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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 UniversityPontoppidanstrde 101, DK-9220 Aalborg East, Denmarke-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

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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. Bendse (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 Mller, 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 Masters thesis in

2001, 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 and

Emilie for bringing me joy and always lifting my spirits after a long days work.Their love and support is invaluable.

Jan StegmannAalborg, May 2005.

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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 composite

structures 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.

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Abstrakt

Formlet med nrvrende arbejde er at udvikle finite element baserede op-timeringsteknikker til laminerede kompositte skalkonstruktioner. Implementer-ingsplatformen er det finite element baserede analyse- og optimeringsvrktjMUST (MUltidisciplinary Synthesis Tool), og en rkke funktioner er blevettilfjet og opdateret. Dette inkluderer en opdateret implementering af elementertil analyse af skaller, vrktjer til undersgelse af ikke-linere effekter i multi-lags topologioptimering samt en ny metode kaldet Diskret Materiale Optimering(DMO), der kan lse med hensyn til optimal fordeling og orientering af materialer.

Robuste elementer er et ndvendigt vrktj i optimering, og finite element

biblioteket i MUST er derfor blevet udvidet med et nyt tre-knuders element samtet opdateret fire-knuders element. Disse benvnes henholdsvis MITC3 og MITC4,idet de bruger Mixed Interpolation of Tensorial Components til at undg problemermed shear locking. For at opn bedre ydelse er SHELLn familien af standardisoparametriske skalelementer i MUST ogs blevet opdateret. Alle elementer kanhndtere laminater og geometriske ikke-lineariteter, og test viser at elementernesydelse er god.

Geometriske ikke-linere effekter undersges med henblik p at afgre, om dissebr medtages ved design af laminerede konstruktioner for maksimal stivhed med

strukturel optimering. Funktioner til ikke-liner topologioptimering af multi-lagsskalstrukturer implementeres ved brug af Newton-Raphson metoden til analyse,adjoint variabel metoden til sensitivitetsanalyse og MMA optimizeren til at lseoptimeringsproblemet. SIMP metoden bruges til lagvis skalering af stivheden,hvilket giver mulighed for at tilfje/fjerne materiale i specifikke lag. Flereeksempler illustrerer effekten af ikke-lineariteterne p den optimale topologi og,afhngig af problemet, kan forbedringen af designets ydelse vre signifikant.

Eksisterende metoder til lsning af optimal materialeorientering lider underproblemer med lokale optima, hvilket inspirerede til udviklingen af DiskretMateriale Optimering (DMO), der er en ny tilgang til samtidig lsning foroptimal materialefordeling og -orientering. DMO metoden anvender en vgtetsum formulering til at lave en parametrisering p elementniveau, der tilladeroptimizeren at vlge et enkelt materiale fra et st af pre-definerede materialer,ved at skubbe vgtene mod 0 og 1. Metodens succes afhnger sledes afoptimizerens evne til at skubbe vgtene til 0 og 1, og flere formuleringer afvgtene er implementeret. Numeriske eksempler viser, at metoden er i standtil at lse det kombinerede fordelings- og orienteringsproblem. Endvidere lseset industrirelevant designproblem med en hovedbjlke fra en vindmllevinge, ogresultaterne er meget lovende. DMO viser sledes potentiale til anvendelse pproblemer i industrien, og der kunne ikke identificeres problemer med lokale optimai de krte eksempler.

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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. 349360

Stegmann J, Lund E (2005): Discrete material optimization of general compositeshell structures. International Journal for Numerical Methods in Engineering,62(14), pp. 20092027.

Lund E, Stegmann J (2005): On structural optimization of composite shellstructures using a discrete constitutive parameterization. Wind Energy, 8(1), pp.109124.

Publications in proceedings

Lund E, Stegmann J (2004): On Structural Optimization of Composite ShellStructures Using a Discrete Constitutive Parameterization. In: The Science ofmaking Torque from Wind, (ed. G.A.M. van Kuik), 19-21 April 2004, DUWind,Delft University of Technology, pp. 556567.

Stegmann J, Lund E (2003): Optimizing General Shell Structures Using a

Discrete Constitutive Parameterization. In: American Society for Composites 18thTechnical Conference, ASC 18, Gainesville, FL, US, 19-22 October 2003, pp. 110.

Stegmann J, Lund E (2003): Discrete Fiber Angle Optimization of General ShellStructures using a Multi-Phase Material Analogy. In: Fifth World Congress onStructural and Multidisciplinary Optimization, WCSMO 5 (ed. C. Cinquini et al),Venice, Italy, 19-23 May 2003, pp. 16.

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. 215218.

Stegmann J, Lund E (2002): Topology Optimization of Multi-Layered Shell

Structures 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. 110.

Publications partially based on Masters 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. 8386.

Stegmann J, Jensen RL, Rauhe JM, Lund E (2001): Finite Element Analysis of

Laminated Composite Shells Undergoing Large Displacements, In: 2nd Max PlanckWorkshop on Structural Optimization (ed. M.P. Bendse et al), Nyborg, Denmark,12-13 October 2001, pp. 6568.

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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

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.1 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

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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 . . . . . . . . . . . . . . . . . . . . . 785.3.7 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

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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 designoptimization 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 passengers

will 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

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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 Bendse 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 the

same 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

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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 defining

the 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 (Newtons laws, laws ofthermodynamics etc.)1. Now, the problem may be stated in mathematical termsas:

Objective : mina

f(a)

Subject to : g(a) Gamin a amaxPhysical 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.

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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 these

topics 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 of

attention 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

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such as locking and hour-glass modes usually associated with shell finite elements.The finite element method has, of course, also benefitted tremendously from the

exponential 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 the

laminate 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), accounting

for 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 Bendse and Kikuchi (1988) who usedthe homogenization technique and Bendse (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), Bendse 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

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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 optimization

is 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). Recent

developments 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 fiber

reinforced 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

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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 used

optimality 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 Mller (Mller,

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.

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8 1.3. Objectives of work

MUST

Inputmodule

(ANSYS, COSMOS, ODESSY)

Finiteelementlibrary(Beam, Solid/fluid 2D/3D, shell)

Sensitivityanalysismodules(Finitedifference, analytical)

Optimizer library(MMA, simplex)

Optimizationmodules(Shape, size, topology, fiber)

Databasemodule(Resultgeneration)

Postprocessor(FEPlot)

Solver library(Linear, nonlinear)

Analysismodules(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 by

colleagues 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 Mller. The major features of the MUST system are depicted in

Fig. 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.

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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.

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10 1.3. Objectives of work

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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

Designanalysis(finiteelement)

Sensitivity

analysis

Improvedesign(optimizer)

Newdesign

variables

Optimize?Preprocessing

(Initialdesign)

Yes

Postprocessing(Finaldesign)

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.

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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 thorough

treatment 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 Newtons 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:

VsT dV

Internal work

Vpbu dV +

Apsu dA

External work

= 0 (2.1)

Here u ui is a displacement increment and s sij is the second Piola-Kirchoff stress, which is work conjugate with the Green-Lagrange strain increment, ij. The external forces are divided into body forces, p

b, 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 = {11, 22, 33, 212, 223, 213}T and

s = {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 = Buwhere u indicates nodal values. Alternatively, the strain-displacement matrix canbe expressed from the variation of the strain =

uu as:

B =

u(2.2)

The form in (2.2) is convenient for deriving the strain-displacement matrix as

will 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

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Chapter 2. Analysis and optimization 13

derived from the internal work term as:

r = VB

T

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)

Subject to : g(a) Gamin a amaxR(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 an

iterative 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)

uu = 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 this

is standard in the literature

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14 2.2. Improving the design

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

KTu

= R

(u

) (2.9)In MUST (2.7) is solved continuously until the ratio R/p is below somespecified value, typically around 108 106. At present, MUST encompasses anumber of nonlinear solvers due to Mller (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)

ukdV

=Ne

k=1

V

BTkuksk +B

Tk

skuk

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

BTCB 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 for

u 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

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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) = pTu(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 ith 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 gradient

information 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 illustrated

in 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

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a

a1

2

g a G( ) Originaldesignspace

Convexapproximationg 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. aand 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)Subject to :

g(a) G

amin a amaxR(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

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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 does

not 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 a

stationary move limit of typically 25% 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.

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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 the

degenerated 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 required

for 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).

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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 of

the 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

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Chapter 3. Multilayered shell finite elements 21

g3

g1g2

x

y

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 = [gT1 ,gT3 ,gT3 ]T,which can be easily determined as the shape functions are known. Defining thekinematics is therefore straightforward.

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(a) (b)

1

2

3

4

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 = { |uax, u

ay, u

az | }

T. The kinematics of a shell element is closely relatedto this expression, the difference being that assumptions regarding the structuralbehavior of the shell will be built into the kinematic description. To do so the

degenerated 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 opposite

the 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 (33 0) and normals remain straight

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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 g2g1 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 adjacent

node 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 v3a v3

; v2 = v3 v1 (3.4)

where a = j if j v3 = 0 and a = k if j v3 = 0. Here, the vectors j and kare global unit vectors along the y- and z-axis, respectively. The key role of thedirector coordinate system is to define the local rotations, and , of the node

director, 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.

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As indicated in Fig. 3.3 a rotation, , ofv3 about v1 causes a linear displacementof magnitude sin in the direction ofv2. In the same way the director vector

tip is displaced in the direction ofv1 a distance sin . Assuming small rotationsthis amounts to a local linear displacement of ua = {a,a, 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 = ava1

ava2 . This expression is essential in shellanalysis 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+ t2

hv1 v2a (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, , ), 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 = Nuwhere u is defined as a vector containing all the nodal degrees of freedom, i.e.

u = { |uax, uay, uaz , a, a| }T. This rather tedious notation will be abandonedand from this point and on, N and u will always imply modified shell quantitiesas defined by (3.5).

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

ij =1

2

tgi tgj 0gi 0gj

=

1

2u

xj

gi +u

xi

gj +u

xj

u

xi(3.6)

where ij 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 = 0tx, which is perhaps a more direct approach:

ij =1

2

FTF I

(3.7)

This can be efficiently evaluated in terms of the displacement gradients, which are

readily available in a finite element framework such that F = 0tu + I. Whichof (3.6) and (3.7) is most convenient depends on the type of finite element beingimplemented.

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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 = C, 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 ofC is then:

C =

C11 C12 0 0 0 0

C22 0 0 0 00 0 0 0

C44 0 0

Sym. C55 0C66

(3.9)

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

C11 =E1

1 1221C22 =

E21 1221

C12 =21E1

1 1221C44 = G12 C55 = G23 C66 = G13

(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 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 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

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a

bv3a

v3b

Fiberdirection

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 as

the 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 g2g1 g2

(3.11)

which is the same definition as used for the director vector, v3, the difference

being 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 d

1and m

3is smaller than 45

, d

1is projected as m

1. Else

d2 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

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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 d3else Default

Use global Cartesian, i.e. d1 = i, d2 = j and d3 = kend if

if

abs(d1 m3) < 45 then Project using either d1 or d2

Compute a = d1 m3else

Compute a = d2 m3

end ifCompute m1 =m3 aCompute 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 single

angle, . 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 thatC = TT()CT() whereT() 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 provides

fewest 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 ESL

description 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 (11, 22 and 12) are

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28 3.2. Laminate description

m1

m2

m3

q 1

2

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

coordinate system base vectors mi.

continuous across the thickness as well and furthermore, that transverse strains(13 and 23) 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 better

result 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 Nl and the layer thickness is hl. Each layer is also associatedwith a material, Cl, as well as an angle, l, 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(l)l (3.12)

and consequently, the stresses will be layer-wise continuous since strains are

continuous 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,

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Layer1Layer2Layer3

LayerN-1

LayerN

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 integrationEvaluation 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

Nli=1

hi hl(1 tl)

(3.13)

where tl = 13

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

becomes:

KT =Ne

k=1

Nl

q=1

BTquqsq +B

Tq

squq

hlh

drdsdtl

k(3.14)

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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

Nlq=1

BTq sq

hlh

drdsdtl

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 Nl sub-elements in thethickness direction and so, the computational cost will increase linearly with the

number 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 ofZienkiewicz 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 to

obtain 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 that

the 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

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element order gets higher. Indeed, the SHELL16 element in MUST has provenvery robust and only exhibits locking behavior under extreme conditions such

as 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 solutions

to 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 low

computational 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 the

number 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 . . . N p,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. ij|p ij(rp, sp, tp). Introducing a set of interpolation functions for each

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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 (AD) used for

transverse shear strain evaluation.

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

ASij =Np

p=1

Nijp (r, s)ij|p (3.16)

where Np is the number of tying points, p, for the ijth strain component. TheMITC interpolation functions, Nijp , must naturally fulfill the relation:

Nijp |q = kq , q = 1 . . . N p (3.17)

so that the pth interpolation function assumes the value 1 in the pth 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 is

the same (see Bucalem and Bathe (1993) for details). For the three- and four-nodeMITC elements there are four tying

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