j=1;j 6=i

[1 − (xe

j)p] ]

︸︷︷︸wi

Ci (5.11)

The difference from (5.7) is the term (1−xej), which is included so that an increase

in xi automatically involves a decrease in all other weights. The behavior of theinterpolation for two materials is illustrated in Fig. 5.8.

The introduction of the (1 − xej) term helps drive the design towards 0/1 and the

method has proven quite effective for the problems tested. The difference between(5.10) and (5.11) becomes clearer when writing out the expression for e.g threematerials (phases) and comparing to (5.9):

Ce = (xe1)

p[1 − (xe2)

p][1 − (xe3)

p]︸︷︷︸w1

C1

+ (xe2)

p[1 − (xe1)

p][1 − (xe3)

p]︸︷︷︸w2

C2 + (xe3)

p[1 − (xe1)

p][1 − (xe2)

p]︸︷︷︸w3

C3

(5.12)

The interpolation is quite simple to implement in a general way from (5.11) andthe weight factors can be computed efficiently for both analysis and sensitivityanalysis as illustrated in Algorithm 5.1.


Algorithm 5.1: Pseudo code for computing weight factors with DMO scheme 4.

wi = 1 for all i = 1 . . . ne . Initializationif (analysis) then

for i = 1 to ne do

wi = xpi

for j = 1 to ne do

if (i 6= j) wj = wj × (1 − xpi )

end for

end for

else if (sensitivity analysis) then

for i = 1 to ne do

if (k = i) then . k is the active variable, i.e. ∂wi/∂xk

wi = p xp−1k . From xp

k termelse

wi = −p xp−1k . From (1 − xp

k) termwi = wi × xp

i . Constant termend if

for j = 1 to ne do

if ((i 6= j) and (i 6= k)) wj = wj × (1 − xpi )

end for

end for

end if

The structure of Algorithm 5.1 is identical for the other interpolation schemes inMUST and consequently, pseudo codes for these have not been included.

The disadvantage of DMO scheme 4 in (5.11) is that the weighting functions ingeneral do not add up to unity, i.e.

∑wi 6= 1, which is illustrated in Fig. 5.9.

Consequently, the element stiffness will be unrealistically low initially and onlyslowly (due to move limits on xi) be pushed towards the physical stiffness. Whenthe problem has converged, and the design variables have been pushed to 0 and 1,the sum of the weights is 1 as indicated in Fig. 5.9.

If the convergence in compliance of the problem considered is monotonic then theinitially low stiffness poses no problem but increases the number of iterations toreach optimum and thus increases the computational cost. For problems relyingon a realistic stiffness, however, the low initial stiffness presents a major problemand can lead to erroneous results. This has been observed when optimizing forfrequency response where the low stiffness regions give rise to very large variationsin the ratio between stiffness and mass (k/m), which ultimately leads to numericalproblems and renders the analysis unsolvable. Furthermore, evaluating the massconstraint or other physical constraints from (5.11) is meaningless due to theartificially small scaling. To circumvent these problems a scaled version of (5.11)has been formulated.


p = 1 p = 3

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Figure 5.9: Sum of weight function, w1 and w2, for two materials, computed with DMO

scheme 4.

5.3.5 DMO scheme 5

DMO scheme 5 is an extension of scheme 4 and introduces a common denominatoron all weights, equal to the sum of the weights. This ensures that the sum of theweights is always one, i.e.

∑wi = 1, and for the general case scheme 5 is written

as:

Ce =ne∑

i=1

wi∑ne

k=1 wk︸︷︷︸wi

Ci where wi = (xei )

p

ne∏

j=1;j 6=i

[1 − (xe

j)p]

(5.13)

This formulation gives faster convergence to a “near optimum” compliance valuebut it cannot converge fully since it is less effective in driving the weights to0/1. The reason is that the scaling to unity alters the effect of penalizing thedesign variables as shown in Fig. 5.10 for two materials. Essentially the scalingincreases the number of favorable combinations of design variables and thus makesthe interpolation less distinct. Incidently, increasing the penalization will nothelp much as this only increases the size of the flat triangular “plateau” in thedesign variable space, as shown in Fig. 5.10. However, scheme 5 is still very usefulfor two reasons. First, it can be used to compute the physical constraints in acombined interpolation scheme where scheme 4 is used for stiffness interpolation.This strategy has been applied successfully to a number of examples. Second,scheme 5 can be used “as is” for problems requiring so, e.g. frequency optimization,and the inability of the scheme to fully push all the design variables to their limitvalues can be accepted as a necessary trade-off. In fact, this issue seems to poseless of a problem than expected. For most tested examples scheme 5 is able topush the design variables to their limit values in 65 − 75% of all elements. In theremaining elements the solutions gets stuck on the triangular plateau in Fig. 5.10but often a material has been selected with e.g. w1 = 0.7 and w2 = 0.3, whichgives strong indication that material 1 should be chosen. Consequently, a “manual”selection subroutine has been implemented in MUST, which can a posteriori pushthe largest weight to 1 and the remaining weights to 0 to complete the materialselection. This has not been employed on any of the examples presented in thiswork but could be used (and have been tested) for practical applications.


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5.3.6 DMO schemes 6 and 7

The two last interpolation schemes, 6 and 7, are identical to schemes 4 and 5,respectively, with the difference that they accommodate combined material andtopology optimization by reintroducing the variable x0 as:

Ce = x0

ne∑

i=1

[(xe

i )p

ne∏

j=1;j 6=i

[1 − (xe

j)p] ]

︸︷︷︸wi

Ci (5.14)

from scheme 4 (5.11) and complementary to this we have the scaled version:

Ce = x0

ne∑

i=1

wi∑ne

k=1 wk︸︷︷︸wi

Ci where wi = (xei )

p

ne∏

j=1;j 6=i

[1 − (xe

j)p]

(5.15)

from scheme 5 (5.13). These two schemes have been implemented recently andtherefore not tested extensively yet, but this will be part of our future work.

5.3.7 Multi layered structures

For multi layered structures the interpolation method described in the previoussection can be applied directly. The only difference is that the interpolation mustbe invoked layer-wise instead of element-wise, i.e. for all layers in all elements.Consequently, the interpolation scheme is written by layer, her illustrated for DMO


1

(a) (b)

1

11

2 2

22

3 3

33

4 4

44

1

1

2

2

3

3

4

4

Figure 5.11: Four patch variables (1–4) used to reduce the total number of design

variables by collecting elements over the structure (a) or layers in an element (b).

scheme 4 from (5.11), as:

Cl =nl∑

i=1

[(xl

i)p

nl∏

j=1;j 6=i

[1 − (xl

j)p] ]

︸︷︷︸wi

Ci (5.16)

where l again denotes “layer” and thus, nl is the number of candidate materialsfor the layer.

The number of element design variables, ne, for multi layered elements is thenthe sum of the number of design variables per layer, nl, over all layers, N l, i.e.

ne =∑

k=1N l

nlk. As before, the total number of design variables in the problem is

Ndv =∑Ne

i=1 nei , which for multilayered structures implies a significant increase in

the total number of design variables. To counter this, patches of design variablesare introduced.

5.3.8 Patch design variables

Collecting design variables in patches reduces the number of total design variablesby merging several design variables from different layers and elements into a singlevariable – a patch design variable. The idea springs from the manufacturing processof laminated composites where fiber mats covering larger areas are often used. Asingle variable could then govern the orientation of the fiber mat even though itcovers several elements, Fig. 5.11(a). Of course, the layout of the patches is left upto the engineer a priori and consequently, the final result will also be dependenton the initial patch layout. Patches of design variables may also be used to enforcelaminate symmetry by assigning the same design variable to opposite laminae, seeFig. 5.11(b). This is a very convenient feature for many practical applicationswhere laminate coupling effects may be unwanted.

The total number of design variables, Ndv, when using patch design variables

80 5.4. The optimization problem

is Ndv =∑Np

i=1 npi where np is the number of candidate materials for the patch

variable and Np is the number of patches. This can provide a significant reductionin the number of design variables but at the same time requires some extra effortand insight on the part of the engineer.

5.4 The optimization problem

The discrete material optimization problem can be stated in the same way as atopology optimization problem, subject to an optional constraint on total mass:

Objective : minx

C(x) = pT u

Subject to : (m ≤ mc)0 ≤ xmin ≤ x ≤ 1

Ku = p

(5.17)

where m is the mass of the structure and mc is the allowable mass, analogous tothe constraint on volume in topology optimization, (4.2). The mass constraint iscompared to the weighted mass of all, nl, candidate materials in all layers, N l,over all elements, Ne, i.e.:

m =

Ne∑

e=1

N l∑

n=1

nl∑

i=1

(wiρiVi)le (5.18)

This constraint is not active when doing pure fiber angle optimization since achange in fiber angle involves no change in mass and consequently, the constraintmay be left out entirely (which in MMA is achieved by setting the constraintbound very high). When doing multi material optimization the mass constraint isimportant since it effectively determines the amount of light material in the finalstructure. Note that in (5.18) the density, ρi, is the physical material density in[kg/m3] of material i and not a scaling as was the case in Chapter 4.

5.4.1 Design sensitivity analysis

Obtaining the gradients of the objective and constraints follows the methodologyof Section 4.2. The compliance sensitivity is again written for the i’th designvariable as:

dC

dxi

= pT du

dxi

=d

dxi

(uT Ku

)(5.19)

and rearranging the derivative of (5.2) it can be shown as in (5.5) that:

dC

dxi

= −(uem)T ∂Ke

m

∂xi

uem (5.20)

which again indicates that the sensitivity can be computed on the element level,thus reducing computational time. However, the expression in (5.20) can be


computed more efficiently by employing the residual (2.4) as:

dC

dxi

= −(uem)T ∂Re

m

∂xi

(5.21)

This expression has been implemented and provides an increased performance ofmore than 300%, depending on the number of stiffness matrix evaluations saved.For patch design variables the same expressions apply by introducing a summationover the number of elements in the patch, Nep:

dC

dxi

= −

Nep∑

j=1

(uej)

T∂Re

j

∂xi

(5.22)

Obtaining the derivative of the residual (or the stiffness matrix) consists ofcomputing the derivative of the internal force vector (2.3) since the forces areassumed independent of the design variables. In turn, the gradient is the derivativeof the weighting functions introduced earlier in Section 5.3. These are relativelysimple polynomial expressions that allow the sensitivity analysis to be implementedanalytically in a general and simple way. The weighting function derivative is alsoused to compute the constraint function sensitivities directly from (5.18).

5.4.2 DMO convergence

To determine whether the optimization has converged to a satisfactory result, i.e.a single candidate material has been chosen in all elements and all other materialshave been discarded, a DMO convergence measure is defined. For each elementthe following inequality is evaluated for all weight factors, wi:

wi ≥ ε√

w21 + w2

2 + · · · + w2ne (5.23)

where ε is a tolerance level, typically 95-99.5%. If the inequality (5.23) is satisfiedfor any wi in the element it is flagged as converged. The DMO convergence, hε, isthen the ratio of converged elements to the total number of elements:

hε =Ne

c

Ne(5.24)

The DMO convergence is denoted h99.5 if the tolerance level is 99.5% (and so forth)and full convergence, i.e. h99.5 = 1, simply means that all elements have a singleweight contributing more than 99.5% to the Euclidian norm of the weight factors.This provides a good measure of the convergence although it is not a rigorousmathematical definition. For multilayered structures the DMO convergence issimply computed layer-by-layer instead of element-by-element, i.e. hε = N l

c/N l.

5.4.3 Explicit penalization

As discussed in Section 5.3, some of the DMO parametrization schemes (3 and 5 inparticular) often have difficulty pushing the design variables to their limit values.


In an attempt to improve the performance of these schemes a penalty functionmethod has been implemented, which explicitly penalizes the objective as:

C(x) = pT u + w

Ndv∑

i=1

xpi (1 − xp

i ) (5.25)

where w is a scaling parameter introduced to ensure that the penalty term does notdominate the problem. The value of w can be set in various ways e.g. according tothe compliance value, the largest sensitivity or some other measure. The penaltyfunction is designed to automatically decrease as the design variables tend towards0/1 by encompassing the term (1 − xp

i ).

An elaborate framework has been implemented in which the penalty function canbe introduced either depending on the iteration number or the level of convergencereached, i.e. when hε reaches a preset value. Furthermore, the penalization canbe gradually increased or decreased as the optimization progresses by additionalscaling of w. However, the method requires an extensive amount of tweakingand for the tested cases it was not able to significantly improve the overallconvergence of the optimization. Eventually the method was abandoned in favorof the combined interpolation scheme in which DMO scheme 4 is used for thestiffness and scheme 5 for the physical constraints, see Section 5.3.5.

5.5 Numerical examples

Now, to demonstrate the capabilities of the discrete material optimization methodseveral numerical examples will be presented. To give an impression of thecomputational effort required to use DMO, the approximate runtime on a desktopPC is stated for each example. A total of four examples will be presented, eachdemonstrating various aspects of the DMO method. The first two examples are2D beam examples that are included mainly to verify the method by comparingto known solutions. The third example is a curved, multilayered shell, which isoptimized for both material layout and fiber angles. Last, an industry relevantexample is included to illustrate the potential practical application of DMO. Allexamples are run using combined DMO scheme 4/5.

5.5.1 Cantilever beam with distributed top load

The cantilever beam with distributed top load has become a standard test forminimum compliance fiber angle optimization (Pedersen, 1991). The beam hasa length to height ratio of 3 and unit thickness and has been meshed using 768MITC4 shell elements with all out-of-plane displacements fixed. The DMO setupallows for 12 candidate materials in each element which results in a model having9, 216 design variables in total. The candidate materials used are glass fiberreinforced epoxy with the orthotropic properties Ex = 54 GPa, Ey = 18 GPa,Gxy = 9 GPa and νxy = 0.25 oriented at [90◦,±75◦,±60◦,±45◦,±30◦,±15◦, 0◦].


Figure 5.12: Optimal fiber angle distribution in cantilever beam with uniformly

distributed top load.

Figure 5.13: Optimal fiber angle distribution in cantilever beam with uniformly

distributed top load. Solved using 48 patches of 4 × 4 elements.

The optimization converges monotonically to full DMO convergence (h99.5 = 1.0)in 157 iterations taking just under 7 minutes on a desktop PC. The optimal fiberangle distribution determined is shown in Fig. 5.12 and agrees very well with theresults obtained by e.g. Pedersen (1991).

To illustrate the patch variable methodology the problem has been solved using48 patches of 4× 4 elements, which reduces the number of design variables to 576and reduces the runtime by approximately 16%. The resulting optimal fiber angledistribution is shown in Fig. 5.13.

5.5.2 Beam subjected to four-point bending

This example demonstrates the ability of the DMO method to simultaneouslychoose material type and material orientation. The domain is as defined for theprevious example and a mesh of 768 SHELL9 elements is used (again, all out-of-plane displacements are fixed). As candidate materials the same orthotropicglass/epoxy as above is taken at [90◦,±45◦, 0◦] and furthermore, an isotropic


sym

met

ry

P

Figure 5.14: Optimal material and fiber angle distribution in beam subjected to four-

point bending (symmetric). Black indicates glass/epoxy, light gray indicates foam and

intermediate values represent unconverged elements.

polymeric foam material having E = 125 MPa and νxy = 0.30 is used. Thisresults in 5 design variables per element and thus 3, 840 in total. The massconstraint is set to mc = 4 kg which effectively means that the foam must accountfor roughly 74% of the material usage when the densities of glass/epoxy andfoam are 1900 kg/m3 and 100 kg/m3, respectively. The optimization convergesmonotonically to h95 = 0.96 in 125 iterations taking just over 30 minutes.

The result of the optimization is shown in Fig. 5.14. The edge of the optimumgeometry (marked by black) is very similar to results obtained with classicaltopology optimization techniques. The difference is that instead of obtaining aframe-like structure, the DMO method uses the polymeric foam material (lightgray in Fig. 5.14) to form a sandwich structure. If the mass constraint is loosenedthe DMO method will tend toward distributing stiff material in a frame structureas well. The intermediate densities found in the area below the point load is alocal effect and roughly outline would-be bars. Tightening the mass constraintwill reduce this effect but the intermediate densities have been allowed here inorder to illustrate the relationship of the DMO methods with classical topologyoptimization.

5.5.3 Hinged 8-layer spherical cap

The doubly-curved shell geometry used here is identical to the geometry usedin Section 4.3.2 and has been used to demonstrate the capabilities of the DMOmethod for a general structure. The thickness of the shell is 8 mm in total, dividedevenly over eight layers. The structure is again loaded by a single load in the centerpoint and the model is hinged (ui = 0) on the four edge curves. The entire shellgeometry is modeled using a 40× 40 mesh of MITC4 shell elements, see Fig. 5.15.


Glass/epoxy (+45,0,-45,90), foam

Glass/epoxy (+45,0,-45,90)}

Glass/epoxy (+45,0,-45,90)

Figure 5.15: Geometry of hinged spherical cap example with actual thickness and

distribution of layers.

0

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0 10 20 30 40 50Iterations

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Oco

nv

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en

ce

(99

.5%

)

Objective

DMO convergence

Figure 5.16: Convergence of objective function (left axis) and DMO convergence ratio

(right axis) for spherical cap example.

As candidate materials we again use a glass/epoxy composite with Ex = 54 GPa,Ey = 18 GPa, Gxy = 9 GPa and νxy = 0.25 and the permissible fiber angles[90◦,±45◦, 0◦] as well as a polymeric foam with E = 125 MPa and νxy = 0.30.The two skin layers are not allowed to choose the polymeric foam but the inner6 layers can be either foam or composite. This results in 38 design variablesper element, distributed as [4, 5, 5, 5, 5, 5, 5, 4] bringing the total number of designvariables for the model to 64000. The mass constraint is mc = 5.0 kg whichmeans that the foam must constitute just under 75% of the total volume. TheSIMP power is increased from 3.0 to 10.0 in steps of 1.0 every 10 iterations andconvergence to h99.5 = 0.992 is reached in 50 iterations using roughly 80 minutesof computational time.


(1) (2)

(3) (4)

Figure 5.17: Optimal material distribution and orientation in layers 1–4 of spherical

cap. White and black represents foam and glass/epoxy, respectively.

The convergence is shown in Fig. 5.16 and the resulting material distributionand fiber orientations are shown for layers 1–4 in Fig. 5.17 and for layers 5–8in Fig. 5.18. The layers are numbered from the outside of the shell, i.e. layer 8 ison the center of curvature side.

As shown in Figs. 5.17 and 5.18 the fiber angle optimization problem has beensolved for the skin layers (layer 1 and 8) and the combined material distributionand orientation problem has been solved for the internal layers (layers 2–7).The result for the skin layers resemble those found in the literature for plates


(5) (6)

(7) (8)

Figure 5.18: Optimal material distribution and orientation in layers 5–8 of spherical

cap. White and black represents foam and glass/epoxy, respectively.

under similar boundary conditions (e.g. Pedersen (1991)). A solution for thematerial distribution problem has not been reported in the literature but thesolution corresponds well to known reinforcement techniques for sandwich panels(Bozhevolnaya and Lyckegaard, 2005).

In the final solution a cone has been formed through the thickness of the shellto support the local, concentrated load. At the bottom of the cone (the centerof curvature side) a wider reinforcement is obtained to distribute the transverseload over a larger area, thus reducing the local deformation in the lower skin. The


Figure 5.19: Internal material distribution of glass/epoxy material at the center (real

geometry is 3D and curved, see Fig. 5.15). The thickness has been magnified 10 times to

better illustrate the cone-like shape.

internal material distribution is shown in Fig. 5.19.

The hinged spherical cap example agrees well with the expected results and, to theextent comparison is possible, with existing results in the literature. The exampledemonstrates that the DMO method is capable of simultaneously solving the fiberangle problem and the material layout problem. The latter is in fact solved bothover the surface of the structure as well as in the thickness direction.

5.5.4 Wind turbine blade main spar

This example is included to illustrate that the DMO method can be used forstructural optimization of laminated composite structures in general. Basicallythis means that the method can be used on any real life structure where the startingpoint is a sufficiently accurate finite element model. In order to demonstrate thisversatility, maximum stiffness design of a generic main spar model provided bythe wind turbine manufacturer Vestas Wind Systems A/S is studied. It should benoted that this is a preliminary study, and thus, the main goal with the exampleis to demonstrate the method on a complicated design problem. A very similarexample has been published in Lund and Stegmann (2005). The wind turbineblade basically consists of two structural components, the main spar and theaerodynamic shell, Fig. 5.20.

Assembly

Leading edge

Trailing edge

Suction side shell

Main spar

Pressure side shell

M

Me

f

Figure 5.20: Composition of a wind turbine blade. The blade is subjected to flapwise

bending, Mf , and edgewise bending, Me. Courtesy of Lennart Kühlmeier, Vestas Wind

Systems A/S.


Figure 5.21: Finite element model used for maximum stiffness design of the load

carrying main spar.

The main spar carries most of the flapwise bending loads, Mf , whereas the shellcarries most of the edgewise bending loads, Me. In this study the main spar issubjected to the most critical load case, which is the flapwise bending load thatarises when the turbine has been brought to a standstill due to high wind, andthe blade is hit by a 50 year extreme wind. In the model used, only the mainspar is considered, i.e., the two shell parts in Fig. 5.20 are removed to reducethe complexity of the model. In order to account for the stiffness contribution ofthe shells to the stiffness of the main spar, the thicknesses of the shells in directcontact with the main spar are added to the top of the flanges of the main spar,see Fig. 5.22. Thus, the local stiffness contribution is included but the supportconditions from the shells are ignored.

The employed finite element model shown in Fig. 5.21 was generated using ANSYS,and the mesh consists of 9600 MITC4 shell elements with 16 layers. The main sparhas a total length of 15 meters and the flapwise bending is applied using two nodalforces at the end as shown in Fig. 5.21. The model is clamped at the root end,i.e. all displacements and rotations are fixed. With these boundary conditions thedominant state will be bending, which results in tension/compression in the topand bottom flanges and shear in the wedges. Furthermore, due to the geometryof the spar, which twists its cross section along the length, the spar will also besubjected to torsion when bend at the tip. These basic considerations will be usedas a guideline for interpreting the results of the optimization.

The design parametrization proceeds in two ways as follows. 1) The main spar isdivided into 77 patches with identical lay-up and thickness to reduce the numberof design variables, Fig. 5.23. The choice of patches follows the constitution ofthe main spar in that elements with identical lay-up and thicknesses are collectedin patches. The main spar model was a priori split up into discrete regions with


Figure 5.22: A closer look at the constitution of the main spar model with discrete

thickness jumps along the length of the spar (left) and over the cross section (right).

Figure 5.23: Main spar model with distribution of patches. Not all patches are shown

due to limitations in the color resolution.

identical thickness and lay-up as indicated in Fig. 5.22. 2) All elements in themain spar are assigned design variables in all layers. This renders the modelmuch more complex but allows for a more detailed material selection. For bothparametrizations there are two materials in play, a stiff glass/epoxy material anda softer isotropic foam core material. To govern the amount of light materialused a mass constraint is included, which allows for 80% of glass/epoxy and 20%foam material. Due to a confidentiality agreement with Vestas Wind System A/Swe cannot disclose the exact material properties, mass constraint, geometry andthickness distribution. For simplicity and to keep the number of design variablesrelatively low, the glass/epoxy material can only be oriented at 0◦, ±45◦, and 90◦.Furthermore, only the interior layers are allowed to consist of foam material sincefoam is not a realistic choice for the skin layers. Consequently, the distributionof design variables for the 16 layers is [4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4], whichbrings the number of design variables per element to 78 and thus the total numberof design variables is 6006 for the patch variable model and 748800 for the full


0

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Oco

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.5%

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Objective (full)

DMO convergence (patch)

DMO convergence (full)

Figure 5.24: Convergence of objective function (left axis) and DMO convergence ratio

(right axis) for wind turbine main spar example with patched and full variables.

variable model.

Both models were solved on a desktop PC. The patch variable model used 131iterations to reach h99.5 = 0.975 taking approximately 33 hours of computationaltime while the full variable model used just 81 iterations to reach h99.5 = 0.997in 21 hours. The vast majority of time, around 98%, is spent on the designsensitivity analysis and increasing this performance is going to be addressed infuture work. Again the MMA algorithm proves very efficient and solves theapproximate subproblem of the full variable model in just under 4 seconds. Theconvergence is shown for the first 50 iterations in Fig. 5.24 where the subsequentiterations have been left out for clarity. The objective of the full variable model isapproximately 5% lower than for the patch variable model.

The result of the optimization using patch variables is shown for all 16 layers inFigs. 5.26, 5.27, 5.28 and 5.29 – 4 layers at a time. Similarly, the result of theoptimization using full variables is shown in Figs. 5.30, 5.31, 5.32 and 5.33. Theoptimized material directions for the glass fiber are shown for all 16 layers, andif no material direction is indicated it is implied that soft core material has beenchosen for that particular layer of a given patch/element. Note that layer 1 is theinner (bottom) layer and layer 16 the outer (top) layer.

The two models yield very similar results and, as expected, most of the foamcore material has been put in the webs in the internal layers 2–15 close to theroot of the main spar in order to increase the bending stiffness where the bendingmoment is strong. Looking at the two models in detail it is apparent that thefull variable model chooses to use foam material in all internal layer of the web,distributed continuously over approximately the first third of the main spar. Incontrast the patch variable model only places foam in layers 2–14, and distributes


(a) (b)

Figure 5.25: Optimal material directions for the full variable main spar model. Shown

at the tip for layer 1 (a) and layer 8 (b).

it in two distinct regions rather than continuously. This illustrates that choice ofmaterial and orientation for each patch is a compromise between all elements inthe patch while in full variable model, the properties of each element can be setindividually. The orientation of the glass/epoxy material also corresponds well towhat was expected from the basic considerations made earlier regarding the loadcarrying mechanisms of the spar. In the flanges all layers are dominated by 0◦

orientation, i.e. along the length of the spar, to account for bending while thewebs are dominated by 45◦ to account for shear. In the full variable model thepattern of 45◦ elements resemble the distribution obtained for the cantilever beam(see Fig. 5.12) in that there is a gradual occurrence of 45◦ elements, starting withthe elements in the center of the web.

One of the characteristics of the full variable model is that local effects are affectingthe results much more than for the patch variable model, which is not surprisingsince the patch variables smear out the design. The phenomenon is most easilyseen by considering the tip of the main spar, Fig. 5.25. Here, the cross sectionhas obviously been reinforced by both introducing fibers in the circumferentialdirection and foam material in the middle layers. Both of these measures aretaken in order to carry the two point loads applied to the cross section. For designpurposes such patterns may in many cases be considered as noise since point loadsare often used for modeling convenience and do not represent a physical occurrence.Consequently, the patch variable solution is by far the easiest to realize, from amanufacturers point of view since, it encompasses large and well defined regionsas opposed to the complex solution of the full variable model.

The results presented here are somewhat crude in that only five candidate materialshave been used, and the natural next step would be to expand the design spaceand allow for a larger variation of fiber angles in order to obtain a more detaileddesign. However, Figs. 5.26 to 5.33 still illustrate very well the potential of theDMO method for designing a wind turbine blade main spar for maximum integralstiffness. As such it has been demonstrated that DMO indeed has potential as apractical design tool for industrial applications, which has been one of the primarygoals of developing the method.


Layer 1)

Layer 2)

Layer 3)

Layer 4)

Figure 5.26: Optimized material directions for the GFRP material in layers 1 to 4.

Patch variable model with 77 patches and 6 006 design variables.


Layer 5)

Layer 6)

Layer 7)

Layer 8)




Layer 9)

Layer 10)

Layer 11)

Layer 12)




Layer 13)

Layer 14)

Layer 15)

Layer 16)




Layer 1)

Layer 2)

Layer 3)

Layer 4)

Figure 5.30: Optimized material directions for the GFRP material in layers 1 to 4. Full

variable model with 748 800 design variables.


Layer 5)

Layer 6)

Layer 7)

Layer 8)

Figure 5.31: Optimized material directions for the GFRP material in layers 5 to 8. Full

variable model with 748 800 design variables.


Layer 9)

Layer 10)

Layer 11)

Layer 12)


Full variable model with 748 800 design variables.


Layer 13)

Layer 14)

Layer 15)

Layer 16)


Full variable model with 748 800 design variables.



In this chapter discrete material optimization (DMO) has been introduced as anovel gradient based technique for maximizing structural stiffness by optimizingmaterial choice and material orientation. The method operates on a fixeddomain, i.e. shape and thicknesses are defined a priori and remain fixed duringoptimization, and as such we deal entirely with solving a laminate lay-up problem.The DMO method is derived from multiphase material optimization in the sensethat the element stiffness is computed from a weighted sum of candidate materials.The aim of the optimization is for each element (or layer) to choose the materialfrom the set of candidate materials that minimizes the objective the most. Thecandidate materials may be either isotropic or orthotropic with a given fiber angle.

The material selection is made at the element level and several options for theweighted sum formulation have been suggested but the most successful is thecombined scheme in which DMO scheme 4 (Section 5.3.4) is used for stiffnessinterpolation and scheme 5 (Section 5.3.5) is used for the physical constraint.This combined scheme has been used in the four numerical examples presentedin Section 5.5. The first example was the cantilever beam, which served asa validation of the DMO methods ability to solve fiber angle optimizationproblems. The results were compared to known solutions in the literature andgood correlation was obtained. The second example was a four-point beam bendingproblem which was solved for material layout and orientation and the result couldin part be validated with known solutions from topology optimization. Goodcorrelation was obtained and the result agreed well with the known physics of theproblem. The third example was a multilayered spherical shell, which was solvedfor material layout and orientation both over the surface and in the thicknessdirection. This combined example has not been studied before in the literaturebut the results for material orientation can be compared to some extent to planesolutions and the DMO solution showed similar patterns. The material layoutproblem was solved and showed a cone-like reinforcement through the thickness,which has been verified as a known technique for sandwich structures. The finalexample was the wind turbine blade main spar, which represented a realisticindustry relevant problem. The problem was optimized for material layout andorientation and the results correlated well with what was expected based on theload carrying mechanisms of the structure. Furthermore, the use of patch designvariables illustrated that relatively simple optimal solutions can be obtained andthus, the manufacturability of the optimal design obtained can be quite high.

None of the tested examples showed behavior similar to that of CFAO methodswhere the optimization will distinctively get stuck in a local optimum. Thisindicates that the method is more robust than existing methods and in general,the results obtained with the discrete material optimization method were veryencouraging. Furthermore, the DMO method shows promising potential forapplication to industry relevant problems.

6

Conclusions

The objective os this work was to develop finite element based optimiza-tion techniques for laminated composite shell structures. The platform of

implementation has been the computer aided analysis and design tool MUST anda number of features have been added and updated. This includes an updatedimplementation of finite elements for shell analysis, tools for investigation ofnonlinear effects in multilayered topology optimization and a novel frameworkcalled Discrete Material Optimization (DMO) for solving the material layout andorientation problem. In the following the main points of these topics will besummarized.

Shell finite elements

Efficient and robust finite elements for performing analysis are an importantprerequisite for doing design optimization and consequently, a substantial amountof work has been invested in this (Chapter 3). Prior to this work a number of bothstabilized and non-stabilized shell finite elements had been implemented in MUSTbut the stabilized elements suffered from errors in the element routines. Thesewere therefore reimplemented using a modular approach based on FORTRANcode generated from the mathematical software MAPLE. As a result MUST nowsupports a new three-node element and an updated four-node element, both withlaminate and geometrically nonlinear capabilities. These elements are designatedMITC3 and MITC4, respectively, since they employ Mixed Interpolation ofTensorial Components, which effectively removes the problem of shear locking.The non-stabilized elements have been updated for better performance and anefficient SHELLn family of elements is now available in MUST, including elementswith three, four, six, nine and sixteen nodes.

The performance of the MITC3, MITC4 and SHELL9 elements was demonstratedas these constitute the working horses among the shell elements in MUST. Thethree elements passed the patch tests and showed good predictive capabilitiesas well as high computational efficiency compared to commercial finite elementpackages. As such, the first objective of the project was reached.

104

Nonlinear topology optimization

Topology optimization for maximum stiffness (minimum compliance) of multi-layered shell structures has been studied before but the influence of including thenonlinear terms of the strain measure had not been subject of investigation prior tothis work. The purpose of studying these effects was to determine if nonlinearitiesshould be taken into account when designing large laminated composite structuresusing structural optimization (Chapter 4). The topology optimization problemwas solved using a SIMP approach with an iterative Newton-Raphson solver forthe analysis, the adjoint variable method for sensitivity analysis and the MMAoptimizer for solving the optimization problem. The design parametrization allowsmaterial to be added/removed in specific layers of the structure using a singledesign variable per element indicating that all voided layers are scaled in thesame way. The layers, which are not scaled, remain fixed and thus continuouslycontribute to the element stiffness. This circumvents problems of near-zero termsin the stiffness matrix. Multiple load cases were handled in a weighted sumformulation.

Several numerical examples with plates and shells were presented and illustratedthat the importance of the nonlinear effects is problem dependent. For all examplesthe change to the optimal topology was dramatic but in some cases the actualperformance gain was very small. However, in some examples the performanceincrease was significant and furthermore, the use of multiple load cases provideddesigns that showed good performance at multiple load levels. The results wereconsistent with the progressive nature of nonlinear effect in that a gradual increasein load lead to a gradual change in optimal topology, i.e. for very light loadingthe linear and nonlinear results were identical while they became increasinglydifferent with higher loading. An interesting observation was that the topologiesobtained at high load with the nonlinear formulation were “global” in the sensethat they covered a large, continuous area over the geometry. This is interestingin regard to manufacturing since such patterns are much easier to realize than thelocal patterns usually associated with topology optimization. Including nonlineareffects can therefore, in most cases, be considered an improvement due to thepotential performance increase and improved manufacturability of the optimaldesigns obtained. This marks the successful arrival at the second goal of thepresent work.

Discrete material optimization

Solving for optimal material orientation and maximum stiffness has been subjectto extensive investigation over the years but most existing methods inherentlysuffer from problems with local optima. This inspired the development of discretematerial optimization, which is a novel approach that allows for simultaneoussolution for material distribution and orientation (Chapter 5). The DMO methodbuilds on ideas from multiphase topology optimization but extends the scope ofapplication by allowing the optimizer to choose from any number of pre-defined

Chapter 6. Conclusions 105

candidate materials, which can be both different materials and the same materialat different orientations. The element level parametrization is a weighted sumformulation of the constitutive equation in which the design variables switch onand off contributions from individual materials. An absolute necessary conditionfor the method is that a single material is ultimately chosen and consequently, thatall elements/layers have a single weight of value one while all other weights arezero. The success of the method is therefore dependent on the optimizers ability topush the weights to 0 and 1 and to assist it, several weighting schemes have beentested and implemented. The most successful has been found to be the combinedDMO scheme where scheme 4 is used for stiffness and scheme 5 for the physicalconstraints.

A number of numerical examples demonstrated the capabilities of the method.Two well known 2D examples were used to validate the DMO methods abilityto solve the material orientation problem and the material distribution problem.Furthermore, two generic shell examples were presented, one of which was anindustry related design problem: a wind turbine blade main spar. The model wassolved and the results correlated well with what was expected from the knowledgeof the load carrying mechanisms of the main spar. The patch variable modelprovided a very distinct solution with large areas of uniform material orientationwhile the full variable model provided a more detailed and complex solution. Froma practical point of view the design obtained with the patch variable model ismore compelling due to its higher degree of manufacturability. None of these shellproblems have been solved in the literature before and the present study thereforerepresents a novel extension to structural design optimization.

In the tested examples local optima could not be distinctively identified whenusing discrete material optimization as it is the case when running CFAO and ingeneral, the obtained results were very encouraging and the DMO method showspromising potential for application to problems of industrial relevance. This bringshome the last goal of the present work.

Suggestions for further work

Considering the promising results with discrete material optimization we wantto continue working on developing this method in the future. Encouraged by theresults with nonlinear topology optimization a natural next step is to extend DMOto geometrically nonlinear problems as well using the same methodology. Havingdone so, the DMO method could be applied to problems of local buckling providedthat an appropriate local objective or constraint can be devised. Furthermore,the introduction of local stress criteria could be considered if developing anappropriate methodology for interpreting the stresses computed from the materialinterpolation. Introducing stress constraints also considerably increases thecomplexity for the optimizer and thus renders the problem more difficult to solve.Entering into the realm of local criteria could also spawn a need for layer-wise

106

or higher-order shell elements with better stress prediction capabilities. A simpleextension of the presents framework is the use the multiple load case capabilitiesin MUST to investigate how DMO performs for such problems.

One of the very interesting aspects of discrete material optimization is toinvestigate the global convergence properties – are we getting global optimumsolutions? The typical local optimum behavior known from CFAO was notobserved but this of course does not guarantee that local optimum solutions are nota problem. Consequently, further understanding of this aspect must be obtained,and such efforts will be made in the near future.

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Index

Adjoint variable method, 45Assumed natural strain, 5, 31

CFAO, 66, 68Classical laminate theory, 29Compliance, 7

definition, 15DMO, 65Topology optimization, 44

Compliance sensitivity, 15DMO, 81fiber angle, 66Topology optimization, 46

CONLIN, 15, 68Constitutive matrix, 25Convex approximation, 16Coordinate system

covariant, 20director, 23material, 26

Deformation gradient, 24Degenerated solid elements, 22Design variables, 2

DMO, 70, 72fiber angle, 66topology, 42

Discrete material optimization, 65Displacement definition, 24DMO, 65, 69

convergence measure, 81explicit penalization, 81implementation, 75parametrization, 69patch variables, 79, 83, 91

DMO interpolation, 72implementation, 75scheme 3, 73scheme 4, 75, 79scheme 4/5, 77, 82

scheme 5, 77

Equation solvers, 14Equivalent single layer, 5, 27

FSDT, 23, 40

GCMMA, 17Genetic algorithms, 69Governing equations, 3

equilibrium, 12

Implementation, 7DMO, 75element, 34

Internal force vector, 13

Laminate, 4description, 28integration, 29layer-wise, 28

Lamination parameters, 68Locking, 5, 30

Manufacturability, 62, 79Material, 25MITC, 31

definition, 32elements, 33implementation, 34patch test, 37stiffness matrix, 35verification, 36

MMA, 15, 44MUST, 7

introduction, 7sandwich option, 25verification, 36

NAND, 13, 46

Index 115

Optimization, 1, 3DMO, 80fiber angle, 66Topology, 42

Optimization problemDMO, 80fiber angle, 66generic, 3topology, 44

Patch test, 37

Reference surface, 22Residual

definition, 13linearization, 13

Sensitivity analysis, 3, 44, 75, 80Adjoint, 45Direct differentiation, 45Finite difference, 45

Shear correction factor, 25Shell

geometry, 20kinematics, 22reference surface, 22rotations, 23

SIMP, 5, 41, 44Solvers, 14Stiffness matrix

AG method, 35implementation, 35linear, 14, 65tangent, 13

StrainGreen-Lagrange, 24

Strain-displacement matrixdefinition, 12Implementation, 35

Stresssecond Piola-Kirchoff, 12, 25

Thickness integration, 29Topology optimization, 5, 41

multilayered, 42multiphase, 69Multiple load cases, 46Parametrization, 42

Volume constraint, 44, 80

Weight factors, 70Weights, 70–78

analysis and optimization of laminated composite shell structures

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