Aerostructural analysis and design optimization of ... Aerostructural analysis and design optimization of composite aircraft Graeme James Kennedy Doctor of Philosophy Graduate Department

Aerostructural analysis and design optimization of compositeaircraft

by

Graeme James Kennedy

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Applied Science and EngineeringUniversity of Toronto

Copyright c© 2012 by Graeme James Kennedy

Abstract

Aerostructural analysis and design optimization of composite aircraft


Doctor of Philosophy

Graduate Department of Applied Science and Engineering

University of Toronto

2012

High-performance composite materials exhibit both anisotropic strength and stiffness prop-

erties. These anisotropic properties can be used to produce highly-tailored aircraft struc-

tures that meet stringent performance requirements, but these properties also present unique

challenges for analysis and design. New tools and techniques are developed to address some

of these important challenges. A homogenization-based theory for beams is developed to

accurately predict the through-thickness stress and strain distribution in thick composite

beams. Numerical comparisons demonstrate that the proposed beam theory can be used to

obtain highly accurate results in up to three orders of magnitude less computational time

than three-dimensional calculations. Due to the large finite-element model requirements for

thin composite structures used in aerospace applications, parallel solution methods are ex-

plored. A parallel direct Schur factorization method is developed. The parallel scalability

of the direct Schur approach is demonstrated for a large finite-element problem with over

5 million unknowns. In order to address manufacturing design requirements, a novel lami-

nate parametrization technique is presented that takes into account the discrete nature of

the ply-angle variables, and ply-contiguity constraints. This parametrization technique is

demonstrated on a series of structural optimization problems including compliance mini-

mization of a plate, buckling design of a stiffened panel and layup design of a full aircraft

wing. The design and analysis of composite structures for aircraft is not a stand-alone prob-

lem and cannot be performed without multidisciplinary considerations. A gradient-based

aerostructural design optimization framework is presented that partitions the disciplines

into distinct process groups. An approximate Newton–Krylov method is shown to be an

ii

efficient aerostructural solution algorithm and excellent parallel scalability of the algorithm

is demonstrated. An induced drag optimization study is performed to compare the trade-off

between wing weight and induced drag for wing tip extensions, raked wing tips and winglets.

The results demonstrate that it is possible to achieve a 43% induced drag reduction with no

weight penalty, a 28% induced drag reduction with a 10% wing weight reduction, or a 20%

wing weight reduction with a 5% induced drag penalty from a baseline wing obtained from

a structural mass-minimization problem with fixed aerodynamic loads.

iii

Acknowledgements

Many people have helped me over the course of my research. In particular, I am deeply

grateful for the support and guidance of my supervisor, Professor Joaquim Martins. His

knowledge, enthusiasm and vision for aircraft design and optimization have been an inspi-

ration to me. It has been a pleasure to work with him and I look forward to our future

collaborations.

I would also like to thank the other members of my doctoral committee, Professor Chris

Damaren and Professor David Zingg, for their insights and challenging questions that helped

me to examine different perspectives. Thier questions and comments greatly enhanced the

quality of the thesis.

I am very grateful to many of my colleagues both past and present from UTIAS. In

particular, the members of the MDO lab have provided a unique and enjoyable atmosphere

for research. Specifically, I would like to acknowledge Sandy Mader, for his perspective

and helpful suggestions, Gaetan Kenway for his aircraft design advice, Kai James for his

perspective on all things related to topology optimization, and Jason Hicken for his help

with iterative methods and preconditioners.

I would not have started down the road of graduate studies without my parents, who

encouraged me in all my academic efforts and instilled in me the importance of working on

problems that deeply interest you.

Finally, I would not have been able to complete my studies without the loving support of

my wife Sabrina. Thank you for keeping me grounded, making me smile, and always being

supportive.


University of Toronto Institute for Aerospace Studies

September, 2012

Contents

List of Figures v

List of Tables vi

List of Symbols and Abbreviations vii

1 Introduction 1

1.1 Thesis outline and contributions . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 A homogenization-based theory for beams 7

2.1 Review of relevant contributions . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 The homogenization-based beam theory . . . . . . . . . . . . . . . . . . . . 13

2.3 A finite-element method for the fundamental states . . . . . . . . . . . . . . 26

2.4 Comparison with three-dimensional results . . . . . . . . . . . . . . . . . . . 32

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Parallel finite-element analysis of shell structures 43

3.1 Finite-element analysis of shell structures . . . . . . . . . . . . . . . . . . . . 44

3.2 Parallel finite-element analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Parallel solution methods for sparse linear systems . . . . . . . . . . . . . . . 55

3.4 Structural sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 Laminate parametrization 77

4.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2 The proposed laminate parametrization . . . . . . . . . . . . . . . . . . . . . 81

4.3 Adjacency constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.4 Structural optimization studies . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.5 Wing-box optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

i

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5 Aerostructural analysis and design optimization 110

5.1 Review of aerostructural optimization . . . . . . . . . . . . . . . . . . . . . . 111

5.2 Aerostructural analysis components . . . . . . . . . . . . . . . . . . . . . . . 114

5.3 Aerostructural solution methods . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.4 Aerostructural gradient evaluation . . . . . . . . . . . . . . . . . . . . . . . . 126

5.5 Aerostructural optimization studies . . . . . . . . . . . . . . . . . . . . . . . 129

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6 Contributions, conclusions and future work 144

6.1 Contributions and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.3 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

References 149

A Shell element tests 162

ii

List of Figures

1.1 Approximate composite mass percentage by year of entry into service . . . . 2

1.2 Degrees of freedom required for the structural analysis of a wing . . . . . . . 3

1.3 Connections between the major thesis topics . . . . . . . . . . . . . . . . . . 5

2.1 A comparison between through-thickness shear stress and strain distributions 8

2.2 Geometry of the reference beam . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 The fundamental states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 The angle section geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Sectional strain energy of the difference between theory and finite-element

results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.6 Comparison of the through-thickness volumetric strain at the cross-section

x = L/2 for the statically indeterminate beam . . . . . . . . . . . . . . . . . 35

2.7 Relative errors of the strain moments for the statically determinate beam . . 36

2.8 Relative errors of the strain moments for the statically indeterminate beam . 37

2.9 Relative errors of the stress moments for the statically indeterminate beam . 38

2.10 The moments of the strain residual for the statically determinate beam . . . 39

2.11 Components of the relative strain correction error for the statically determi-

nate beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 The initial and deformed geometry of a shell segment . . . . . . . . . . . . . 47

3.2 The interpolation scheme for the MITC shell elements . . . . . . . . . . . . . 52

3.3 Condition number of a plate problem for various slenderness ratios . . . . . . 54

3.4 Domain decomposition and matrix for the approximate Schur preconditioner 58

3.5 Domain decomposition and matrix for the direct Schur method . . . . . . . . 61

3.6 The block cyclic matrix data format . . . . . . . . . . . . . . . . . . . . . . . 64

3.7 The annular disk and transonic transport wing finite-element problems . . . 66

3.8 Level of fill and parallel scaling studies for a plane stress problem . . . . . . 67

3.9 Factorization times for the direct Schur approach using various orderings . . 69

iii

3.10 Factor time as a function of the ideal factor time . . . . . . . . . . . . . . . 70

3.11 Solution and assembly times for the direct Schur approach . . . . . . . . . . 72

3.12 Adjoint computational cost assessment . . . . . . . . . . . . . . . . . . . . . 74

4.1 Illustration of the spherical constraint . . . . . . . . . . . . . . . . . . . . . . 84

4.2 Convergence history for the plate compliance problem . . . . . . . . . . . . . 92

4.3 Convergence history for the plate compliance problem with adjacency constraints 92

4.4 The lamination sequences for the compliance minimization problem . . . . . 93

4.5 The lamination sequences for the compliance minimization problem with ad-

jacency constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.6 The geometry of the buckling optimization problem formulation. . . . . . . . 94

4.7 Optimal lamination sequences for the stiffened panel optimizations . . . . . . 97

4.8 Convergence history of the stiffened panel optimizations . . . . . . . . . . . . 98

4.9 Function evaluations required for the stiffened panel optimizations . . . . . . 98

4.10 The number of plies for the top and bottom skin and stiffener, and the top

and bottom the stiffener heights and stiffener base-widths. . . . . . . . . . . 103

4.11 An illustration of the global-local wing-box analysis . . . . . . . . . . . . . . 104

4.12 The continuation history of the load factor, λ, and the infeasibility ||cs(x∗n)−e||1 for the wing-box optimization. . . . . . . . . . . . . . . . . . . . . . . . 106

4.13 The number of function evaluations and gradient evaluations required for the

wing-box optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.14 Ply angle sequences for the wing-box optimization problem. Only the top half

of the symmetric laminate is shown. . . . . . . . . . . . . . . . . . . . . . . . 109

5.1 Timing results for analysis and sensitivity evaluation using TriPan with a

14 400 surface panel mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.2 Verification of TriPan against SUmb for the ONERA M6 at M = 0.5 and

α = 3.06. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.3 The initial and final geometry obtained using FFD approach . . . . . . . . . 122

5.4 Comparison of solution times for the ANK 1 and ANK 2 aerostructural solu-

tion methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.5 Comparison of computational times for different parts of the aerostructural

sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.6 Aerostructural sensitivity verification . . . . . . . . . . . . . . . . . . . . . . 129

5.7 Summary of the results from all the induced drag minimization studies . . . 137

5.8 A comparison of the optimal planforms . . . . . . . . . . . . . . . . . . . . . 138

iv

5.9 The t/c distribution for the span extension results . . . . . . . . . . . . . . . 139

5.10 Thickness and stiffener height distributions for the span extension cases . . . 142

5.11 Twist and clc distributions for the span extension cases . . . . . . . . . . . . 143

A.1 Displacement and MITC-based shell element accuracy study for a plate . . . 163

A.2 Displacement and MITC-based shell element accuracy study for a cylinder . 163

A.3 A test of the MITC shell elements for the snap-through of a partial cylinder

and the pressure-buckling of a full cylinder . . . . . . . . . . . . . . . . . . . 164

v

List of Tables

2.1 Representative orthotropic stiffness properties . . . . . . . . . . . . . . . . . 32

4.1 Representative IM7/3501-6 stiffness and strength properties. . . . . . . . . . 90

4.2 Design problem summary for the buckling optimization studies . . . . . . . . 96

4.3 Wing-box mass breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.1 Levels of parallelism and process groups within the aerostructural optimiza-

tion framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2 Summary of the design variables in the aerostrucutral induced drag study . . 132

5.3 Summary of the constraints in the aerostrucutral induced drag study . . . . 133

vi

List of Symbols and Abbreviations

Abbreviations

AMD Approximate Minimum Degree

BILU Block-Incomplete LU-factorization

BWB Blended Wing Body

CAD Computer Aided Design

CLT Classical Lamination Theory

DMO Discrete Material Optimization

FFD Free Form Deformation

F-GCROT Flexible-GCROT

F-GMRES Flexible-GMRES

FSDT First-order Shear Deformation Theory

GA Genetic Algorithm

GCROT Generalized Conjugate Residual with inner-Orthogonalization

and outer Truncation

GMRES Generalized Minimum RESidual

ILU Incomplete LU-factorization

KS Kreisselmeier–Steinhauser (constraint aggregation technique)

LICQ Linear-Independence Constraint Qualification

MDO Multidisciplinary Design Optimization

MFCQ Mangasarian–Fromovitz Constraint Qualification

MITC Mixed Interpolation of Tensorial Components

MTOW Maximum TakeOff Weight

ND Nested Disection

OML Outer Mould Line

RCM Reverse Cuthill–McKee

SIMP Solid Isotropic Microstructure with Penalization

vii

SNOPT Sparse Nonlinear OPTimizer (software)

SUmb Stanford University multi-block solver

TACS Toolkit for the Analysis of Composite Structures

Chapter 2

A The cross-sectional area

C The constitutive matrix

Cs The strain moment correction matrix

D The homogenized stiffness matrix

E The homogenized flexibility matrix

ε The strain

ε The strain residuals

ε(k)F The primary fundamental states

ε(k)LF The load-dependent fundamental states

e The strain moments

e The moments of the strain produced by the displacement residuals

e(k)FL The load-dependent strain moment correction

Iz, Iy The second moments of area

Ks The stress moment correction matrix

L0 The normalized displacement operator

Ls The stress and strain moment operator

Lε The average strain operator

σ The stress

σ The stress residuals

σ(k)F The primary fundamental states

σ(k)LF The load-dependent fundamental states

s The stress moments

s(k)FL The load-dependent stress moment correction

u The displacements

u The residual displacements

u0 The normalized displacement moments

viii

Chapter 3

A A sparse matrix

b, x The right-hand-side and solution vector

C The constitutive tensor

ε The Green strain

η The shell volume parameters

f(u,x) A vector of functions of interest

φ Small rotations in the global Cartesian frame

ψ The adjoint vector

∇xf The total derivative of a vector of functions of interest

Q The shell-normal rotation matrix

r The mid-surface of the shell

R The shell volume position vector

S The second Piola–Kirchhoff stress tensor

u The displacement of the mid-surface

U The through-thickness displacement

ω The rate of change of the displacements through the thickness

ξ1, ξ2 The shell surface parameters

ζ The through-thickness parameter

Chapter 4

A(i), B(i), D(i), A(i)s The constitutive matrices

Aw The ply-angle selection variable weighting matrix

cs(x) The spherical constraints

d(x) The grouped adjacency constraints

∆cr The critical end shortening

f(x) The design objective

F(i)KS The aggregated failure envelope

γ The penalization parameter

Ik The set of excluded design variable selections

τ The regularization parameter

xijk The ply-angle selection variables

ix

x The full set of ply-angle selection variables

Chapter 5

b The wing span

C(a, ϕ) A rotation matrix

Di Induced drag

e The span efficiency factor

F Consistent structural force vector due to aerodynamic loads

L Aerodynamic lift

ψ The aerostructural adjoint vector

q The aerostructural state variables

R The aerostructural residuals

RS The structural residuals

RA The aerodynamic residuals

u The structural state variables

w The aerodynamic state variables

x The aerostructural design variables

Xs The aerodynamic surface nodes

x

Chapter 1

Introduction

Composite materials are fabricated using a macroscopic combination of two or more con-

stituent materials such that the overall structural properties of the composite are superior

to the properties of the individual component materials. In aerospace applications, high-

performance composites are often made from carbon fibers set in an epoxy matrix. These

high-performance composite materials exhibit both highly anisotropic strength and stiff-

ness properties, making the analysis and design of composite structures more challenging.

However, the anisotropic properties of composite materials can also be exploited to obtain

tailored structures that meet stringent design requirements, yet are lighter than equivalent

metallic structures. Fully exploiting the anisotropic nature of composites often requires new

analysis methods and new design methodologies.

Over the years, new aircraft designs have employed increasing amounts of composite

materials. Figure 1.1 shows the usage of composite materials in both civil and military

aircraft over several decades. This trend is due in part to the higher stiffness-to-weight

and strength-to-weight ratios exhibited by composite materials when compared to metallic

structures. Additionally, composites may have manufacturing benefits. For example, using

composite manufacturing techniques, a large complex structural component can be manu-

factured in a single piece. This part-count reduction can save both weight and maintenance

costs [Baker et al., 2004]. Currently, composite materials are being used for highly-loaded,

primary structures in civil aircraft, including the Bombardier CSeries wing, the Boeing 787

wing and fuselage, and the Airbus A350 wing and fuselage. Yet these applications have not

come without considerable investment in research and development over the past 50 years.

The design and analysis of composite structures for aircraft, however, is not a stand-alone

problem and thus cannot be performed in isolation. Aircraft are complex, coupled systems

that require the simultaneous consideration of multiple disciplines, such as structures, aero-

1

Chapter 1. Introduction 2

0

20

40

60

80

100

1970 1980 1990 2000 2010 2020

Approximate year of entry into service

Approximate

composite mass

percentage

Civil

Business

Military

F-15A F-16AF-18A

AV8BB2

F-18E/F

V-22F-22

F-35

767737-300

A320

MD11A340

A330

777

787 A350

CSeries

HondaJet

Lear85

Figure 1.1: Approximate composite mass percentage by year of entry into service. Sources: Baker

et al. [2004], compositesworld.com, airbus.com, cseries.com

dynamics, stability and control, and propulsion, amongst others. Multidisciplinary design

optimization (MDO) techniques are frequently applied to aircraft design problems to address

this complexity. MDO methods are designed to address both the strong interdisciplinary cou-

pling that exists amongst disciplines, and the difficulties of integrating many disciplines with

various consistency constraints and discipline design variables into a single, consistent design

optimization problem [Sobieszczanski-Sobieski and Haftka, 1997, Martins and Lambe, 2012].

As with any emerging technology, there is a need to assess the relative benefits of new

composite systems, new construction techniques, and new structural configurations that

have the potential to improve overall aircraft performance. A promising way to analyze

these new technologies more reliably is with high-fidelity analysis tools. However, these

tools are more computationally intensive than conventional analysis techniques. In addition,

a fair assessment of the potential benefits of novel technologies can only be performed if

the performance of each new technology is maximized and the optimum results compared.

Therefore, when considering new structural concepts and novel aircraft designs, there is a

need for efficient high-fidelity MDO methods. High-fidelity MDO methods are essential for

a reliable analysis of unconventional designs, such as the blended wing body (BWB), where

no existing knowledge base exists to help guide design decisions [Liebeck, 2004].

There are several barriers to the full adoption of high-fidelity MDO methods, includ-

ing the high computational cost, and the difficulty and complexity of coupling disciplinary

analyses. In addition to the high computational cost of analysis, high-fidelity methods of-

ten require detailed design parametrizations, frequently resulting in optimization problems

with thousands to tens of thousands of design variables. Large high-fidelity MDO problems

http://www.compositesworld.com/articles/boeing-787-update

http://www.airbus.com/aircraftfamilies/passengeraircraft/a350xwbfamily/

http://cseries.com/en/#/cseries/technology/advancedmaterial/


Model complexity

Degreesoffreedom

102

104

106

108

1010

1012

Beam model

Shell model

Full 3D model

Metallic wing

Composite wing

Figure 1.2: Number of degrees of freedom required for the finite-element analysis of a Boeing

777-size wing.

can only be solved in a reasonable time using a gradient-based optimization algorithm in

conjunction with efficient gradient evaluation techniques. Implementing an efficient gradient

evaluation technique for a coupled, multidisciplinary system, however, requires significant

time and effort.

The analysis of composite structures may require more refined meshes and as a result,

more computational time than an equivalent isotropic structure. Finer meshes are required

when modeling non-symmetric composites, or composites bonded to metallic components,

where stress concentrations can arise near boundaries and joints between structural members.

Furthermore, composites exhibit complex through-thickness stress distributions that must

be analyzed using large finite-element models. Figure 1.2 shows the approximate number

of degrees of freedom required for the analysis of both a metallic and a composite wing for

increasingly complex structural models: a simple beam model, a shell model, and a full three-

dimensional model. These ranges of values are based on a Boeing 777-size wing with the

following set of assumptions: the beam model ranges are based on using approximately 100

beam elements for the span; the shell model lower bound is based on an analysis with smeared

stiffeners, while the upper bound is based on an analysis with discrete stiffeners; the full

three-dimensional model lower bound is based on four through-thickness nodes for the skin,

while the upper bound is based on four through-thickness nodes for each composite layer.

While the larger meshes that use three-dimensional elements may currently be impractical


for design, they illustrate an upper bound on the size of the structural model.

As Figure 1.2 illustrates, using high-fidelity models for analysis and design optimiza-

tion involves solving large finite-element problems that require significant computational

resources. These large problems can only be solved in a practical time frame if efficient

parallel methods can be employed. Furthermore, gradient-based optimization methods for

structural design optimization with parametrizations that require thousands to tens of thou-

sands of design variables are only practical if accurate and efficient derivative evaluation

methods, such as the adjoint method, are employed. In order to address these requirements

I have developed a parallel finite-element code specifically designed for multidisciplinary de-

sign optimization of composite structures. This finite-element code, called the Toolkit for

the Analysis of Composite Structures (TACS), includes routines that enable accurate and

efficient analytic derivative computation, an essential tool for gradient-based design opti-

mization. In addition, TACS is designed to efficiently couple with other disciplines for both

analysis and multidisciplinary derivative computations.

The goal of this thesis is to address challenging problems in the areas of composite struc-

tural analysis and design, and in the area of aerostructural design optimization of composite

aircraft structures. In order to make progress towards this goal, I have developed new

analysis methods for composites, I have refined numerical algorithms for solving large finite-

element problems, and I have applied multidisciplinary design analysis and optimization

methods to the design of composite aircraft structures. In this thesis, I have focused on the

analysis and design of composite wing box structures for large transport aircraft. However,

the techniques presented within this thesis are more broadly applicable to other composite

design problems. Furthermore, I have also focused on applications that use high-strength

carbon epoxy composite systems, which are commonly used in transport aircraft structures.

However, many of the results could also be applied to other laminated composite systems

with different material properties.

1.1 Thesis outline and contributions

The connections between the major topics addressed in this thesis are illustrated in Fig-

ure 1.3. The central goal is to enhance aerostructural analysis and design optimization

methods for composite aircraft. This topic, however, cannot be approached without also

addressing other topics that are closely related.

Accurate stress and strain distributions are required to predict the failure properties

of composite structures. In Chapter 2, I present a novel beam theory for isotropic and


Aerostructuralanalysis and design

optimization

Structural analysisand design

optimization

Laminateparametrization

Parallelfinite-element

solution methods

Beam theory Geometricparametrization

Aerostructuralsolution methods

Aerostructuralderivative evaluation

Figure 1.3: An illustration of the connection between the major topics addressed in this thesis.

composite beams. This beam theory can be used to accurately capture the through-thickness

distributions of all components of stress and strain in isotropic and composite sections.

Conventional beam theories often cannot be used to accurately determine the stress and

strain at ply interfaces or stress concentrations in the presence of edge effects. If present,

these effects are likely to dominate the failure behavior of thick sections [Pagano and Pipes,

1971]. In addition, the theory provides a consistent definition of the shear strain correction

matrix as well as higher-order pressure corrections that provide additional refinement in the

presence of externally applied loads. Finally, I demonstrate that the beam theory can be

used to obtain stress and strain distributions with a high-degree of accuracy when compared

to full three-dimensional results but in 3 orders of magnitude less computational time. The

accuracy of the stress and strain distributions as well as the computational efficiency, make

this theory a powerful tool for analysis and design.

Thin, stiffened shell structures, such as aircraft wings, are frequently used in aerospace

applications due to their high stiffness-to-weight ratios. In Chapter 3, I present a detailed

description of the analysis of thin composite shell structures. First, I present a high-order

shell element formulation using both a displacement-based approach and a mixed interpola-

tion of tensorial component (MITC) element that is not susceptible to shear and membrane

locking [Dvorkin and Bathe, 1984, Bathe et al., 2000]. Next, I present the parallel solution

methods used to solve the large, sparse, linear systems resulting from the finite-element dis-

cretization of thin shell structures. Finally, I present the sensitivity analysis methods used

to compute the derivatives of objectives of interest. The high-order elements prove to be

effective for analysis, yielding the most accurate solutions for a fixed computational cost.

On the other hand, the computational cost of the derivatives of these higher-order elements

increases dramatically with element order. Therefore, there is a trade-off between element


order and accuracy of the solution and the computational cost of the gradients for design

optimization.

Design optimization of composite structures cannot be performed without a flexible de-

sign parametrization that can take into account important manufacturing requirements.

In Chapter 4, I present a parametrization technique for laminated composite structures.

This parametrization takes into account the discrete nature of the ply-angle variables that

may arise due to manufacturing constraints. Often these ply parametrization problems are

solved with gradient-free approaches [Haftka and Walsh, 1992, Le Riche and Haftka, 1993,

Adams et al., 2004], however, this parametrization results in a continuous formulation that

is amenable to gradient-based design optimization. The proposed parametrization uses an

exact penalty function to ensure that there are no intermediate plies in the final design. I also

present additional constraints that can be used to enforce other manufacturing requirements

such as a restriction on the number of contiguous plies at the same angle, or that adjacent

ply angles be restricted to a reduced set of values.

In Chapter 5, I present an aerostructural optimization framework, focusing in partic-

ular on the parallel computational aspects of the approach. Previous authors have used

high-fidelity aerodynamic models coupled to low or medium fidelity structural finite-element

models [Martins et al., 2004, Maute et al., 2001]. This imbalance may be acceptable if the pri-

mary interest is the aerodynamic performance of the flying, displaced shape of a conventional

wing. However, more detailed effects, such as the skin-bulge due to the internal pressure for

a BWB [Liebeck, 2004, Hansen et al., 2008], can only be assessed using high-fidelity models.

In addition, low-fidelity models such as a simple beam model, cannot always be relied on to

provide detailed stress distributions or accurate weight estimates for novel configurations or

even novel structural composite technologies. For instance, it would be difficult to assess the

benefits of a composite system such as the new structural concept PRSEUS [Jegley et al.,

2002, Velicki and Thrash, 2008, Li and Velicki, 2008], for a BWB configuration using only

a beam model, or even a coarse shell model. As a first step towards this goal, I examine

methods in which high-fidelity, finite-element structural analysis is coupled to a medium-

fidelity aerodynamic tool. In particular, I examine solution methods for problems in which

both the aerodynamic and structural analyses are performed in parallel and where both re-

quire significant computational time. Finally, using the proposed aerostructural framework,

I present results for a series of non-planar configurations and draw conclusions about their

relative benefits.

Chapter 2

A homogenization-based theory for

beams

Beam theories are developed based on a set of assumptions that are used to reduce the

complex behavior of a slender, three-dimensional body to an equivalent one-dimensional

problem. The usefulness of a beam theory should be assessed based on its range of appli-

cability, the accuracy of its results, and the complexity of the analysis required to obtain

results. In this chapter, I present a homogenization based theory for anisotropic beams. This

homogenization-based theory is based on a series of novel contributions to beam theory orig-

inally conceived by Hansen and Almeida [2001] and Hansen et al. [2005] and applied to the

analysis of layered beams under conditions of plane stress. While the assumptions used to

derive this homogenization-based theory differ significantly from classical assumptions, the

proposed beam theory takes a form similar in many respects to classical Timoshenko beam

theory [Timoshenko, 1921, 1922]. This homogenization-based theory, however, is specifically

designed for composite beams. In the homogenization-based approach, the stiffness prop-

erties, shear strain correction matrix, and load-dependent corrections within the theory are

calibrated based on a hierarchy of solutions called the fundamental states. The fundamen-

tal states are accurate sectional stress and strain solutions to a series of carefully-chosen,

statically determinate beam problems. Since it is difficult to obtain exact solutions for the

fundamental states for an arbitrary section, I formulate a finite-element solution technique

to obtain approximate solutions.

There are several difficulties that arise when developing a beam theory for the analysis of

composite beams. To illustrate the most significant challenges, consider the four layer beam

illustrated in Figure 2.1. This two-dimensional beam is composed of alternating layers of two

materials, where one material has a shear modulus that is 10 times higher than the other.

7

Chapter 2. A homogenization-based theory for beams 8

Actual

strain stress

Timoshenko

strain stress

Higher-order

strain stressz

0.1 G

G

0.1G

G

Figure 2.1: A comparison between the actual through-thickness shear stress and strain distribu-

tions, and the distributions use in Timoshenko and higher-order beam theories.

A classical approach to developing a beam theory is to assume a polynomial distribution of

the through-thickness displacements. For composite beams, this displacement distribution

leads to a continuous distribution of the through-thickness shear strain and a discontinuous

shear stress. In reality, as shown in Figure 2.1, the shear stress should be continuous and

the shear strain should be discontinuous. While a post-processing integration can be used

to obtain a continuous shear stress distribution, the original, incorrect distributions are used

to evaluate the shear stiffness of the beam. This inconsistency leads to a poor shear stiffness

prediction. To account for this discrepancy, shear correction factors are often introduced

but the precise value and definition of these factors is frequently not defined rigorously. As

a result, the predictions of these types of theories can be somewhat arbitrary, and depend

on the value of the correction factor employed in the analysis.

In the following Chapter, I develop a beam theory based on integrals of the displace-

ments, stresses and strains through the thickness of the beam. This avoids the use of as-

sumptions about the through-thickness displacement distribution, and correctly accounts for

non-smooth or discontinuous through-section stress, strain and displacement distributions.

In addition, the fundamental states, which are accurate through-thickness stress and strain

solutions, are used to evaluate the stiffness of the beam. As a result, the stiffness is pre-

dicted based on the actual stress and strain distribution in the beam, rather than an assumed

distribution. Finally, correction factors that arise naturally within the theory are precisely

defined and, as a result, there is no ambiguity about their value.

This chapter is organized as follows: Section 2.1 contains a review of contributions to

beam theory in the literature relevant to the homogenization-based approach. Section 2.2

contains the presentation of the homogenization-based theory. Section 2.3 presents a finite-

element method for the determination of the fundamental states for beams with arbitrary


cross-sections. Finally, a comparison with three-dimensional results is presented in Sec-

tion 2.4. The material from this chapter is based on the publications Kennedy et al. [2011]

and Kennedy and Martins [2011].

2.1 Review of relevant contributions

In this section, I present a review of various contributions to the literature that are most

relevant to the proposed beam theory. A comprehensive review of all beam theories is not

practical here due to the volume of literature that has been produced on the subject over

several decades.

In two influential papers, Timoshenko [1921, 1922] developed a beam theory for isotropic

beams based on a plane stress assumption. Timoshenko’s theory takes into account shear

deformation and includes both displacement and rotation variables. In addition, Timoshenko

introduced a shear correction factor that modifies the relationship between the shear resultant

and the shear strain at the mid-surface. The definition and value of the shear correction

factor has been the subject of many papers, some of which are discussed below.

Later, Prescott [1942] derived the equations of vibration for thin rods using average

through-thickness displacement and rotation variables. Like Timoshenko, Prescott intro-

duced a shear correction factor to account for the difference between the average shear on a

cross-section and the expected quadratic distribution of shear.

Cowper [1966], independently from Prescott, developed a reinterpretation of Timoshenko

beam theory based on average through-thickness displacements and rotations. Using these

variables and integrating the equilibrium equations through the thickness, Cowper developed

an expression for the shear correction factor, which he evaluated using the exact solution to

a shear-loaded cantilever beam excluding end effects. Cowper obtained values for the shear

coefficient for beams with various cross-sections, but his approach was limited to symmetric

sections loaded in the plane of symmetry. Mason and Herrmann [1968] later extended the

work of Cowper to include isotropic beams with an arbitrary cross-section.

Stephen and Levinson [1979] developed a beam theory along the lines of Cowper’s, but

recognized that the variation in shear along the length of the beam would lead to a modifica-

tion of the relationship between bending moment and rotation. Therefore, they introduced a

new correction factor to account for this variation, and obtained its value based on solutions

to a cantilever beam subject to a constant body force given by Love [1920].

More recently, Hutchinson [2001] introduced a new Timoshenko beam formulation and

computed the shear correction factor for various cross-sections based on a comparison with


a tip-loaded cantilever beam. For a beam with a rectangular cross-section, Hutchinson

obtained a shear correction factor that depends on the Poisson ratio and the width-to-depth

ratio. In a later discussion of this paper, Stephen [2001] showed that the shear correction

factors he had obtained in earlier work [Stephen, 1980] were equivalent to Hutchinson’s

results.

Various authors have developed analysis techniques specifically for composite beams.

Capturing shear deformation effects is often more important for a composite beam than for

a geometrically equivalent isotropic beam, due to the significantly lower ratio of the shear to

extension modulus exhibited by composite materials. As a result, Timoshenko-type beam

theories are often used to model composite beams. This direct extension of Timoshenko

beam theory to the analysis of composite beams is presented by many authors, such as

Librescu and Song [2006] or Carrera et al. [2010b]. Other authors have developed extensions

to Cowper’s approach. Dharmarajan and McCutchen [1973] extended Cowper’s work to

orthotropic beams, obtaining results for circular and rectangular cross-sections. Later, Bank

[1987] and Bank and Melehan [1989] used Cowper’s approach to develop expressions for the

shear correction for thin-walled open and closed section orthotropic beams.

Numerous authors have developed refined beam and plate theories that are designed to

better represent the through-thickness stress distribution behavior for both isotropic and

composite plates and beams. For instance, Lo et al. [1977a,b] developed a higher-order plate

theory for isotropic and laminated plates using a cubic through-thickness distribution of the

in-plane displacements and quadratic out-of-plane displacements. Reddy [1987] developed a

high-order plate theory for laminated plates based on a cubic through-thickness distribution

of the in-plane displacements and obtained the equilibrium equations using the principle

of virtual work. More recently, Carrera and Giunta [2010] developed a refined beam theory

based on a hierarchical expansion of the through-section displacement distribution. This the-

ory, which presents a unified framework, is more accurate than classical approaches [Carrera

and Petrolo, 2011] and can be used for arbitrary sections composed of anisotropic materials.

A finite-element approach using this refined beam theory has also been developed for both

static [Carrera et al., 2010a] and free-vibration analysis [Carrera et al., 2011].

Although these higher-order theories are more accurate than classical Timoshenko beam

theory, one drawback is their additional analytic and computational complexity. Further-

more, for laminated plates and beams, these theories predict a continuous through-thickness

shear strain and discontinuous shear stress, whereas the exact distribution is discontinuous

shear strain and continuous shear stress. Zig-zag theories address these through-thickness

compatibility issues by employing a C0, layer-wise continuous displacement. These types of


theories were first developed by Lekhnitskii [1935]. An extensive historical review of these

theories was performed by Carrera [2003].

Many authors have used three-dimensional elasticity solutions as a way to improve the

modeling capabilities of beam theories. Following the variational framework of Berdichevskii

[1979], Cesnik and Hodges [1997] and Yu et al. [2002a] developed a variational asymptotic

beam sectional analysis approach for the analysis of nonlinear orthotropic and anisotropic

beams. In their approach, cross-sectional solutions containing all stress and strain compo-

nents are used to calibrate the stiffness properties and reconstruct the stress distribution

for a Timoshenko-like beam. The stiffness properties are recovered using an asymptotic

expansion of the strain energy. Popescu and Hodges [2000] used this approach to examine

the stiffness properties of anisotropic beams, focusing in particular on the shear correction

factor. Yu et al. [2002b] validated the approach of Cesnik and Hodges [1997] and Yu et al.

[2002a] using full three-dimensional finite-element analysis.

Ladeveze and Simmonds [1998] and Ladeveze et al. [2002] presented an “exact” beam

theory that uses three-dimensional Saint–Venant and Almansi–Michell solutions for the cali-

bration of the stiffness properties of the beam and stress reconstruction. Using the framework

set out by Ladeveze and Simmonds [1998] and Ladeveze et al. [2002], El Fatmi and Zenzri

[2002] and El Fatmi and Zenzri [2004] developed a method for determining the Saint–Venant

and Almansi–Michell solutions required by the “exact” beam theory using a computation

only over the cross-section of the beam. El Fatmi [2007a,b] developed a beam theory based

on non-uniform warping of the cross-section, using the framework of Ladeveze and Simmonds

[1998]. Their theory incorporated the Saint–Venant and Almansi–Michell solutions obtained

by El Fatmi and Zenzri [2002, 2004].

Dong et al. [2001], using the techniques presented by Iesan [1986a,b], developed a tech-

nique to solve the Saint–Venant problem for a general anisotropic beam of arbitrary con-

struction. Kosmatka et al. [2001] determined the sectional properties, including the stiffness

and shear center location, based on the finite-element technique of Dong et al. [2001].

Other authors have also used full three-dimensional solutions within the context of a

beam theory. Gruttmann and Wagner [2001], following the work of Mason and Herrmann

[1968], performed a finite-element-based analysis of isotropic beams with arbitrary cross-

sections. Dong et al. [2010] used a semi-analytical finite-element formulation to compare

shear correction factors for general isotropic sections computed using the methods of Cowper

[1966], Hutchinson [2001], Schramm et al. [1994] and Popescu and Hodges [2000].


2.1.1 Features of the homogenization-based approach

The single most important feature of the present theory is the use of the fundamental states.

The fundamental states are obtained from solutions to certain statically determinate beam

problems. These fundamental state solutions are used to construct a relationship between

stress and strain moments, and to reconstruct the stress and strain solution in a post-

processing step. The fundamental states are the axially invariant components of what are

known in the literature as the Saint–Venant and Almansi–Michell solutions. The key com-

ponents of the proposed theory include:

• The use of normalized displacement moments as a representation of the displacement

in the beam, as used by Prescott [1942] and Cowper [1966].

• The use of strain moments as a representation of the strain state in the beam.

• The homogenization of the relationship between stress and strain moments as used by

Guiamatsia [2010] for plates.

• The representation of the full stress and strain field by an expansion of the solution

using the fundamental state solutions.

• The strain moment correction matrix that corrects the strain predicted from the dis-

placement moments.

• The use of load-dependent strain and stress moment corrections that modify the re-

lationship between stress and strain moments in the presence of externally applied

loads.

Hansen and Almeida [2001] and Hansen et al. [2005] developed a theory with these same

ideas for laminated and sandwich beams, using a plane stress assumption. An extension of

this theory to the analysis of plates was presented by Guiamatsia and Hansen [2004] and

Guiamatsia [2010].

These features of the present theory address several issues commonly encountered in

conventional beam theories. The proposed theory contains a self-consistent method to obtain

the equivalent stiffness of the beam and any correction factors required. In addition, all

results from the theory, including the predicted strain moments, can easily be compared

with three-dimensional results. This is due to the fact that all components of the theory

rely on an averaging process that is well-defined for a beam of any construction, which is

not always the case with conventional beam theories. These properties, in addition to the


relatively inexpensive cost of analysis, make the proposed theory a powerful technique for

analysis and design.

2.2 The homogenization-based beam theory

In this section, I present the theoretical development of the homogenization-based beam

theory. The starting point is a description of the geometry of the beam under consider-

ation. Next, I develop a kinematic description of the beam using averaged displacement

and rotation-type variables, based on the work of Prescott [1942] and Cowper [1966]. At

this point, I introduce the fundamental states and use the properties of these solutions to

develop expressions for the homogenized stiffness, stress and strain moment correction ma-

trices, and load-dependent corrections. I conclude with a discussion of the benefits of the

present approach.

y

z

x

L

Figure 2.2: Geometry and reference coordinates for the beam composed of arbitrarily oriented

composite layers.

The geometry of the beam under consideration is illustrated in Figure 2.2. The beam

is aligned with the x-axis and the geometry and construction of the cross-section do not

vary along the length of the beam. The primary purpose of this theory is to analyze layered

composite beams with arbitrarily oriented plies. This type of beam construction results in

an anisotropic constitutive relationship that exhibits coupling amongst all stress and strain

components. As a result of these assumptions, the constitutive equation may be expressed

as

σ(x, y, z) = C(y, z)ε(x, y, z), (2.1)


where σ(x, y, z) and ε(x, y, z) are the full states of stress and strain, and C(y, z) is the

constitutive relationship.

The beam of length L is subject to distributed surface tractions applied in the plane

perpendicular to the x-axis and is subject to axial forces, bending moments, shear forces

and torques at its ends. Shearing tractions applied on the surface of the beam in the x

direction are excluded from consideration.

The reference axis is located at the geometric centroid of the section and the coordinate

axes are aligned with the principal axes of the section. As a result, the moments of area are

defined as follows:

A =

∫Ω

dΩ, Iz =

∫Ω

z2 dΩ, Iy =

∫Ω

y2 dΩ,∫Ω

y dΩ = 0,

∫Ω

z dΩ = 0,

∫Ω

yz dΩ = 0.

The restriction to principal coordinate axes simplifies many of the expressions that are

required below.

2.2.1 The displacement representation

Following the work of Prescott [1942] and Cowper [1966], the exact displacement field can

be expressed in terms of an average representation of the displacement field and residual

displacements. The residual displacements capture the part of the displacement field that

deviates from the average representation. This decomposition of the displacement field is

expressed as

u(x, y, z) =

u(x, y, z)

v(x, y, z)

w(x, y, z)

=

u0(x) + zuz(x) + yuy(x) + u(x, y, z)

v0(x)− zθ(x) + v(x, y, z)

w0(x) + yθ(x) + w(x, y, z)

, (2.2)

where u(x, y, z), and u(x, y, z) =[u v w

]Tare the displacements and residual displace-

ments, respectively. The x-component of the residual displacement u(x, y, z) represents the

warping of the section in the axial direction. For convenience, I collect the variables, u0, v0,

θ, uz and uy in a vector u0(x), defined as follows:

u0(x) =[u0 v0 w0 θ uz uy

]T=

∫Ω

[u

A

v

A

w

A

(yw − zv)

Iy + Iz

zu

Iz

yu

Iy

]TdΩ

= L0u(x, y, z).

(2.3)


Here, u0, v0, and w0 are average displacements in the x, y and z directions. The terms uz,

uy and θ are normalized first-order displacement moments about the z, y and x directions,

respectively. Note that uz, uy and θ represent rotation-type variables, but are not equal to

the average rotations of the section. The vector of variables u0(x) are called the normal-

ized displacement moments, since these variables represent zeroth and first-order normalized

moments of the displacement field u(x, y, z). In addition, the operator L0 is introduced in

Equation (2.3). This operator takes the full three-dimensional displacement field, u(x, y, z),

and returns the normalized moments of displacement. Note that the action of L0 removes

the y-z dependence of the displacement field.

At this point it should be emphasized that the displacement field decomposition (2.2)

ensures that the normalized displacement moments of the residual displacement field are

identically zero, i.e.,

L0u(x, y, z) = 0.

This property of the residual displacement field will be required later to simplify expressions

for the strain moments.

The strain produced by the displacements (2.2) is:

ε(x, y, z) =

εx

εy

εz

γyz

γxz

γxy

=

u0,x + yuy,x + zuz,z + u,x

v,y

w,z

v,z + w,y

uz + w0,x + yθ,x + u,z + w,x

uy + v0,x − zθ,x + u,y + v,x

, (2.4)

where the comma convention has been used to denote differentiation. Note that the exact

pointwise strain distribution requires knowledge of the residual displacements u(x, y, z).

Instead of using pointwise-strain directly, the homogenization-based approach uses mo-

ments of the strain across the section of the beam. This choice has the advantage that

the strain moments are defined regardless of the through-thickness behavior of the pointwise

strain. This property is important since some pointwise strain components are discontinuous

at material interfaces. It is important to recognize, however, that these interfaces are always

parallel to the x direction. As a result, differentiation with respect to x can commute with

integration across the section in the regular manner.


The strain moments are defined as follows:

e(x) =[ex κz κy et exz exy

]T=

∫Ω

[εx zεx yεx (yγxz − zγxy) γxz γxy

]TdΩ

= Lsε(x, y, z).

(2.5)

Here another operator Ls is introduced that takes the full strain field ε(x, y, z) and returns

the moments of strain e(x).

The next step in the development of the theory is to express the strain moments in terms

of the displacement representation (2.2). Using the strain-displacement relationships (2.4),

the definitions of the displacement moments (2.3), and the moments of area, the strain

moments can be written as follows:

e(x) =

Au0,x

Izuz,x

Iyuy,x

(Iy + Iz) θ,x

A (uz + w0,x)

A (uy + v0,x)

+ e(x) = ALεu0(x) + e(x), (2.6)

where e(x) are the moments of the strain produced by the residual displacement. Here A is

a diagonal matrix given by

A = diag A, Iz, Iy, (Iy + Iz), A,A .

The operator Lε takes the vector of average displacements and normalized displacement

moments u0(x), such that ALεu0 produces the first term on the right hand side of Equa-

tion (2.6). Note that action of the operator Lε on the normalized displacements, Lεu0(x), pro-

duces terms that are identical in form to the center-line strain used in classical Timoshenko

beam theory. However, here the variables u0(x) are interpreted as normalized displacement

moments taken from Equation (2.3), not as center-line displacements and rotations.

The term e(x) in the strain moment expression (2.6), is a function of the axial residual


displacement u(x, y, z) and is defined as follows:

e(x) =

∫Ω

u,x

zu,x

yu,x

y (u,z + w,x)− z (u,y + v,x)

u,z + w,x

u,y + v,x

dΩ =

∫Ω

0

0

0

yu,z − zu,yu,z

u,y

dΩ

= Lu(x, y, z),

(2.7)

where the relationship L0u = 0 is used to simplify the expression on the right-hand side

of the above equation. An additional linear operator L has been introduced that takes the

residual axial displacement u(x, y, z) and returns the moments e(x).

The strain moments corresponding to torsion et and shear exz and exy involve terms from

both the normalized displacement moments and the residual axial displacement, u(x, y, z).

These extra terms cannot be evaluated unless u(x, y, z) is known. The approach taken below

is to account for the effect of the residual displacements while formulating the theory in

terms of the average displacement variables, u0(x).

2.2.2 The equilibrium equations

The equilibrium equations are formulated based on the classical approach of integrating

moments of the three-dimensional equilibrium equations over the cross-section of the beam.

The axial, bending, torsion and shear resultants are defined as follows,

s(x) =[N Mz My T Qz Qy

]T=

∫Ω

[σx zσx yσx (yσxz − zσxy) σxz σxy

]TdΩ

= Lsσ(x, y, z).

(2.8)

Here, Ls is the same operator that was introduced for the strain moments (2.5). The vari-

ables s(x) are the stress resultants or stress moments. Integrating the three-dimensional


equilibrium equations over the section results in the following equilibrium equations:

N,x

My,x −Qz

Mz,x −Qy

T,x

Qy,x

Qz,x

+

0

0

0

Px

Py

Pz

= 0. (2.9)

The torque Px(x) and forces Py(x) and Pz(x) are defined as follows:

Px(x) =

∫S

ytz − zty dS,

Py(x) =

∫S

ty dS,

Pz(x) =

∫S

tz dS,

(2.10)

where ty and tz are the y and z components of the surface traction. The integrals above are

carried out over the boundary of the cross-section S.

2.2.3 The fundamental states

In this section, I present a decomposition of the stress and strain distribution within the

beam. This stress and strain decomposition is based on a linear combination of axially-

invariant stress and strain solutions called the fundamental states. The use of the fundamen-

tal states leads to a consistent method for deriving the constitutive relationship between the

stress resultants and the strain moments. Furthermore, the fundamental states can be used to

reconstruct the approximate stress and strain distribution in the beam in a post-processing

step. This representation of the solution is similar to the stress representation presented

by Ladeveze and Simmonds [1998] and used by El Fatmi [2007a,b]. Unlike these authors

however, I also use an analogous representation of the strain solution that is later used to

construct the homogenized stiffness relationship. In this section I describe the properties of

the fundamental states and how they are used in the present theory.

The fundamental states are the axially-invariant, or x-independent, stress and strain

solutions. These solutions are obtained from specially-chosen, statically determinate beam

problems. The loading conditions leading to the fundamental states are shown in Figure 2.3.

These beam problems are sometimes referred to as the Saint–Venant problem [Iesan, 1986a],

for axial, bending, torsion, and shear loads, and the Almansi–Michell problem [Iesan, 1986b],


Primary fundamental states Stress resultants

xy

z

First N = 1

Second Mz = 1

Third My = 1

Fourth T = 1

Fifth Qz = 1 Mz = x

Sixth Qy = 1 My = x

Load-dependent fundamental state

FirstPz = 1 Qz = −x

Mz = −x2/2

Figure 2.3: An illustration of the primary fundamental states and the distribution of the stress

resultants. Forces are denoted by a single arrow and moments by a double arrow.

for a beam subject to a distributed surface load. The beam used to calculate the fundamental

states has the same cross-section and construction as the beam under consideration, but must

be long enough that the end effects do not alter the solution at the mid-plane of the beam.

The fundamental states are extracted from these solutions by taking the distribution of stress

and strain at the mid-plane of the beam. As a result, the fundamental state stress and strain

distributions are solutions in the y-z plane and have no x-dependence.

It is necessary to distinguish between two types of fundamental state solutions: primary

fundamental states, which are labeled σ(k)F (y, z) and ε

(k)F (y, z), and load-dependent funda-

mental states, which are labeled σ(k)LF (y, z) and ε

(k)LF (y, z). The six primary fundamental

states correspond to axial resultant, bending moments about the y and z axes, torsion, and

shear in the z and y directions, respectively. The load-dependent fundamental states are

associated with loads applied to the beam. The fundamental states are used here to form

an approximation of the stress and strain field within the beam. To complete the stress and

strain representation, I also introduce stress and strain residuals, σ(x, y, z) and ε(x, y, z),

that account for the discrepancy between the approximate stress and strain representation


and the exact distribution.

Using these definitions, the stress and strain in the beam may be expressed as follows:

σ(x, y, z) =6∑

k=1

sk(x)σ(k)F (y, z) +

N∑k=1

Pk(x)σ(k)FL(y, z) + σ(x, y, z), (2.11a)

ε(x, y, z) =6∑

k=1

sk(x)ε(k)F (y, z) +

N∑k=1

Pk(x)ε(k)FL(y, z) + ε(x, y, z). (2.11b)

The magnitudes of the primary fundamental states are given by the components of the vector

s(x) and represent axial force, bending moments, torsion, and shear resultants. Individual

components of s(x) are written as sk(x). Note that the magnitudes of the load-dependent

fundamental states Pk(x) are known from the loading conditions and that the fundamental

state magnitudes link the stress and strain distribution.

For consistency between the stress resultants and the stress distribution, the primary

fundamental states must satisfy the relationship,

Lsσ(k)F (y, z) = ik, k = 1, . . . , 6, (2.12)

where ik is the kth Cartesian basis vector. This relationship ensures that the stress resul-

tants of the stress distribution (2.11a) are equal to sk(x). Furthermore, the load-dependent

fundamental states must satisfy

Lsσ(k)FL(y, z) = 0, k = 1, . . . , N. (2.13)

The load-dependent fundamental states do not contribute to the stress resultants. In addi-

tion, the stress moments of the stress residuals must be zero, i.e.,

Lsσ(x, y, z) = 0.

An important benefit of the stress and strain distributions (2.11) is that they can capture

all components of stress and strain. Typically, beam theories retain only a few components

of the stress and strain and assume that the remaining components are negligible. These

neglected components can sometimes be determined using a post-processing integration of

the equilibrium equations through the thickness. For composite materials, however, it can

be important to retain all components of stress and strain, since singularities can arise at

ply interfaces and both strength and stiffness vary significantly between different material

directions [Pagano and Pipes, 1971].


2.2.4 The constitutive relationship

With these definitions, it is now possible to derive the relationship between the stress resul-

tants and the strain moments. The starting point for the derivation is the expression for the

strain field (2.11b). Using the moment operator Ls, the strain moments of Equation (2.11b)

become,

e(x) =6∑

k=1

sk(x)Lsε(k)F (y, z) +

N∑k=1

Pk(x)Lsε(k)FL(y, z) + Lsε(x, y, z). (2.14)

Note that the strain moments have contributions from all fundamental states and the strain

residuals.

Next, I introduce a square flexibility matrix E whose kth column contains the strain

moments from the kth primary fundamental state. The components of the matrix E are:

E∗k = Lsε(k)F (y, z), k = 1, . . . , 6, (2.15)

where E∗k is the kth column of the matrix E. Note that the matrix E is constant for a given

beam construction and is independent of x.

The contributions to the strain moments from the primary fundamental states are the

product of the matrix E and the primary fundamental state magnitudes s(x). Rearranging

the strain moment relationship (2.14) and using the flexibility matrix E yields

Es(x) = e(x)−N∑k=1

Pk(x)Lsε(k)FL(y, z)− Lsε(x, y, z). (2.16)

The stiffness form of the constitutive relationship can be found by inverting the matrix of

strain moments D = E−1, to obtain

s(x) = D

(e(x)−

N∑k=1

Pk(x)Lsε(k)FL(y, z)− Lsε(x, y, z)

). (2.17)

For a section composed of a single isotropic material the relationship between stress and

strain moments simplifies to

D = diag E,E,E,G,G,G .

Equation (2.17) is exact in the sense that the stress moments can be determined exactly if

the strain moments, load-dependent strain moments and strain residuals ε(x, y, z) are known.

Unfortunately, evaluating the strain residuals ε(x, y, z) requires a full three-dimensional so-

lution of the equations of elasticity.


At this point, an assumption must be made about the contribution to the strain moments

from the term Lsε. Since three-dimensional solutions are typically not available, I assume

that the contribution from term Lsε is small and can thus be neglected. This assumption

introduces an error in the predicted strain moments, and as a result, also introduces an

error in the predicted stress resultants. Typically, the magnitude of Lsε is highest near the

ends of the beam where the solution must adjust to satisfy the end conditions. In situations

where these disturbed regions require precise modeling, a beam theory is not appropriate.

However, at a sufficient distance from the ends of the beam, the strain representation (2.11b)

is accurate and thus Lsε should be small.

2.2.5 The stress and strain moment corrections

Next, a relationship between strain moments and the normalized displacement moments is

required. Initially, I limit the analysis to conditions where no external loads are applied to

the beam. Starting from the stiffness form of the constitutive equations (2.17), and assuming

that the strain residual moments are negligible Lsε = 0, the stress moments may be expressed

in terms of the normalized displacement moments u0(x) and the moments of the warping

strain e(x) using Equation (2.6),

s(x) = D (ALεu0(x) + e(x)) . (2.18)

To proceed, an expression for e(x) must be obtained. Following the arguments presented

by Cowper [1966], this term should be linearly dependent on the magnitudes of the primary

fundamental states in regions sufficiently far removed from end effects or rapidly varying

loads. This dependence can be written as

e(x) = Es(x) + er, (2.19)

where E is a flexibility matrix defined below. Here er, is a warping residual term that

accounts for the deviation of the warping moment in disturbed regions of the beam, called

the strain correction error.

Using the operator L from Equation (2.7), the matrix E can be written as

E∗k = Lu(k)F (y, z), k = 1, . . . , 6, (2.20)

where u(k)F (y, z) is determined from the residual displacement of the kth primary fundamental

state. Note that due to the nature of the operator L, the matrix E only has entries in the

last three rows. All other entries in E are zero.


An expression for the stress resultants in terms of the normalized displacement moments

can be obtained by using the simplified form of the constitutive relationship (2.18), and the

moments of the strain due to warping (2.19), yielding

s(x) = (E− E)−1ALεu0(x) + (E− E)−1er. (2.21)

In the remainder of this section I assume that the strain correction error is negligible, i.e.,

er = 0.

In order to isolate the effect of the terms E, the strain moment correction matrix may

be introduced as follows:

Cs = (I− ED)−1, (2.22)

such that Equation (2.21), with er = 0, simplifies to

s(x) = DCsALεu0(x).

Here, the strain moment correction matrix (2.22) provides a correction to the strain mo-

ments predicted from the average displacements that accounts for e(x). Note that the strain

moment correction matrix Cs has a specific structure. The first three rows of Cs are always

equal to the identity matrix, while the last three rows may contain non-zeros in any location

due to the definition of the matrix E.

A stress moment correction matrix may also be defined as follows:

Ks = (I−DE)−1, (2.23)

such that Equation (2.21), with er = 0, simplifies to

s(x) = KsDALεu0(x).

The stress moment correction matrix (2.23) provides a correction to the stress moments that

accounts for e(x). In general, the stress moment correction matrix Ks is fully populated.

In the case of a doubly symmetric, isotropic section, the stress and strain corrections

matrices are diagonal and equal. In this case, Cs and Ks take the form

Ks = Cs = diag1, 1, 1, kt, kxz, kxy,

where kt = J/(Iy + Iz) is the strain correction associated with torsion, and J is the torsional

rigidity of the section. The shear strain correction factors kxz and kxy are identical to those


obtained by Cowper [1966] and Mason and Herrmann [1968],

kxz =2(1 + ν)Iz

ν

2(Iy − Iz)−

A

Iz

∫Ω

z2y2 + zχz dΩ

kxy =2(1 + ν)Iy

ν

2(Iz − Iy)−

A

Iy

∫Ω

z2y2 + yχy dΩ

where χz and χy are classical Saint–Venant flexure functions [Love, 1920].

2.2.6 The load-dependent corrections

The constitutive relationship (2.21) derived above explicitly excluded the effect of externally

applied loads. At this point, I derive load-dependent corrections that account for the effect

of external loads. Again, the starting point is the flexibility form of the constitutive equa-

tions (2.16). Neglecting the moments of the strain residuals, Lsε = 0, results in the following

expression for the strain moments:

e(x) = Es(x) +N∑k=1

Pk(x)Lsε(k)FL(y, z). (2.24)

The next step is to obtain an expression for the strain moments e(x) as a function of the

normalized displacement moments u0(x). The externally applied loads produce additional

moments of the warping strain. In an analogous manner to the primary fundamental state

contributions, I assume that these moments of the warping strain are predicted by the load-

dependent fundamental states and are proportional to the applied load. These assumptions

result in the following expression:

e(x) = ALεu0(x) + Es(x) +N∑k=1

Pk(x)Lu(k)FL(y, z) + er. (2.25)

Here, u(k)FL(y, z) denotes the warping function associated with the kth load-dependent funda-

mental state and er, is the strain correction error.

Again, assuming that er = 0, the flexibility form of the constitutive equations (2.24) and

the strain moment expression (2.25) can now be combined into a constitutive relationship

that takes the following form:

s(x) = (E− E)−1ALεu0(x) +N∑k=1

Pk(x)s(k)FL, (2.26)


where the load-dependent stress moment corrections s(k)FL are defined as

s(k)FL = (E− E)−1

(Lu(k)

FL(y, z)− Lsε(k)FL(y, z)

). (2.27)

In a similar fashion, it can be shown that the strain moments take the modified form

e(x) = CsALεu0 +N∑k=1

Pk(x)e(k)FL, (2.28)

where the load-dependent strain moment corrections e(k)FL are defined as

e(k)FL = Cs

(Lu(k)

FL(y, z)− Lsε(k)FL(y, z)

)+ Lsε(k)

FL(y, z). (2.29)

The load-dependent stress moment corrections (2.27) and the load-dependent strain mo-

ment corrections (2.29) take into account the change in the relationship between the stress

and strain moments and the normalized displacement moments as a result of externally ap-

plied loads. The externally applied loads do not directly produce stress moments; rather,

these loads produce strain moments that must be taken into account in the constitutive

relationship (2.26). The main assumptions required for the derivation of the constitutive

expression are that the moments of the strain residuals, Lsε, and the strain moment correc-

tion, er, can be neglected. These assumptions are examined below in the numerical results

section.

2.2.7 The asymmetry of the constitutive relationship

In general, the homogenized stiffness matrix D, and the matrix product DCsA are not sym-

metric. This is not a classical result and deserves attention. Linear constitutive relationships

between pointwise stress and pointwise strain expressed in the form of Equation (2.1) are

symmetric due to the existence of the strain energy density. However, the homogenized

stiffness matrix D that relates the stress resultants to the strain moments cannot be derived

from a strain energy density, since D relates integrated quantities. The integral of the point-

wise strain energy density across the section cannot be related directly to the product of the

integrals of stress and strain. As a result, D is not guaranteed to be symmetric. The matrix

product DCsA that relates the normalized displacement moments to the stress resultants

is not symmetric based on the same argument. Therefore, symmetry of the constitutive

relationship cannot be assumed within the context of a finite-element implementation of the

present beam theory.


2.3 A finite-element method for the fundamental states

The fundamental states play an important role within the beam theory presented in Sec-

tion 2.2. In principle, full three-dimensional solutions for each of the fundamental states are

required before any analysis can be performed. It is possible to derive some exact solutions to

the fundamental states. However, these exact solutions can only be obtained for a small set

of geometries and beam constructions of interest. In order to solve more general problems,

it is necessary to develop a finite-element approach to determine the fundamental states for

cross-sections of arbitrary geometry and construction.

Conventional three-dimensional finite-elements can be used to obtain the fundamental

state solutions. However, this approach is computationally expensive due to the large, three-

dimensional mesh requirements. Instead, I have developed a technique to obtain the funda-

mental states that only requires computations in the plane of the section, eliminating the

need to discretize the axial direction. This approach is possible due to the fact that the

fundamental states are far-field solutions.

In developing the following finite-element method, I follow the work of Pipes and Pagano

[1970], who used a semi-inverse approach to obtain the stress and strain distributions in a

long beam subject to an axial load. I modify the form of the assumed displacement field

proposed by Pipes and Pagano [1970], but retain the terms that account for the effects of

axial force, bending, shear, and torsion. El Fatmi and Zenzri [2002] developed a similar

technique to obtain the Saint–Venant and Almansi–Michell solutions based on the work of

Ladeveze and Simmonds [1998]. Dong et al. [2001] also developed a finite-element solution

technique for the Saint–Venant problem based on the work of Iesan [1986a].

In the following section, all variables refer to a single fundamental state calculation. Rela-

tionships with the beam theory are described explicitly in Section 2.3.2. In this finite-element

approach, I develop a displacement-based solution to the three-dimensional equations of elas-

ticity based on the following expansion of the displacement field in the axial direction:

u(x, y, z) =M∑k=1

xk

k!

(c

(k)1 + c

(k)2 z + c

(k)3 y)

+xk−1

(k − 1)!U (k)(y, z)

,

v(x, y, z) =M∑k=1

xk

k!

(c

(k)6 − c(k)

4 z − c(k)3

x

k + 1

)+

xk−1

(k − 1)!V (k)(y, z)

,

w(x, y, z) =M∑k=1

xk

k!

(c

(k)5 + c

(k)4 y − c(k)

2

x

k + 1

)+

xk−1

(k − 1)!W (k)(y, z)

,

(2.30)

where the displacements U (k)(y, z), V (k)(y, z) and W (k)(y, z) are sometimes collected in the


vector u(k)(y, z) =[U (k) V (k) W (k)

]T, and are only functions of y and z. The terms c

(k)1

through c(k)6 are constant across the section, and are called the section invariants. It is

convenient to collect c(k)1 through c

(k)6 into a vector denoted c(k) =

[c

(k)1 . . . c

(k)6

]T. The

number of terms M , retained in the expansion is discussed in more detail below. Pipes and

Pagano [1970] used a similar form of Equation (2.30) with M = 1 to determine the stresses

in the vicinity of the free edge of a laminated composite beam subjected to an axial force.

As demonstrated below, the displacement field above can also be used to predict the stress

and strain fields due to bending, torsion, shear, and applied loads.

When M > 1, the representation of the displacement field (2.30) is not unique. The

invariants c(k)1 through c

(k)6 define displacements that can be represented by U (k+1), V (k+1)

and W (k+1). Furthermore, displacement boundary conditions must be imposed on the dis-

placement field (2.30) to remove rigid body translation and rotation modes. In order to

handle both of these issues, the following constraint is imposed:

L0u(k)(y, z) = 0, k = 1, . . . ,M, (2.31)

where L0 is the operator defined in (2.3). This constraint removes the rigid body translation

and rotation modes for k = 1, and ensures that the displacements are uniquely defined for

k > 1. A different method for imposing the boundary conditions could be applied, but

Equation (2.31) simplifies later results in relation to the beam theory.

The strain produced by the displacement field (2.30) is most clearly expressed in the

form,

ε(x, y, z) =M∑k=1

xk−1

(k − 1)!ε(k)(y, z), (2.32)

where ε(k)(y, z) is a strain distribution in the y-z plane. In Equation (2.32), the coefficient

ε(k) is given by,

ε(k)(y, z) =

ε(k)x

ε(k)y

ε(k)z

γ(k)yz

γ(k)xz

γ(k)xy

=

c(k)1 + c

(k)2 z + c

(k)3 y + U (k+1)

V(k),y

W(k),z

V(k),z +W

(k),y

c(k)5 + c

(k)4 y + U

(k),z +W (k+1)

c(k)6 − c(k)

4 z + U(k),y + V (k+1)

, k = 1, . . . ,M (2.33)

with U (M+1) = V (M+1) = W (M+1) = 0.

From the expression for the strain (2.32), it is clear that the stresses in the beam take


the form

σ(x, y, z) =M∑k=1

xk−1

(k − 1)!σ(k)(y, z). (2.34)

Using this polynomial expansion for the stresses, the three-dimensional equilibrium equations

areσ(k)xy,y + σ(k)

xz,z + σ(k+1)x = 0,

σ(k)y,y + σ(k)

yz,z + σ(k+1)xy = 0,

σ(k)yz,y + σ(k)

z,z + σ(k+1)xz = 0,

k = 1, . . . ,M (2.35)

with σ(M+1) = 0. These are the same equations used by Love [1920] for the solution of a

tip-loaded cantilever, and a beam subject to gravity load. Here, the next highest-order terms

in the expansion appear as body forces for the current equilibrium equations. For the kth

coefficient, the body force is equivalent to

b(k) =

σ

(k+1)x

σ(k+1)xy

σ(k+1)xz

.Using the expressions for the strain (2.33) in conjunction with the constitutive relation-

ship (2.1) and the equilibrium equations (2.35) results in 3M partial differential equations

for the displacements u(k). The next task is to determine equations that can be used to

determine the values of c(k).

At this point, it is necessary to use the property that the fundamental states are statically

determinate. As a result, the moment equilibrium equations (2.9) can be integrated to obtain

the stress moment distribution in the beam. Furthermore, it is necessary to limit the load-

dependent fundamental states to loads that are polynomials in the x direction. (Note that

this restriction applies to the integrated pressure loads (2.10), but does not apply to the

distribution of the tractions across the section.) With this additional assumption, it is

always possible to obtain a solution for the stress moments in the form of a polynomial,

s(x) =M∑k=1

xk−1

(k − 1)!s(k)c , (2.36)

where s(k)c is the kth coefficient in the polynomial. Clearly, the value of M must be chosen such

that M − 1 is equal to the degree of the polynomial stress-moment distribution (2.36). The

primary fundamental states corresponding to axial force, torsion and bending moments can

be determined with M = 1. The primary fundamental states corresponding to shear require

a solution with M = 2 corresponding to a linearly varying bending moment and constant


shear. The load-dependent fundamental state corresponding to a distributed surface loads

requires M = 3, with a quadratically varying bending moment and linearly varying shear.

The following additional set of constraints must be imposed to ensure that the moments of

the stress expansion (2.34) to match the coefficients of the stress moment polynomial (2.36):

Lsσ(k)(y, z) = s(k)c , k = 1, . . . ,M. (2.37)

These constraints represent an additional 6M equations that are used to determine c(k).

To summarize, there are 3M , u(k)(y, z) coefficients defined in the y-z plane, and an

additional 6M constants c(k) that are required in the displacement field (2.30). These vari-

ables can be determined from the 6M moment constraints (2.37) and the 3M equilibrium

equations (2.35) used in conjunction with the strain expressions (2.33) and the constitutive

relationship (2.1).

Note that this system of equations can be solved in a sequential fashion. The coefficients

of the highest order k = M , u(M), and c(M), are independent of the lower order coefficients

k < M . The k = M terms couple with the next terms, k = M − 1, u(M−1), and c(M−1),

through the equivalent body-force terms in the equilibrium equations (2.35). Thus the

k = M − 1 order terms may be determined once the k = M order terms are known. This

sequential process can continue until all the coefficients, u(k), and c(k), for k = 1, . . . ,M , have

been determined. This same solution sequence was employed by Love [1920] for isotropic

beams.

2.3.1 Finite-element implementation

The above set of equilibrium equations admit a straightforward, displacement-based finite-

element discretization. The implementation presented here shares many similarities with the

approach of Dong et al. [2001]. To discretize the system of equations discussed above, I em-

ploy conventional isoparametric displacement-based elements with bi-cubic Lagrangian shape

functions in the plane for the 3M displacement field components u(k)(y, z), k = 1, . . . ,M .

I write the nodal displacements of u(k)(y, z) in the vector d(k). It can be shown that the

constraints on the stress moments (2.37) arise naturally using the principle of stationary

total potential energy. The unconventional displacement boundary conditions (2.31) are im-

posed by adding Lagrange multipliers and using a Gauss quadrature approximation of the

constraint (2.31). The discrete form of the displacement constraint is written

L0d(k) = 0,


where L0 is the discrete analogue of L0. The Lagrange multipliers associated with the

displacement constraints are denoted as λ(k).

The discrete approximation of the kth coefficient of the strain expansion (2.33) is written

as

ε(k) = Bd(k)e + Bcc

(k) + Bud(k+1)e ,

where B, Bc and Bu take the kth nodal displacements, kth invariants and (k+ 1)th displace-

ment and produce the pointwise strain. Here, the subscript e has been used to denote the

element displacements from the vector d(k). The matrices B and Bu are defined as follows:

B =

0 0 0 . . .

0 N1,y 0 . . .

0 0 N1,z . . .

0 N1,z N1,y . . .

N1,z 0 0 . . .

N1,y 0 0 . . .

, Bu =

N1 0 0 . . .

0 0 0 . . .

0 0 0 . . .

0 0 0 . . .

0 0 N1 . . .

0 N1 0 . . .

,

where Ni are the shape functions, and the comma notation has been used to denote differ-

entiation. The pattern in the matrices B and Bu, repeats itself for each node. The matrix

Bc is given by

Bc =

1 z y 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 y 1 0

0 0 0 −z 0 1

.

For convenience, the element matrices may be written in a block format as follows:Kedd

Kecd Ke

cc

Keud Ke

uc Keuu

=

∫Ωe

BTCB

BTc CB BT

c CBc

BTuCB BT

uCBc BTuCBu

dΩe, (2.38)

where Ωe is the element domain and the element matrices are denoted with a superscript e.

The superscript e is omitted for the assembled form of the matrix.

The assembled finite-element equations are:Kdd KT

cd LT0

Kcd Kcc 0

L0 0 0

d(k)

c(k)

λ(k)

=

f (k)

f(k)c

0

. (2.39)


Care must be exercised when solving equation (2.39), since the matrix Kdd is singular. This

is due to the fact that no conventional Dirichlet boundary conditions are applied to Kdd.

However, the final row of the system of equations (2.39) imposes a constraint that removes

this singularity.

The two terms on the right hand side of Equation (2.39) are

f (k) = f (k)s + f

(k)b −KT

udd(k+1),

f (k)c = s(k)

c −KTucd

(k+1),(2.40)

where terms with superscripts greater than M are zero. The term fs is the surface traction

contribution to the right hand side and fc is the right hand side for the invariants. The force

vector, f(k)b , represents a body force associated with the (k+ 1)th fundamental state, defined

as follows:

f(k)b = Kudd

(k+1) + Kucc(k+1) + Kuud

(k+2).

Note that the left hand side of Equation (2.39) is the same for each coefficient k. Therefore,

only the right hand sides (2.40) needs to be recomputed for each subsequent solution.

2.3.2 Relation to beam theory

In this section, I outline the connection between the finite-element approach described above

and the proposed beam theory.

The computations outlined above are performed for each fundamental state. First, the

polynomial stress resultant coefficients from Equation (2.36) are determined. These polyno-

mials are summarized for each of the fundamental states in Figure 2.3. Next, the unknowns

d(k) and c(k), k = 1 . . .M , are determined using Equation (2.39). The fundamental state

stress and strain solutions are the lowest-order terms of the polynomial expressions for the

stress in Equations (2.34) and (2.32), respectively. Therefore, the fundamental states are

σ(1)(y,z) and ε(1)(y, z) in the y-z plane.

With this definition, the strain moments of the fundamental state can be computed using

e = Lsε(1)(y, z), (2.41)

where Ls is a discrete analogue of the operator Ls computed using Gaussian quadrature. The

strain moments are required to compute the flexibility matrix E (2.15) and for components

of the stress and strain moment corrections in Equations (2.27) and (2.29), respectively.

Another key quantity required for the beam theory is the axial warping u(x, y, z). The

x-independent component of axial warping is precisely U (1)(y, z) due to the imposition of


Property Value Property Value

E1 164 000 G12, G13 21 000

E2, E3 8300 G23 12 000

ν12, ν13 0.34 ν23 0.21

Table 2.1: The representative orthotropic stiffness properties used in the finite-element calcula-

tions. The relative stiffnesses are chosen to be representative of a graphite-epoxy composite system.

the displacement moment constraint (2.31). The moments of the warping strain can be

evaluated using:

e = LU (1)(y, z), (2.42)

where L is the discrete analogue of L and is computed using Gaussian quadrature. The

terms e are required for computing the flexibility matrix e (2.20), and the stress and strain

moment correction matrices Ks (2.23) and Cs (2.22), respectively.

2.4 Comparison with three-dimensional results

In this section, I present a comparison between three-dimensional finite-element calculations

and results obtained from the beam theory using the finite-element technique presented in

Section 2.3. This study uses the layered composite angle-section shown in Figure 2.4, with

sectional dimensions a = 3/2, b = 3/4, r = 1/2, and overall beam length L = 50. The ply

angles used for this case are θ = [45o,−35o, 35o,−45o], which is a balanced, anti-symmetric

laminate. The tip of the beam at x = L, is loaded with axial, bending, torque and shear

loads, s = [10,−625, 1250, 50,−25, 12.5], and a constant traction is applied to the beam such

that Pz = 1, Py = 0. The traction is distributed on the outer surface of the section and is

only applied in the z direction with tz = 2/(π(r+b)+4a) and ty = 0, as shown in Figure 2.4.

There is also a non-zero Px torque due to the distribution of the traction on the section.

The material properties for the beam are listed in Table 2.1. The relative magnitudes of

the stiffnesses properties are chosen to be representative of a high modulus graphite-epoxy

system. Note that the dimensions of the beam are selected to facilitate three-dimensional

modeling using finite-elements and are not representative of a physical beam. This test case

should be viewed as a convenient model for comparison purposes.

To test different aspects of the present beam theory, I impose two different sets of bound-


Cr

ba

a

Pz = 1

θ1θ2θ3θ4

Figure 2.4: The angle-section geometry. The centroid of the section is marked with a C.

ary conditions on the same finite-element model. These two sets of boundary conditions

result in two separate finite-element problems, denoted B1 and B2. The boundary condi-

tions for case B1 are statically determinate. All displacement components at the beam root,

x = 0, are completely fixed, while the displacement components at the tip, x = L, remain

free. The boundary conditions for case B2 are statically indeterminate. In this case, the

beam root is completely fixed, while at the tip only the axial displacement u = 0 is fixed.

For case B2, the axial force and bending moments at the tip are not applied.

To model the beam using three-dimensional finite-elements, I use a mesh with 289×97×25

nodes, where these three dimensions are the x-direction, the direction along the contour of

the section, and through the thickness, respectively. This results in a problem with just over

2.102 million degrees of freedom. The finite-element model consists of 96× 32× 8 tri-cubic

elements with two elements through the thickness of each ply. This large high-order finite-

element model is employed to accurately model the through-section stresses and limit the

effect of discretization error. Such a large high-order finite-element problem must be solved

by a specialized, parallel, finite-element code. For these problems I use the Toolkit for the

Analysis of Composite Structures (TACS), a parallel finite-element code that is described in

Chapter 3. I solve the three-dimensional beam problems using 48 processors. The solution

time for each case is approximately 10 minutes of wall time corresponding to 8 hours of CPU

time.

To model the beam described above using the present beam theory, I employ a funda-

mental state analysis with a sectional nodal mesh of 97 × 25 nodes along the contour of


the section and through the thickness, with a 32 × 8 bi-cubic element mesh. This problem

contains 7 287 degrees of freedom for the section, including nodal degrees of freedom, in-

variants, and the Lagrange multipliers. For the beam analysis, I use 96, displacement-based,

Timoshenko-type cubic elements along the length of the beam. These elements have been

modified to use load-dependent strain and stress moment corrections and to accept the non-

symmetric stiffness relationship. All beam theory computations, including the determination

of the fundamental states and solution of the beam problem, take less than 15 seconds on

a desktop computer with a single processor. This is a vast difference in computation effort:

the full three-dimensional problem requires approximately 1920 times more computational

time compared to the beam theory calculations.

There are three main objectives in this study:

1. To assess the accuracy of the stress and strain reconstruction obtained using the beam

theory,

2. To assess the accuracy of the stress and strain moments obtained from the theory,

3. To assess the assumption that the strain correction error, er, and the moments of the

residual strain, Lsε may be neglected.

In the remainder of this section, I address each of these three objectives in turn.

x

∆SE

rel

0 10 20 30 40 5010

15

1013

1011

109

107

105

103

101

(a) Case B1

x

∆SE

rel

0 10 20 30 40 5010

6

105

104

103

102

101

100

(b) Case B2

Figure 2.5: ∆SErel as a function of the distance along the beam for the cases B1 and B2.


2.4.1 Stress and strain accuracy

To address the first objective listed above, I examine the difference between the full three-

dimensional finite-element results and the beam theory results for the cases B1 and B2.

Instead of plotting the error in each of the components of stress and strain, I use the following

strain energy error measure in order to concisely quantify the discrepancy in the stress and

strain distributions per unit length of the beam:

∆SErel(x) =

∫Ωε3D · σ3D dΩ∫

Ωε3D · σ3D dΩ

. (2.43)

In the above equation, σ3D and ε3D are the stress and strain solutions from the three-

dimensional finite-element problem, while σ3D and ε3D are the differences between the three-

dimensional solution and the beam theory reconstruction, and therefore represent approxi-

mations of the true stress and strain residuals, σ and ε, in Equation (2.11). The quantity

∆SErel is the strain energy of the difference between the beam theory and the full three-

dimensional solution, per unit length of the beam, normalized by the sectional strain energy

at the current x position. An error in one component of the stress or strain produces a

measurable error in ∆SErel. As a result, ∆SErel shows the accuracy of all components of the

stress and strain reconstruction.

Y

Z

1 0.5 0 0.5

1.5

1

0.5

0

0.5

1

1.5

Y

Z

1 0.5 0 0.5

1.5

1

0.5

0

0.5

1

1.5

0.0017

0.0008

0.0001

0.001

εV

Figure 2.6: A comparison of the through-thickness volumetric strain εV = εx + εy + εz at the

cross-section x = L/2 for the statically indeterminate beam. The beam theory solution is shown

on the left, while the full three-dimensional solution is shown on the right.

Figure 2.5a and Figure 2.5b show the strain energy error measure ∆SErel as a function of

x-location for cases B1 and B2 respectively. For case B1, ∆SErel is largest at the ends of the


beam and decreases rapidly towards the center of the beam. The beam theory and three-

dimensional stress and strain solutions are effectively the same at the center of the beam.

For the case B2, ∆SErel decreases rapidly away from the beam ends, but only falls to between

10−4 and 10−5 over the center portion of the beam. Clearly, the beam theory reconstruction

and the finite-element results do not match as closely as the statically determinate case B1.

The source of this error will be investigated in the following sections.

Figure 2.5 shows that the difference between the boundary conditions in the cases B1 and

B2 has influenced the accuracy of the beam theory reconstruction. However, the stress and

strain reconstruction for case B2 is still quite good. Figure 2.6 shows a comparison of the

volumetric strain for the three-dimensional and beam theory solutions at the middle of the

beam, x = 25, for the case B2. The beam theory and three-dimensional results are nearly

indistinguishable. Furthermore, the beam theory captures the edge effects that occur at the

edge of the beam section between ply interfaces. Depending on the material properties, these

effects may be important in determining the failure properties of the beam.

2.4.2 Strain moment and stress moment accuracy

x

strain

momentrelativedifference

0 10 20 30 40 50

108

107

106

105

104

103

102

101

ex

κz

κy

(a) ex, κz, κy

x

strain


0 10 20 30 40 50

108

107

106

105

104

103

102

101

et

exz

exy

(b) et, κxz, κxy

Figure 2.7: The relative error between the beam theory prediction and the full three-dimensional

calculations for the statically determinate beam.

The second objective of this study is to examine the accuracy of the stress and strain

moments predicted by the theory and the full three-dimensional calculations.


For the statically determinate case B1, the stress moments are known based on the con-

ditions of equilibrium. As a result, the results must be interpreted from the perspective of

known stress moments and unknown strain moments. Figure 2.7b shows the relative error

between the strain moments predicted using the beam theory and the strain moments com-

puted from the three-dimensional finite-element results for case B1. The error of each strain

moment component is normalized by the maximum absolute value of the strain moment

over the length of the beam, such that erel = |e− e3D|/max(|e3D|). The comparisons shown

in Figure 2.7 demonstrate agreement to a relative tolerance of 10−6 between the full three-

dimensional finite-element results and the present beam theory, over the middle portion of

the beam. The differences near the ends of the beam cannot be predicted without recourse

to full three-dimensional calculations. From Equation (2.24), the accuracy of the strain

moments depends on the flexibility matrix E and the load-dependent term Ls ε(k)FL(y, z),

since the stress moments are known. The strong agreement shown in Figure 2.7 verifies the

accuracy of these terms.

x

strain


0 10 20 30 40 50

105

104

103

102

101

100

ex

κz

κy

(a) ex, κz, κy

x

strain


0 10 20 30 40 50

105

104

103

102

101

100

et

exz

exy

(b) et, κxz, κxy

Figure 2.8: The relative error of the strain moments between the beam theory prediction and the

full three-dimensional calculations for the statically indeterminate beam.

The case B2 is statically indeterminate and therefore more challenging than the case B1.

For case B2, the stress and strain moments are unknown and recourse must be made to the

displacement representation to determine both the stress moments and the strain moments.

Figure 2.8 shows the relative error between the strain moments predicted using the beam


x

stressmomentrelativedifference

0 10 20 30 40 5010

4

103

102

101

N

Mz

My

(a) N , Mz, My

x

stressmomentrelativedifference

0 10 20 30 40 50

109

108

107

106

105

104

103

102

T

QzQy

(b) T , Qz, Qy

Figure 2.9: The relative error of the stress moments between the beam theory prediction and the

full three-dimensional calculations for the statically indeterminate beam.

theory and the strain moments computed from the three-dimensional finite-element results

for case B2. This relative error decreases away from the beam ends but reaches a constant

value over the middle portion of the beam. The largest relative error occurs for ex and is

approximately 2%.

Figure 2.9 shows the relative error in the stress moment components over the length of

the beam. In an analogous fashion to the strain moments, the relative error of each stress

moment is normalized by the maximum absolute value of the stress moment over the entire

length of the beam, such that srel = |s − s3D|/max(|s3D|). The relative error in the stress

moment components N , Mz and My is nearly constant over the entire length of the beam,

while the relative error in the remaining torsion and shear components, T , Qz and Qy falls

below 10−5 over the center portion of the beam. Note that the components of the stress

moments in error, N , Mz and My, are associated with the statically indeterminate boundary

condition u = 0 at x = 0 and x = L. The error in these stress moments produces the

discrepancy between the stress and strain reconstruction measured in ∆SErel in Figure 2.5b.

As is demonstrated in the next section, the source of the discrepancy in the strain mo-

ments and stress moments for the case B2 is the error introduced by neglecting both the

strain correction error er and the moments of the strain residual Lsε.


2.4.3 Verification of assumptions

The final objective of this study is to assess the errors introduced by neglecting the shear

correction error, er, and the moments of the strain residual, Lsε. In this section, I use a

combination of beam theory and three-dimensional finite-element results to calculate er and

Lsε approximately. These calculations provide insights into the behavior of these terms and

the errors introduced by neglecting them.

x

relativemoments

ofthestrain

residual

0 10 20 30 40 5010

9

108

107

106

105

104

103

102

101

100

ex

κz

exz

(a) Case A

x

relativemoments

ofthestrain

residual

10 20 30 40

108

107

106

105

104

103

102

101

100

ex

κz

exz

(b) Case B

Figure 2.10: Components of the relative moments of the strain residual Lsε|3D normalized by the

maximum absolute values of the strain moment over the domain. These quantities are computed

from a combination of the finite-element solution and the beam theory using Equation (2.44).

One of the key assumptions in the beam theory presented above is that the moments

of the residual strains, Lsε are small and can be neglected. I test this assumption using

the full three-dimensional finite-element solutions to the cases B1 and B2. To evaluate Lsε,I first use the beam theory to determine the flexibility matrix, E, and the strain moment

contributions from the externally applied loads:

eP = PzLsε(1)FL(y, z),

where Pz = 1 is the magnitude of the applied load and Ls is the discrete analogue of Ls.Based on Equation (2.16), the discrete analogue of Lsε can be determined using

Lsε|3D = Lsε3D − ELsσ3D − eP , (2.44)


where the ε3D and σ3D are the three-dimensional finite-element stress and strain fields re-

spectively.

Figure 2.10 shows the ex, κz and exz components of Lsε|3D normalized by the maximum

absolute value of the strain moment component over the domain for cases B1 and B2. The

remaining components of the moments of the residual strain exhibit similar behavior. At

the ends of the beam the contribution of the moments of the strain residuals are significant,

however their influence decreases rapidly towards the center of the beam. Note that in

Figure 2.10 the oscillations at the center of the beam in the ex and exz components are due

to the finite precision of the finite-element solutions. In these regions, the moments of the

strain residuals are essentially zero.

The strain correction error er from Equation (2.25) represents the difference between the

actual strain moments and the corrected strain moments. I examine an approximation of

the strain correction error obtained from the full three-dimensional finite-element solution

of problem B1. This case verifies the accuracy of the correction flexibility matrix E and

the load-dependent strain correction contribution Lu(k)FL terms. Note that no correction

is required for the strain moments ex, κz and κy, so I examine only the behavior of the

components et, exz and exy.

In order to obtain an approximation for er, I first compute the flexibility correction

matrix E (2.20) and the load-dependent strain correction L U(1)FL using the beam theory.

Rearranging Equation (2.25), results in the following expression:

er|3D = Lsε3D −ALεu03D − ELsσ3D − PzLU (1)FL(y, z), (2.45)

where er|3D is the finite-element approximation of er. Here, u03D is the finite-element ap-

proximation of the normalized strain moments u0(x) and ε3D is the finite-element strain

distribution.

Instead of plotting er|3D directly, I plot the relative values in Figure 2.11, normalized by

the maximum absolute value of the strain moment along the length of the beam. The results

shown in Figure 2.11 are similar in many respects to the moments of the strain residuals

shown in Figure 2.10. The strain correction error is greatest near the ends of the beam

and quickly decays towards the middle of the beam. The largest relative error is in the exy

component of the relative strain correction error. However, all components fall below 10−5

over the center portion of the beam. This suggests that it is reasonable to neglect er at a

sufficient distance from the ends of the beam.

These results demonstrate that it is reasonable to neglect the terms er and Ls ε, but that

these assumptions produce measurable errors in disturbed regions of the beam where the far-


x

relativestrain

correctionerror

0 10 20 30 40 5010

9

108

107

106

105

104

103

102

101

100

et

exz

exy

(a) Case A

x

relativestrain

correctionerror

10 20 30 40

107

106

105

104

103

102

101

100 e

t

exz

exy

(b) Case B

Figure 2.11: Components of the relative strain correction error. The strain correction error is

normalized by the maximum absolute values of the strain moment over the domain.

field assumptions are invalid. In particular, neglecting er and Ls ε, produces discrepancies

in the strain moments and, as a result, the normalized displacement moments, u0(x). For

statically determinate cases, such as case B1, this does not affect either the stress moments

or the stress and strain reconstruction. However, for statically indeterminate cases, such as

case B2, these errors produce measurable errors in the strain moments, stress moments and

the stress and strain reconstruction. This is the cause of the discrepancy between the stress

moments and strain moments shown in Figure 2.8 and Figure 2.9, which in turn produces

errors in stress and strain reconstruction shown in Figure 2.43. These errors, however, are

small and can only be predicted through recourse to computationally expensive, full three-

dimensional finite-element calculations.

2.5 Conclusions

In this chapter I have presented a homogenization-based theory for three-dimensional beams.

The theory uses a kinematic description of the beam based on normalized displacement mo-

ments. The stress and strain distribution in the beam is approximated based on a linear

combination of a hierarchy of axially-invariant stress and strain solutions called the funda-

mental states. The fundamental state solutions are used to construct a constitutive relation-


ship between moments of stress and moments strain. The fundamental states are also used

to determine a strain correction matrix that modifies the strain moments predicted by the

normalized displacement moments. For isotropic beams with symmetric cross-sections, the

present beam theory takes the form of classical Timoshenko beam theory with additional

load-dependent stress and strain corrections. For arbitrary, anisotropic sections, the homog-

enized stiffness matrix becomes fully populated and all components of the stress resultants

are coupled.

In addition, I presented a finite-element based method for the calculation of the fun-

damental state solutions, and verified this approach with three-dimensional finite-element

calculations. I demonstrated excellent agreement between the stress and strain distributions

for statically determinate and statically indeterminate problems, achieving extremely high

accuracy away from the ends of the beam. For statically determinate problems, the relative

error of all strain moment components at the center of the beam was less than 10−6, while for

the statically indeterminate beam, the maximum relative error was 2%. The larger error for

the statically indeterminate case was attributed to the moments of the strain residuals and

the strain correction error. Despite this error, the stress and strain reconstruction remains

sufficiently accurate for engineering purposes. In addition, the finite-element based beam

theory calculations required three orders of magnitude less computational time compared to

three-dimensional finite-element computations. These characteristics make the beam the-

ory an attractive approach for accurate through-thickness stress and strain prediction in

composite beams.

Chapter 3

Parallel finite-element analysis of shell

structures

Thin, stiffened shell structures are frequently used in aerospace applications due to their

high stiffness-to-weight and strength-to-weight ratios. Stiffened structures, such as wings

and fuselages, are primarily designed to carry in-plane loads that produce a more uniform

distribution of stress and strain through the thickness than bending loads, resulting in a more

efficient use of material. As with any slender, compressively-loaded structure, structural

instability is often a critical design case and the sizing and placement of stiffeners is often

dictated by buckling considerations.

In this section, I present methods that may be used to efficiently analyze and design

stiffened shell structures appropriate for aerospace applications. In particular, I present

a higher-order shell element formulation for accurate stress and strain prediction, parallel

solution methods for the analysis of large structural models, and efficient and accurate sensi-

tivity analysis techniques for gradient-based optimization. The material in this chapter has

been developed into a unified computational framework called the Toolkit for the Analysis

of Composite Structures (TACS). TACS is written in C++, with a Python-level interface,

and has been built with multidisciplinary analysis and gradient evaluation in mind.

This chapter is organized as follows: in Section 3.1, I present a higher-order shell element

formulation used for the analysis of thin, composite structures. In Section 3.2, I present an

overview of the domain-decomposition-based parallel finite-element approach used in TACS.

In Section 3.3, I present methods to solve the sparse linear systems that arise from the finite-

element discretization of structural problems. Finally, in Section 3.4, I present efficient design

sensitivity methods for use in structural design optimization.

43

Chapter 3. Parallel finite-element analysis of shell structures 44

3.1 Finite-element analysis of shell structures

There are numerous issues that need to be addressed when developing general purpose shell

elements for use on a wide range of shell problems. These issues range from the method

used to formulate the shell element, to problems with the numerical behavior under certain

analysis conditions. I will address the most significant issues below and describe the methods

used to address them.

There are essentially two main approaches that may be used to formulate shell elements.

The first approach is to derive a shell theory, based on a set of kinematic assumptions about

the behavior of the shell, and then apply a finite-element discretization to the resulting shell

theory. There are several examples of this approach in the literature. Perhaps the most

extensive efforts in this area have been undertaken by Simo and Fox [1989], Simo et al.

[1989, 1990a,b] and Simo [1993]. One of the main difficulties with this approach is obtaining

an element that contains the necessary rigid-body rotation and translation modes [Hansen

and Heppler, 1985]. The second approach, first developed by Ahmad et al. [1970], is to

formulate a continuum-based shell element by reducing the full three-dimensional equations

of elasticity to the mid-surface using shell-theory-like assumptions about the distribution of

the displacements through the thickness. This approach is usually referred to as the degener-

ated solid approach. Bathe and Bolourchi [1980], Parisch [1978] and Hughes and Liu [1981]

also extended the degenerated solid approach to nonlinear geometric and nonlinear material

problems. Buechter and Ramm [1992] performed an extensive comparison of shell-element

formulations and showed that both the degenerated solid and shell-formulation approaches

can lead to a mathematically equivalent formulation if the same modeling assumptions are

used.

Another significant challenge associated with the analysis of thin shells, as well as plates

and beams, is a numerical issue known as locking [Babuska and Suri, 1992]. Shear and

membrane locking may occur when using a conforming low-order displacement-based shell

element that employs some form of shear deformation theory to account for the rotation of

the shell-normal during deformation. These types of shell elements do not allow a state of

pure bending and as a result, over-predict the shear strain energy within the shell [Chapelle

and Bathe, 2003]. This additional shear strain energy makes the element stiffer than it should

be, producing small-than-expected displacements. Membrane locking occurs in shells when

coupling exists between in-plane and shear loads. Like shear locking, membrane locking

produces stiffer-than-expected results, even for extremely fine meshes.

Shear and membrane locking are usually addressed by using one of three approaches:


higher-order displacement-based elements (typically at least fourth-order), reduced or se-

lective integration of the shear strain energy terms, or the use of mixed formulations. For

instance, Hughes et al. [1977, 1978] used selective, reduced-order integration of the shear

strain energy to alleviate shear locking for low-order elements. However, this approach can

suffer due to the introduction of spurious zero-energy modes, especially for higher-order

elements. Heppler and Hansen [1986] developed a locking-free shell element by using fourth-

order Lagrange interpolation functions for thick shells using Reissner–Mindlin kinematic

assumptions.

Various authors have presented mixed shell element formulations based on mixed energy

principles. In these approaches certain stress or strain components, or the entire stress or

strain field are also included in the finite-element discretization. One of the most successful

mixed-interpolation approaches for shell elements is the technique of using mixed interpola-

tion of tensorial components (MITC) originally developed by Dvorkin and Bathe [1984] and

Bathe and Dvorkin [1985]. In the MITC approach, shear and membrane strains are interpo-

lated based on an appropriate lower-order polynomial. The strain interpolation is forced to

match the displacement-based strain values at the strain interpolation points, called tying

points. While the element formulation is based on an a mixed-energy principle, the element

nodal variables remain the classical displacement and rotation variables. This approach can

also be extended to higher-order element formulations [Bathe and Dvorkin, 1986, Bucalem

and Bathe, 1993]. The MITC formulation was put on a rigorous mathematical foundation

by the analysis of Bathe et al. [2000] and Chapelle and Bathe [2003] using the framework of

Brezzi and Fortin [1991].

I have implemented both a displacement-based shell element and an MITC-based shell el-

ement using a degenerated solid approach. However, this formulation can also be interpreted

as a shell theory formulation based on the work of Buechter and Ramm [1992]. Following

Milford and Schnobrich [1986], the formulation uses an explicit integration of the strain en-

ergy through the thickness, enabling the direct use of the classical first-order deformation

theory (FSDT) constitutive relationships. This explicit integration approach, however, in-

troduces a modeling error on the order of the ratio of the thickness to radius of curvature. As

a result, the shell element is not appropriate for thick shells or shells with high-curvature. In

the present implementation, a small angle approximation is used to represent the rotations

and the drilling degree of freedom is included in the element formulation using the approach

outlined by Hughes and Brezzi [1989].


3.1.1 Shell formulation

In this section, I derive the finite-element formulation for a shell using a total Lagrangian

approach based on the degenerated solid technique. The starting point for the derivation is

the principle of virtual work, written here as follows:∫Ω

tr (δεS) dΩ = δW ext, (3.1)

where W ext is the work done by external forces, Ω is the volume of the shell, tr(·) is the

trace operator, S ∈ R3×3 are the components of the second Piola–Kirchhoff stress tensor,

and ε ∈ R3×3 are the components of the Green strain, also referred to as the Green–Lagrange

strain. The constitutive equations, in general, can be written in rate form S′ = C · ε′, where

C ∈ R3×3×3×3 is a fourth-order constitutive tensor that relates the rate of change of strain

ε′ to the rate of change of stress, S′ [Bathe, 1996]. At present, the analysis is limited to

a linear constitutive relationship such that C remains constant, regardless of the stress or

strain fields. As a result, the constitutive relationship is written as follows:

S = C · ε. (3.2)

Note that this is not a matrix-matrix product, but rather an inner product of a fourth and

second order tensors. The Green strain is written as follows:

ε =1

2

(U,x + UT

,x + UT,xU,x

), (3.3)

where U ∈ R3 are the Cartesian displacement components between the initial and deformed

geometry, and x ∈ R3 are the global Cartesian coordinates.

The initial, undeformed geometry of the shell may be described in terms of a reference

surface r(ξ) ∈ R3, the shell thickness h(ξ) ∈ R+ and the unit surface normal vector n(ξ) ∈R3. Here, ξ = [ξ1 ξ2]T is the parametric location on the reference surface. The thickness of

the shell is accounted for with a through-thickness coordinate ζ that is the distance along

the unit normal from the surface, ζ ∈ [−h/2, h/2]. With these definitions, the entire volume

of the shell has the following parametric description:

R(ξ, ζ) = r(ξ) + ζn(ξ), (3.4)

As a result of these definitions, r(ξ) is the mid-surface of the shell. At times it is also

convenient to group the shell parametrization into the vector η = [ξ1 ξ2 ζ]T . The initial and

the deformed geometry of a shell segment are illustrated in Figure 3.1.


x

y

z

r(ξ)

R(ξ, ζ)

n(ξ)

u(ξ)

U(ξ, ζ)

Q(ξ)Tn(ξ)

Initial geometry

Deformed geometry

Figure 3.1: The initial and deformed geometry of a shell segment. This figure illustrates the

geometric parameters required for the shell element formulation.

At this point it is necessary to make a kinematic assumption about the through-thickness

behavior of the displacements in the shell. Here, the displacements are assumed to vary

linearly through the thickness. This kinematic assumption is compatible with either the

Kirchhoff–Love assumption that the surface normal remains normal to the deformed sur-

face, or with the FSDT assumption that the normal remains straight but rotates from the

initial geometry. With either of these kinematic assumptions, the shell displacement field is

expressed as follows:

U(ξ, ζ) = u(ξ) + ζω(ξ), (3.5)

where U(ξ, ζ) is the displacement at any parametric point in the shell and u(ξ) is the

displacement at the mid-surface of the shell. The term ω(ξ) ∈ R3 represents the rate of

change of the displacements through the thickness.

The exact form of ω(ξ) has a significant impact on the shell formulation. Specifying the

rate of change of the displacement in terms of the in-plane displacement quantities, through a

Kirchhoff–Love assumption, results in a Donnel or Koiter-type shell theory. Specifying ω(ξ)

in terms of small-rotation angles using an FSDT or Reissner–Mindlin assumption yields a

Naghdi-type theory, while specifying the values in terms of a rotation matrix yields a large-

rotation theory. The choice of parametrization has a significant impact on the complexity of


the strain expressions. In general, ω(ξ) can be expressed as

ω(ξ) =(Q(ξ)T − I

)n(ξ),

where Q(ξ) ∈ R3×3 is a rotation matrix or an approximation of a rotation matrix, I ∈ R3×3

is the identity and n(ξ) is the unit normal of the undeformed geometry. Here, a small-angle

approximation of the rotation matrix is employed, such that Q(ξ)T = I +φ(ξ)×, where (·)×

is the cross-product operator [Hughes, 2004]. The vector φ(ξ) ∈ R3 are small rotations in

the global Cartesian reference frame:

φ(ξ) =[φx φy φz

]T. (3.6)

As a result of this choice of parametrization, the rate of change of the displacements through

the thickness of the shell is

ω(ξ) = φ(ξ)×n(ξ). (3.7)

Throughout the remainder of this section, the variable ω(ξ) is retained in the formulation.

The next step in the derivation is to use the assumed displacement field (3.5) to arrive

at an expression for the strain in the shell. The approach taken here is to express the

displacements in the global Cartesian reference frame while working with the strains in a

local Cartesian reference frame attached to the shell surface. This technique avoids the use of

non-Cartesian tensors that must be employed when the shell displacements are expressed in a

local non-orthogonal reference frame constructed on the shell surface [Buechter and Ramm,

1992]. The transformation between the global and the shell-attached reference frames is

denoted T(ξ) ∈ R3×3, such that the Green strain (3.3) transforms as follows:

εs(η) = T(ξ)ε(η)T(ξ)T , (3.8)

where εs(η) is the Green strain in the shell-attached coordinate frame. The strain in the

locally-attached reference frame takes the form

εs(η) = ε(0)s (ξ) + ζε(1)

s (ξ) + ζ2ε(2)s (ξ), (3.9)

where quadratic coefficient ε(2)s is neglected in all further formula in order to retain consis-

tency with the FSDT assumptions.

Using the relationship for the through-thickness strain distribution (3.9), the strain energy


expression in the principle of virtual work (3.1) is written∫A

∫ h/2

−h/2tr((δε(0)

s + ζ δε(1)s

) [C ·(ε(0)s + ζε(1)

s

)])|R,η| dζ dA ≈∫

A

tr(δε(0)

s A · ε(0)s + δε(0)

s B · ε(1)s + δε(1)

s B · ε(0)s + δε(1)

s D · ε(1)s

)|R,η|ζ=0 dA = δW ext,

(3.10)

where A is the area of the shell. Note that the determinant of the Jacobian |R,η| varies

through the thickness of the shell. Instead of computing this integral exactly, the determi-

nant value is fixed to the value at the mid-surface such that |R,η| ≈ |R,η|ζ=0. This is a

reasonable assumption for thin shells, with moderate curvature and simplifies the integra-

tion considerably [Milford and Schnobrich, 1986]. The fourth order tensors A, B and D are

defined as follows:

A =

∫ h/2

−h/2C dζ, B =

∫ h/2

−h/2ζC dζ, D =

∫ h/2

−h/2ζ2C dζ. (3.11)

While all components of the constitutive tensor could be integrated through the thickness

of the shell, in practice conditions consistent with the FSDT assumptions are made when

calculating these integrals. In particular, a plane-stress assumption is imposed, such that

the through-thickness stress is zero, and only those constitutive terms that are associated

with the classical FSDT approach are retained: the in-plane normal and shear terms in A,

B and D, and the out-of-plane shear terms in A. In addition, a shear correction factor is

applied to the out-of-plane shear terms in A.

The main task now is to obtain the Green strain in the global reference frame (3.3), and

transform it to the local reference frame using Equation (3.8). In order to evaluate the strain,

the derivative of the displacements in the global reference frame U,x must be expressed in

terms of the shell parametrization variables η. This derivative is expressed as follows:

U,x = U,ηη,x = U,ηR,η(η)−1. (3.12)

This expression for U,x, is a complicated, nonlinear function of the through-thickness coor-

dinate ζ, due to the inverse Jacobian term R,η(η)−1. Belytschko et al. [1989] and Buechter

and Ramm [1992] found that a linearization of Equation (3.12) is sufficient to retain an

accuracy of order h/R in the strain expressions, where h is the shell thickness and R is the

smallest radius of curvature. Linearizing Equation (3.12) about the mid-surface of the shell

gives

U,x ≈ U(0),x + ζU(1)

,x , (3.13)


where U(0),x and U

(1),x are the first and second terms in a Taylor series expansion. These terms

are defined as follows:

U(0),x = U,ηη,x

∣∣ζ=0

= [u,ξ1 u,ξ2 ω]η(0),x ,

U(1),x =

U,ηζη,x + U,ηη,xζ

∣∣ζ=0

= ω,ηη(0),x + [u,ξ1 u,ξ2 ω]η

(0),xζ ,

(3.14)

with η(0),x = R,η(η)−1|ζ=0 and η

(0),xζ = −η(0)

,x n,ηη(0),x . Note that U

(0),x and U

(1),x are independent

of the through-thickness coordinate ζ. With these definitions, the terms in the expansion of

the strain through the thickness (3.9) can be written as follows:

ε(0)s =

1

2T(U(0),x + U(0)

,x

T+ U(0)

,x

TU(0),x

)TT ,

ε(1)s =

1

2T(U(1),x + U(1)

,x

T+ U(1)

,x

TU(0),x + U(0)

,x

TU(1),x

)TT ,

ε(2)s =

1

2TU(1)

,x

TU(1),x TT .

(3.15)

This completes the derivation of the strain and energy expressions required for a displacement-

based shell element. The principle of virtual work (3.10), in conjunction with the strain

expressions (3.14) and (3.15), and the integrated constitutive expressions (3.11) are used to

evaluate the element residuals and, with standard techniques, the geometric and tangent

stiffness matrices [Bathe, 1996, Zienkiewicz et al., 2005]. The assumptions made during the

derivation of the strain expressions are as follows:

1. The displacements vary linearly through the thickness of the shell (3.5)

2. The rate of change of displacements in the through-thickness direction obeys the

Reissner–Mindlin kinematic assumption (3.7)

3. The through-thickness variation of the determinant of the Jacobian |R,η| can be ne-

glected when computing the work integral (3.10)

4. The through-thickness stress is zero resulting in a state of plane-stress in for the inte-

grated constitutive relationship (3.11)

5. The higher-order strain terms ε(2)s from Equation (3.15) can be neglected in the work

integral (3.10)

6. The gradient of the displacement through the thickness can be linearized from Equa-

tion (3.13) with an order h/R error


In the next two subsections, I discuss additional details of two shell-element implementa-

tions: a displacement-based formulation and an MITC formulation. Both shell elements are

quadrilateral and use bi-Lagrangian shape functions where the interpolation varies between

second and fourth order.

Displacement-based formulation

Using bi-Lagrange shape functions of order p, the mid-surface displacements from Equa-

tion (3.5) and the rotation variables (3.6), are expressed in terms of the nodal displacements

and rotations ue ∈ Rne , as follows: [u(ξ)

φ(ξ)

]= N(ξ) ue (3.16)

where ne = 6p2 and N ∈ R6×ne are the shape functions. Next, an expression for the element

residual is obtained by using the displacement interpolation (3.16) in conjunction with the

strain expressions derived above and the method of virtual work (3.10). This element residual

takes the form

δuTe Re(ue) = δW exte , (3.17)

where Re(ue) are the element residuals, and δW exte is the external virtual work done on the

element. The tangent stiffness matrix is obtained by calculating K(ue) = ∂Re/∂ue.

MITC formulation

The MITC shell element uses the same strain expressions for the bending strain, ε(1)s ,

from Equations (3.15) and (3.14), and the same bi-Lagrange displacement interpolation

scheme (3.16) as the displacement-based shell element. The main difference between the

displacement-based element and the MITC element, is that the strain components suscepti-

ble to locking are interpolated using a lower-order scheme. These interpolated strain compo-

nents are then used within the principle of virtual work expression (3.10). Note that in the

MITC formulation, all parts of the virtual work integral are integrated to full order, unlike

shell formulations that employ selective-reduced integration [Hughes et al., 1977, 1978]. To

avoid poor results on highly skewed meshes, the covariant components of the Green strain

in the natural shell coordinates are used in the interpolation, rather than the components

of the strain in the shell-attached local Cartesian frame [Dvorkin and Bathe, 1984]. The

covariant components of the Green strain, ε, are

ε =1

2

(RT,ηU,η + UT

,ηR,η + UT,ηU,η

). (3.18)


Element Nodes ε11 and ε13 ε12 ε22 and ε23

2nd order

ξ2

ξ1

(−1,−1)

(1, 1) ξ2

ξ1

ξ2

ξ1

ξ2

ξ1

3rd order

ξ2

ξ1

(−1,−1)

(1, 1) ξ2

ξ1

δ3 = 1/√3

δ3

δ3

ξ2

ξ1δ3

δ3

ξ2

ξ1

δ3

δ3

4th order

ξ2

ξ1

(−1,−1)

(1, 1)ξ2

ξ1

δ4

δ4

δ4 =√

3/5

ξ2

ξ1δ4

δ4

ξ2

ξ1

δ4

δ4

ξ2

ξ1

Figure 3.2: The interpolation scheme for the second through fourth order MITC shell elements.

Note that the in-plane strains and out-of plane shear strains are interpolations.

The covariant Green strain components can be transformed to the global coordinate system

using the following transformation [Bathe, 1996]:

ε = R−T,η εR−1,η (3.19)

The strain components susceptible to locking are the in-plane normal and shear strains,

and the out-of-plane shear strains. These strain components correspond to ε(0)s in Equa-

tion (3.15). In the MITC formulation, the strain expression ε(0)s is replaced by the interpo-

lated strain ε(0)as and these strain variables are interpolated from the tying points, ξ

(t)k , within

the element such that

ε(0)as (ξ) = T(ξ) R−T,η

∣∣ζ=0

nt∑k=1

N(as)k (ξ)ε(0)(ξ

(t)k )

R−1,η

∣∣ζ=0

T(ξ)T , (3.20)

where ε(0)as is the assumed strain distribution. Here N(as) are the nt shape functions for

the assumed strain distribution. The interpolation schemes for the second through fourth

order elements are summarized in Figure (3.2). Note that different tying points are used for

different components of the strain. The quantity ε(0)(ξ) are the mid-surface values of the


covariant components of the Green strain given by

ε(0) =1

2

(RT,ηU,η + UT

,ηR,η + UT,ηU,η

)∣∣∣∣ζ=0

.

Within the context of the MITC shell element formulation, the virtual work expression

is now∫A

tr(δε(0)

as A · ε(0)as + δε(0)

as B · ε(1)s + δε(1)

s B · ε(0)as + δε(1)

s D · ε(1)s

)|R,η|ζ=0 dAR = δW ext,

(3.21)

where ε(0)as is evaluated from Equation (3.20). Like the displacement-based element (3.17),

the residual of the MITC element, based on the virtual work expression (3.21), takes the

form

δuTe RMITCe (ue) = δW ext

e ,

where RMITCe (ue) are the MITC element residuals and δW ext

e is the external virtual work done

on the element. Note that the vector of element displacements and rotations is identical to the

displacement-based element. Again, the tangent stiffness matrix is obtained by calculating

KMITC(ue) = ∂RMITCe /∂ue.

3.1.2 Shell element tests

Tests of the shell element implementations outlined above are presented in Appendix A. I

have verified the linear shell-element formulations against the standard set of shell tests sug-

gested by MacNeal and Harder [1985]. For the nonlinear shell formulation, I have compared

the buckling results to those obtained by Sobel [1964]. I have also examined the post buck-

ling behavior using the classic snap-through problem presented by Horrigmoe and Bergan

[1978].

3.1.3 The condition numbers of shell problems

It is frequently mentioned in the literature that shell elements produce poorly conditioned

linear systems. However, a systematic assessment of the variation of the condition number

with element order, slenderness ratio and problem size appears to be lacking. In this section,

I briefly present results for the condition of a square plate for increasing slenderness ratio,

element order and decreasing mesh spacing. For a symmetric positive definite matrix, with

the induced `2 norm, the condition number is the ratio of the largest to smallest eigenvalues

of a matrix. In order to obtain an accurate estimate of the condition number it is essential


nodes per edge

conditionnumber

20 40 60 80 10010

5

106

107

108

109

1010

4thorder

3rdorder

2ndorder

L/t = 10

L/t = 100

L/t = 1000

Figure 3.3: A comparison of the condition number of a square plate for different numbers of nodes

per edge, different slenderness ratios and increasing element order.

to obtain an accurate estimate of the minimum eigenvalue, and less important to obtain an

accurate estimate of the maximum eigenvalue.

I determine the condition number of the stiffness matrix by first evaluating the largest

condition number using the power-iteration method [Saad, 1992]. I determine the lowest

eigenvalue using the inverted form of the eigenvalue problem in conjunction with a Lanczos

algorithm with full orthogonalization [Grimes et al., 1994].

Figure 3.3 shows the variation of the condition number for a series of square, clamped

plates with varying numbers of nodes along each edge. The plate is modeled using 2nd, 3rd

and 4th order MITC shell elements described above. The condition number is also shown

for different values of the slenderness ratio, L/t where L is the edge length and t is the

plate thickness. The results show that the condition number varies from 105 to nearly 1010.

Note that realistic slenderness ratios for aerospace structures are typically between 102 and

104. These results demonstrate that the condition number depends most strongly on the

slenderness ratio, followed by element order and number of degrees of freedom.

3.2 Parallel finite-element analysis

Accurate finite-element analysis of aerospace structures often requires the use of extremely

large meshes with millions of nodes, which may be required to ensure that the discretization

error is at acceptable levels. Higher-order finite-element methods are especially important


when accurate stress and strain prediction is required, due to the fact that for displacement-

based finite-element formulations, the stress and strain fields converge at a rate that is

one order lower than the convergence rate of the displacement field [Strang and Fix, 1973].

Large finite-element models require specialized solvers that can accurately and robustly

find a solution, even to very poorly conditioned problems. Finite-element problems with

millions of degrees of freedom can only be solved within a time frame that is practical for

optimization purposes using parallel solution methods. As a result, I only consider parallel

solution techniques.

To perform finite-element analysis and design optimization in a parallel computing envi-

ronment, it is necessary to perform the following three operations in parallel:

1. Assemble the global residual vector and global finite-element matrices,

2. Solve linear systems arising from the finite-element discretization,

3. Evaluate functions of interest, and their gradients.

In order to parallelize these three types of operations, I use an element-based partition of

the finite-element mesh. In this approach, each element is assigned to a unique processor.

The variables and nodes required by the elements on a processor are stored locally and as

a result, nodes and variables on domain boundaries are duplicated on multiple processors.

This duplication allows the element residual and element matrix computations to take place

concurrently without communication. Furthermore, functions of interest that require cal-

culations for each element in the mesh can be calculated in parallel. The disadvantage of

the element-based partition approach is that residual components must be communicated

to other processors during the final global residual assembly. However, typically the ratio of

non-local to local residual components is small, and the parallel performance is excellent.

I have carefully implemented the three types of operations outlined above in parallel. Of

these three operations, the parallel solution of linear systems is by far the most complex and

is the exclusive topic of Section 3.3. The evaluation of the gradients of functions of interest

in parallel is discussed in Section 3.4.

3.3 Parallel solution methods for sparse linear systems

Sparse linear systems of the form,

Ax = b, (3.22)


arise in the analysis of linear and nonlinear static finite-element problems 1. Furthermore, ac-

celeration techniques used in eigenvalue solvers for modal and buckling analysis, often require

the solution of sparse linear systems of the form (3.22) [Saad, 1992]. Therefore, depending on

the context, the matrix A ∈ Rn×n may be the stiffness matrix, tangent stiffness matrix, an

approximate Jacobian or some linear combination of matrices as required by an eigenvalue

solver or nonlinear solution algorithm. The vectors x, b ∈ Rn are the solution vector and

right hand side, respectively. The sparse matrices that are derived from the finite-element

analysis of structural problems are typically symmetric due to underlying energy principles.

However, follower forces, such as fluid pressure loads, are non-conservative and generate non-

symmetric terms in the structural Jacobian [Elishakoff, 2005]. Since aerostructural problems

are a primary focus of this thesis, I have exclusively implemented techniques that do not

assume that the matrix A is symmetric.

Solution methods for sparse linear systems may be divided into two categories: iterative

methods and direct methods. Direct methods use different variants of Gaussian elimination

to factor the matrix A into a lower-triangular matrix L and an upper-triangular matrix

U such that A = LU. Different variants of Gaussian elimination that employ pivoting

strategies for numerical stability use permutation matrices that modify this simple formula.

Iterative methods find an approximate solution to the linear system (3.22) using an iterative

algorithm [Saad, 2003]. Iterative methods have the advantage that they may satisfy a given

solution criterion in less computational time and typically require significantly less memory

than direct methods. However, direct methods have the advantage that they are robust and

can find a solution to Equation (3.22), even for very poorly conditioned linear systems.

In this work, I have investigated both iterative Krylov subspace methods and direct so-

lution methods for solving the linear system of equations (3.22). In order to unify these

different solution techniques, I always use an iterative Krylov subspace method, with either

a conventional preconditioning method, or a direct solver as a preconditioner. Direct solution

methods are not preconditioners in the classical sense since they are expensive to compute

and expensive to apply to a right-hand-side when compared to other preconditioning tech-

niques. However, Krylov subspace methods are frequently used for iterative refinement of

solutions obtained from direct solvers [Li and Demmel, 2003]. In these applications, the

direct solver is a preconditioner. This unified approach has two main advantages: First, it is

transparent to any multidisciplinary solver that may itself use an iterative technique. Sec-

ond, frequently it is necessary to solve a series of linear systems that are slightly perturbed

1Note that in this section I employ the standard notation for linear systems used in the literature whereA is the matrix, x are unknowns, and b is the right-hand-side.


from equation (3.22). In this case, it is often more efficient to solve the perturbed linear

system using a Krylov subspace method without performing an additional factorization.

The remainder of this section is structured as follows: first, I briefly outline relevant

work on direct solution algorithms. Next, I describe the implementation of two parallel

preconditioning methods: the additive Schwarz and approximate Schur preconditioners, and

the implementation of a direct solver: the direct Schur method. In the last section, I examine

the parallel performance of these methods on two large finite-element problems.

Direct solvers

Relatively recently, there has been a renewed interest in direct solution methods for sparse

linear systems, with a focus on parallel implementations. Amestoy et al. [2000], developed

MUMPS (MUltifrontal Massively Parallel Solver), that uses a multi-frontal matrix factor-

ization approach with a right-looking pivot strategy to ensure numerical stability. Once the

frontal matrices reach a certain size, they are factored across groups of processors using a

row-oriented storage format. Finally, the root frontal matrix is factored in parallel using

a 2D block-cyclic dense factorization algorithm. The direct solver SPOOLES (SParse Ob-

ject Oriented Linear Equations Solver) developed by Ashcraft and Grimes [1999] also uses a

multi-frontal factorization approach. SPOOLES achieves parallelism by performing a recur-

sive nested dissection algorithm on the original matrix graph, determining the parts of the

matrix that can be factored independently.

Demmel et al. [1999], Li and Demmel [2003] developed SuperLU and SuperLU DIST,

sequential and parallel distributed direct solvers, respectively, for both symmetric and non-

symmetric linear systems. SuperLU DIST uses a block-cyclic format where the size of the

blocks are determined based on a super-node analysis of the initial matrix. SuperLU DIST

does not perform pivoting for stability, but instead uses various pre-processing operations

to enhance the numerical properties of the matrix. Parallelism during the factorization is

achieved by distributing the trailing-matrix update and performing the update in a process

queue.

3.3.1 Additive Schwarz and approximate Schur preconditioners

The additive Schwarz and approximate Schur preconditioners both employ a row-oriented

matrix storage scheme. In this scheme, each processor is assigned a contiguous segment of

the rows from the linear system (3.22), while the unknowns are assigned to one of three

groups:


B1 E1

F1 C1 F12 F13 F14

B2 E2

F21 F2 C2 F23 F24

B3 E3

F31 F32 F3 C3 F34

B4 E4

F41 F42 F43 F4 C4

Figure 3.4: The domain decomposition and matrix for the approximate Schur preconditioner for

a four processor case. Each colour represents a different processor. The dashed line indicates the

boundary between the internal unknowns and the internal interface unknowns.

1. Internal unknowns that are only coupled to variables on processor i,

2. Internal interface unknowns assigned to processor i, that are coupled to variables on a

different processor,

3. External interface unknowns that are assigned to a different processor but are coupled

to processor i.

As a result of these definitions, all the unknowns associated with elements on the domain

boundary must be included as internal interface unknowns or external interface unknowns.

Furthermore, increasing the order of the finite-elements results in a larger interface problem.

Figure 3.4 shows the domain decomposition, and resulting matrix for a four processor case

with third-order elements.

Using this decomposition of the unknowns, the linear system (3.22) can be written as

follows:

Aixi + Biyi = bi, (3.23)

where Ai ∈ Rni×ni and Bi ∈ Rni×pi are the diagonal and off-diagonal components. The

matrix Bi represents the coupling of the equations between processors. The vectors xi ∈ Rni

and yi ∈ Rpi are the internal unknowns and the external interface unknowns, respectively.


The unknowns and the right hand side are distributed in the same fashion so that only local

vector components are referenced on the right hand side of Equation (3.23).

Additive Schwarz preconditioner

Given the input vector bi on each processor, block-Jacobi or additive-Schwarz precondition-

ing with no overlap is obtained by computing an approximate solution xi of the equation

Aixi = bi. (3.24)

This approximate solution could be obtained using a Krylov method, but I have found

that this approach is usually not competitive. Instead, I find an approximate solution of

Equation (3.24), using a single application of a block incomplete-LU factorization with level

of fill p (BILU(p)) [Saad, 2003].

Approximate Schur preconditioner

Saad and Sosonikina [1999] first developed the approximate Schur preconditioner as an alter-

native to additive Schwarz techniques. Later, Hicken and Zingg [2008] proposed a refinement

to the method to reduce the computational cost of the application of the preconditioner. This

refinement has been incorporated into the current implementation.

For the approximate Schur preconditioner, it is necessary to order the internal interface

unknowns last such that the local unknowns may be partitioned as follows: xTi =[zTi yTi

],

where zi are the internal unknowns and yi are the internal interface unknowns. With this

partitioning of the unknowns, the global system of equations (3.23) can be written[Bi Ei

Fi Ci

][zi

yi

]+

[0∑

j 6=i Fijyj

]=

[fi

gi

]. (3.25)

Solving for the internal degrees of freedom on each process yields, zi = B−1i (fi−Eiyi). This

expression can then be used to form a system of equations for all the interface unknowns

Siyi +∑j

Fijyj = gi − FiB−1i fi, (3.26)

where Si = Ci − FiB−1i Ei is the local Schur complement. The left hand side of Equa-

tion (3.26) couples the interface unknowns from all processors and is referred to as the

global Schur complement. Instead of computing the local Schur complement Si exactly, and

assembling and solving solving the global Schur system (3.26) exactly, the approximate Schur

preconditioner obtains, an approximate solution for the interface unknowns yi.


An approximate global Schur complement system can be obtained by considering an

incomplete LU factorization of the local matrix Ai from Equation (3.25):[Bi Ei

Fi Ci

]=

[LBi 0

FiUB−1i LSi

][UBi LB

−1i Ei

0 USi

]+ R, (3.27)

where R is a matrix of residual entries. Here, I perform this approximate factorization using

BILU(p) [Saad, 2003]. The BILU(p) factorization produces an approximate factorization

of the local Schur complement Si ≈ LSiUSi as a by-product, when the internal interface

variables are ordered last [Saad and Sosonikina, 1999]. The approximate factorization of the

local Schur complement can be used to construct a system of equations closely approximating

Equation (3.26) as follows:

yi + US−1i LS

−1i

∑j

Fijyj = g′i = US−1i LS

−1i

(gi − FiB

−1i fi). (3.28)

In the approximate Schur preconditioner, this system is solved to a loose tolerance using

GMRES [Saad and Schultz, 1986]. Typically, I use GMRES(10) with no restart with a

relative solution tolerance of 10−3. After the interface unknowns yi have been obtained, the

internal unknowns zi are obtained by calculating zi = U−1Bi

L−1Bi

(fi − Eiyi).

The steps in the application of the approximate Schur preconditioner are:

1. Determine the right hand side of Equation (3.28), g′i = US−1i LS

−1i (gi − FiB

−1i fi)

2. Approximately solve Equation (3.28) for the interface unknowns yi using GMRES.

3. Determine the internal unknowns zi = UB−1i LB

−1i (fi − Eiyi)

Note that the cost of factoring the approximate Schur and additive Schwarz precondi-

tioners is the same if the same ordering of Ai is used in both cases.

3.3.2 The direct Schur method

In the following section, I describe the implementation of the direct Schur method. This

method is a domain-decomposition based approach that shares some similarities with multi-

frontal direct factorization methods such as MUMPS [Amestoy et al., 2000] and SPOOLES

[Ashcraft and Grimes, 1999]. In contrast to these methods, however, the global elimination

tree is constrained by the element-based decomposition. While this restricts the parallelism

within the algorithm, it enables better parallel performance of other operations required for

finite-element analysis. In addition, the direct Schur approach also uses a sparse block-cyclic


B1 E1

B2 E2

B3 E3

B4 E4

F1 F2 F3 F4 C

Figure 3.5: The domain decomposition and matrix for the direct Schur method for a four processor

case. Each colour represents a different processor. The dashed line indicates the boundary between

the internal unknowns and the interface unknowns.

format to store and factor the global Schur complement matrix in an approach similar to

SuperLU DIST [Li and Demmel, 2003].

In the direct Schur approach, the unknowns in the matrix are split into two disjoint sets:

1. Internal unknowns that are required by only one processor,

2. Interface unknowns that are required by two or more processors.

These sets of variables are different than the partitioning used for the approximate Schur

preconditioner. Note that there is no differentiation between internal and external interface

variables and as a result the size of the interface problem does not increase with increasing

element order. Figure 3.5 illustrates the domain decomposition and resulting matrix for a

finite-element problem discretized with third-order elements, split between four processors.

In the direct Schur method, the full matrix factorization is split into two steps: the

calculation of the local Schur complement contribution, followed by the formation and fac-

torization of the global Schur complement. To describe these steps, it is first necessary to

introduce the required notation.

All unknowns referenced by processor i, both the internal unknowns and interface un-

knowns, are called the local unknowns and are denoted xi ∈ Rni . The local unknowns


for processor i are be obtained from the global state vector x using a restriction operator

Ri ∈ Rni×n, such that xi = Rix. The local contributions to the matrix A are assembled

from the finite-elements assigned to each processor. These contributions form a local matrix,

Ai ∈ Rni×ni that is related to the global system of equations (3.22) in the following manner:

Ai = RiARTi . (3.29)

In practice, the matrix Ai is assembled on each processor from the local finite-elements

without any communication.

The internal unknowns, denoted zi ∈ Rmi , and the interface unknowns, denoted yi ∈ Rpi ,

are obtained from the local unknowns, xi, using the restriction operators Pi ∈ Rmi×ni and

Qi ∈ Rpi×ni , such that zi = Pixi and yi = Qixi. Instead of forming Ai in an arbitrary

order, the local matrix Ai is split into four block matrices in the following manner:[Bi Ei

Fi Ci

]=

[Pi

Qi

]Ai

[PTi QT

i

], (3.30)

where the dimensions of the matrices Bi, Ei, Fi and Ci can be inferred from the sizes of the

vectors of internal and interface unknowns.

The permutation matrices Pi and Qi are determined to reduce the fill-in during factor-

ization. I have tested three different ordering algorithms: the approximate minimum degree

(AMD) algorithm of Amestoy et al. [1996], and the nested dissection (ND) algorithm in

METIS of Karypis and Kumar [1998]. I have also implemented my own variant of AMD,

based on the work of Amestoy et al. [1996], that takes into account the off-diagonal fill-

ins not considered by the AMD and ND algorithms. I refer to this alternative scheme as

AMD-OD. The AMD and ND orderings determine the permutation matrices Pi and Qi

independently by working only with the diagonal matrices Bi and Ci, respectively. In the

AMD-OD method, Pi and Qi are computed concurrently and the fill-ins produced in the

off-diagonal matrices Ei and Fi are taken into account. This non-standard modification

results in a lower overall fill-in during the factorization process than independently ordering

the internal and interface unknowns.

Using the LU-factorization of the block matrix Bi = LBiUBi

, the local Schur complement

Si can be formed as follows:

Si = Ci − FiU−1Bi

L−1Bi

Ei. (3.31)

Note that both the LU-factorization of the block matrix Bi, and the computation of the local

Schur complement (3.31) can be computed independently, without communication between

processors.


The local interface unknowns yi, can be obtained from the global vector of interface

unknowns, denoted y ∈ Rm, using the restriction operator Ti ∈ Rpi×m such that yi = Tiy.

The global Schur complement matrix S, can be expressed as the sum of contributions from

the local Schur complement (3.31), using the permutation matrices Ti, as follows:

S =

Np∑i

TTi SiTi, (3.32)

The permutation matrices Ti are selected to ensure a low fill-in for the LU factorization of

the global Schur complement based on the AMD ordering. Note that the calculation of the

global Schur complement matrix, S, requires communication amongst all processors. Once

the global Schur complement is formed, it is factored such that

S = LSUS. (3.33)

Based on the above discussion, factoring the global matrix A requires the following steps:

1. Factor the block matrix, Bi = LBiUBi

,

2. Form the local Schur complement, Si = Ci − FiU−1Bi

L−1Bi

Ei,

3. Assemble and factor the global Schur complement, S = LSUS.

Steps 1 and 2 require no communication with other processors, while step 3 requires the

accumulation of the Schur complement contributions from each processor. The factored

matrices in steps 1 and the local Schur complement in step 2, are significantly more sparse

than the global Schur complement matrix S (3.32). Furthermore, to achieve good parallel

performance for the overall factorization, the global Schur complement factorization must

also be performed in parallel. As a result of these considerations, I use a sparse 2D block-

cyclic format to store S in a distributed manner across all processors. The sparse 2D block-

cyclic matrix format has two main advantages: it enables parallelism during the matrix

factorization and it distributes the memory required to store the factored matrix.

Once the factorization of the global Schur complement is complete, the solution to the

system of linear equations, Ax = b, can be obtained using the following steps:

1. Compute an update to the internal variables, zi ← L−1Bi

PiRib;

2. Compute the local contribution to the global Schur complement right-hand-side, yi =

QiRib− FiU−1Bi

zi;

3. Solve the global Schur complement system, y← U−1S L−1

S y;

4. Complete the full update to the internal variables, zi ← U−1Bi

(zi − L−1Bi

Eiyi).


Block-cyclic matrix format

In the block-cyclic storage format, the Schur complement matrix S from Equation (3.33)

is split into a regular 2D pattern of block matrices denoted Sij. These block matrices are

assigned to the available processors in a cyclic pattern that is repeated over the entire matrix

until all blocks are assigned. The processor assignment pattern is called the process mesh.

Figure 3.6 illustrates a 2 × 3 process mesh for a sparse matrix, where each colour in the

matrix corresponds to one of six processors. Note that the sparsity pattern of the matrix is

also taken into account such that blocks with no corresponding entry are not stored. The

size of the block matrices are determined based on the matrix structure, however, there is

a performance tradeoff between the communication latency and memory cache size when

computing the matrix factorization [Li and Demmel, 2003]. As a result, it may be beneficial

to split large blocks.

S11 S14 S16

S22 S25 S26 S29

S33 S35 S36 S37

S41 S44 S46 S47 S48

S52 S53 S55 S56 S57 S59

S61 S62 S63 S64 S65 S66 S67 S68 S69

S73 S74 S75 S76 S77 S78 S79

S84 S86 S87 S88 S89

S92 S95 S96 S97 S98 S99

Figure 3.6: An illustration of the block-cyclic matrix format for a 2 × 3 grid. Each colour

corresponds to a different processor.

The algorithm used to compute the LU-factorization for a block-cyclic matrix is show in

Algorithm 1. Here, the function is block owner(i, j) returns true if the block matrix Sij is

non-zero and is assigned to the current processor, otherwise it returns false. The main loop

of the algorithm consists of three main computational steps. First, the block diagonal is

factored and inverted in place, U−1ii = S−1

ii . Second, the column update is computed by post-


Algorithm 1: Factorization of a matrix stored in a block-cyclic data format

Given S, compute L and U such that S = LU;

for i = 1 to n do

Diagonal update if is block owner(i, i) then

compute U−1ii = S−1

ii and send U−1ii to each processor in column i;

Column update if processor in column i then

receive U−1ii ;

for j = i+ 1 to n do

if is block owner(j, i) then

compute Lji = SjiU−1ii and send Lji to processors in row j;

if processor in row i then

for j = i+ 1 to n doif is block owner(i, j) then send Uij to processors in column j;

receive Uji and Lij;

GEMM update for j = i+ 1 to n do

for k = i+ 1 to n doif is block owner(j, k) then compute Sjk ← Sjk − LjiUik;

multiplying the blocks in column i by U−1ii . Third, the GEMM update2 must be applied to

the remaining un-factored portion of the matrix. Note that the column and GEMM updates

require block factors that are not stored locally. For the column update, U−1ii must be sent

to processors that own a block in column i. For the GEMM update, the lower factor Lji

and upper factor Uik must be sent to the processors in row j and the processors in column

k respectively.

The parallel performance of Algorithm 1 depends on the degree to which the column and

GEMM updates can be parallelized. The diagonal and column update involve only a subset

of the processors in column i and row i. As a result, these operations can only be distributed

across these subsets of processors, resulting in sub-optimal parallel performance. However,

these two steps constitute a small portion of the computational time of the algorithm. The

GEMM update, on the other hand, constitutes the main cost of the algorithm and utilizes

all processors. This step can be implemented efficiently in parallel.

2This short form comes from the BLAS [Lawson et al., 1979] where GEMM routines are used to computea matrix update of the form C← αAB + βC


(a) Annular disk (b) Transonic transport wing

Figure 3.7: The domain decompositions for the quarter section of an annular disk and the tran-

sonic transonic wing on 32 and 64 processors, respectively.

3.3.3 Parallel solution performance

In this section, I examine the parallel solution performance of the additive Schwarz and ap-

proximate Schur preconditioners, and the direct Schur method for two finite-element prob-

lems: an analysis of a quarter section of an annular disk using plane-stress elements, and a

structural analysis of a transonic transport wing using shell elements.

The plane-stress problem consists of a quarter section of an isotropic annular disk with

an inner radius of 4 and an outer radius of 10. The Young’s modulus of the disk is 70 and

the Poisson ratio is 0.3. The annular disk is loaded with a uniform, body force with unit

magnitude in both coordinate directions. The finite-element model consists of a mesh of

1350 × 750 second-order elements in the circumferential and radial directions respectively,

resulting in a discretization with approximately 2.029 million degrees of freedom. Figure 3.7a

shows domain decomposition for the annular disk problem on 32 processors.

The transonic aircraft wing is roughly based on the geometry of a Boeing 777-200 wing.

The wing has a semi-span of 30.45 m, a root chord of 13.6 m and a taper ratio of 0.2. The

crank in the wing occurs at a station at 30% of the semi-span. The wing structure consists

of 44 chord-wise ribs spaced evenly out the wing and two span-wise spars. The wing is

modeled with an isotropic material with a Young’s modulus of 70 GPa and a Poisson’s ratio

of 0.3. For simplicity, the skin thickness of the wing is set to 5 mm uniformly over the entire

structure.

The wing is discretized using either 2nd, 3rd or 4th order MITC shell elements with 907 388

nodes resulting in just over 5.44 million degrees of freedom. The 2nd order discretization con-

tains 912 384 elements, the 3rd order discretization contains 228 096 elements, and the 4th

order discretization contains 101 376 elements. The wing is loaded with a set of aerodynamic

forces computed at a 2.5g maneuver flight condition. Figure 3.7b shows the domain decom-

position corresponding to a 64 processor case where each contiguously coloured segment


level of fill

time[s]

5 10 15 20

25

50

75

100

125

150

175

200direct Schur

natural approximate Schur

RCM approximate Schur

AMD approximate Schur

natural additive Schwarz

RCM additive Schwarz

(a) 32 processor case

number of processors

time[s]

8 16 24 32 40 48

50

100

150

200

250

300350400

ideal

direct Schur

natural ILU(20) app. Schur

RCM ILU(20) app. Schur

AMD ILU(20) app. Schur

natural ILU(8) add. Schwarz

RCM ILU(8) add. Schwarz

(b) Plane stress scaling study

Figure 3.8: Level of fill study for the 32 processor case and the parallel scalability of precondi-

tioning methods for the plane stress problem.

corresponds to a domain belonging to a different processor.

The results in this section are based on calculations performed on the General Purpose

Cluster (GPC) at SciNet [Loken et al., 2010]. Each node of the GPC is an Intel Xeon E5540

with a clock speed of 2.53GHz, with 16GB of dedicated RAM and 8 processor cores. In these

comparisons, I only use nodes connected with non-blocking 4x-DDR InfiniBand.

Annular disk results

First, I assess the parallel performance of the additive Schwarz, approximate Schur and direct

Schur preconditioners for the annular disk problem using 8, 16, 24, 32 and 48 processors.

For the additive Schwarz and approximate Schur preconditioners, I use GCROT(150, 10)

and F-GCROT(150, 10) respectively [Hicken and Zingg, 2010a], with 150 inner iterations of

GMRES or F-GMRES and 10 outer GCR vectors. For the direct Schur approach, I use the

AMD-OD ordering with GMRES(5), which converges in 2 GMRES iterations for each case

presented here. In all cases, I use a relative convergence tolerance of 10−10.

To simplify the discussion, I examine the total time to set up and solve the linear finite-

element problem. Included in this total time is the time to calculate and assemble the global

stiffness matrix, factor the preconditioner, and perform a single linear system solution. A


requirement to perform multiple solutions for different right-hand-sides would significantly

alter the results in favor of the direct Schur approach since the time to factor the precondi-

tioner could be amortized over each linear system solve.

Figure 3.8a shows the solution time for the 32 processor case with natural, RCM and

AMD ordering, with either additive Schwarz or approximate Schur preconditioning, for the

following levels of fill: 2, 4, 8, 10, 15 and 20. Additionally, the direct Schur solution time

is also plotted in Figure 3.8a. For RCM ordering, increasing the level of fill significantly

beyond 20 violates memory constraints for the 8 processor case. For the additive Schwarz

preconditioner, the computational time begins to increase beyond a level of fill of 10 for

either the natural or RCM ordering. This suggests that the additional computational time

required to apply a preconditioner with higher level of fill is not offset by an improvement

in the numerical performance. As a result, the overall computational time increases. For

the approximate Schur preconditioners, the natural, RCM and AMD orderings improve with

increasing level of fill. This suggests that the increasing computational complexity of the

preconditioners results in better preconditioner performance. Note that for this case, the

direct Schur method out-performs any of the iterative methods.

Figure 3.8b shows the parallel scaling results for the plane stress problem using BILU(8)

for the additive Schwarz preconditioners and BILU(20) for the approximate Schur precon-

ditioners. Note that all preconditioning options scale well. The additive Schwarz precon-

ditioners, however, are not competitive with either the direct Schur or approximate Schur

approaches. Note that the approximate Schur method, with AMD ordering improves faster

than the ideal rate. This is due to the reduction in domain size with increasing numbers of

processors, which reduces the size of the domain preconditioners and the time required to

apply the approximate factors, without negatively impacting their effectiveness. This results

in a faster-than-ideal speedup.

Transonic wing results

The transonic wing case is a much more challenging problem than the plane stress case

presented above due to the poor condition numbers exhibited by the stiffness matrices for

these problems. I found that neither the additive Schwarz nor the approximate Schur pre-

conditioners were sufficiently powerful for this case. I did not find a combination of ordering

method and fill-level, within the available memory limits, that provided a sufficiently pow-

erful preconditioner to solve this problem within a time frame competitive with the direct

Schur approach. As a result, I present only results for the direct Schur approach. The most



time[s]

32 48 64 80 9630

40

50

60

70

80

90

100

110

120

ideal

AMD

ND

AMDOD

(a) 2nd order


time[s]

32 48 64 80 9630

40

50

60

70

80

90

100

110

120

ideal

AMD

ND

AMDOD

(b) 3rd order


time[s]

32 48 64 80 9630

40

50

60

70

80

90

100

110

120

ideal

AMD

ND

AMDOD

(c) 4th order

Figure 3.9: The factorization times for the transonic wing case with the direct Schur approach

using the AMD, ND and AMD-OD orderings, for the 2nd, 3rd and 4th order discretizations.

computationally intensive part of the direct Schur method is computing the matrix factor-

ization. However, the time to apply the factorization becomes more important for problems

with multiple right-hand-sides. In this section, I examine both the time to factor the matrix

and the solution time and assess their parallel scalability behavior.

Figure 3.9 shows the factorization times for the finite-element wing model using AMD,

ND and AMD-OD orderings for 24, 32, 48, 64 and 96 processors for the 2nd, 3rd and 4th order

problems. Due to memory constraints, the problem must be run on at least 32 processors for

most orderings. Note that there is no significant difference in the computational time required

for either the 2nd, 3rd or 4th order problems. For the AMD-OD ordering, the computational

time actually decreases slightly, in some cases, with increasing order. Both AMD and AMD-

OD ordering methods are very effective at taking advantage of the matrix structure that

exists in the higher-order problems through super-node identification [Amestoy et al., 1996].

These super nodes help produce re-ordered matrices that are faster to fully factorize.

From Figure 3.9 it is clear that the AMD-OD ordering scheme results in the fastest fac-

torization times for the 24, 32, and 48 processor cases. However, the AMD-OD factorization

times do not scale as well as ND or AMD. The AMD-OD ordering is effective at reducing the

off-diagonal fill-ins, but these fill-ins have the greatest impact when there are fewer proces-

sors and off-diagonal matrices are large. As a result, AMD-OD tends to do better for fewer

numbers of processors, and for larger numbers of processors tends to do as well or slightly


0.2

0.4

0.6

0.8

1.0

1.2

1.4

32 48 64 96 32 48 64 96 24 32 48 64 96

AMD ND TACS AMD


fraction

ofidealfactor

time

max local time

average local time

min local time

global Schur time

communication time

0.2

0.4

0.6

0.8

1.0

1.2

1.4

32 48 64 96 32 48 64 96 24 32 48 64 96

AMD ND TACS AMD


fraction

ofidealfactor

time

max local time

average local time

min local time

global Schur time

communication time

0.2

0.4

0.6

0.8

1.0

1.2

1.4

32 48 64 96 32 48 64 96 32 48 64 96

AMD ND TACS AMD


fraction

ofidealfactor

time

max local time

average local time

min local time

global Schur time

communication time

Figure 3.10: The fraction of time spent in different parts of the factorization process for the

AMD, ND and AMD-OD orderings as a function of the number of processors, shown from top to

bottom are the 2nd, 3rd and 4th order discretizations.


better than AMD. In all cases, the ND ordering seems to scale the most consistently.

The factorization times for increasing numbers of processors do not behave in a simple

manner when compared with a linear speed up. Figure 3.10 presents a detailed study of

the fraction of time spent in different factorization operations normalized to the ideal line

presented in Figure 3.9. Figure 3.10 shows the fractions of time for the processors with the

least idle time, the average idle time and the most idle time. These correspond to the pro-

cessors that require the maximum local time, average local time and minimum local time,

respectively. The local factorization time corresponds to the time to compute the block fac-

torization of Bi, and the local contribution to the Schur complement Si from Equation (3.31).

The communication time is the time required to communicate the local Schur complement

contributions to the required processors for the global Schur complement. Finally, the global

Schur complement time is the time required to factor the global Schur complement in the

block-cyclic data format. Note that the proportion of time spent in each stage of the fac-

torization changes as the number of processors increases. In particular, the fraction of time

to factor the global Schur complement increases. This behaviour makes it difficult to obtain

an ideal speed up consistently.

The efficiency of the ordering techniques can be judged from Figure 3.10 based on the

discrepancy between the processors with the maximum local time and average local time.

Large gaps are produced when one processor takes significantly longer than any of the others.

Note that this gap is smallest for the AMD-OD ordering for the 24, 32 and 48 processor cases,

but increases between 48 and 64 processors. For the ND ordering, this gap does not grow

as rapidly as either AMD or AMD-OD with increasing numbers of processors, while the

discrepancy between the maximum and minimum local times for the ND ordering becomes

smaller. The gap between maximum and average local times is largest for the AMD ordering,

but does not increase significantly.

While the factorization time is the largest single contributor to the time required to

assemble, factor, and solve a linear finite-element problem, the assembly and solution times

are extremely important for nonlinear solution methods and for linear problems with multiple

right-hand-sides. Figure 3.11a shows the solution time and the assembly time, which includes

the time required to compute and store both the residual and the stiffness matrix. These

times are shown for the 2nd, 3rd and 4th order transonic wing problems with ND ordering. The

other ordering schemes exhibit nearly identical behavior. The assembly time scales ideally to

machine precision. This is due to the fact that the operations involved are computationally

intensive, but require very little communication overhead. The higher-order problems require

additional computational time, where the assembly times for the 4th, and 3rd order problems



time[s]

32 48 64 80 96

0.5

1

1.5

2

2.5

3

3.5

4thorder

3rdorder

2ndorder

solution time

matrix and residual assembly time

(a) Solution and assembly times


time[s]

32 48 64 80 96

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45total backsolve time

max local time

max Schur time

ideal

(b) ND 2nd order back-solve times

Figure 3.11: Time required to compute and assemble the stiffness matrix and residual, and solve

the linear system. Note that the assembly time speeds up ideally.

are a factor of approximately 4.1 and 1.8 times the assembly time of the 2nd order problem.

The solution time, on the other hand, does not scale ideally. In all cases, GMRES requires

2 iterations to solve the linear system with 2 matrix-vector products and 2 applications of

the direct Schur preconditioner. Figure 3.11b shows a breakdown of two components of the

computational time required for the application of the factored direct Schur preconditioner:

the time required to assemble and apply the local contribution to the right hand side, la-

beled local time in Figure 3.11b, and the time required to apply the factored global Schur

complement matrix stored in the block-cyclic format, labeled Schur time in Figure 3.11b.

Clearly, the application of the block-cyclic matrix does not scale ideally and is driving the

poor computational performance, most critically for the 96 processor case. All other meshes

and ordering types exhibit similar behavior.

3.4 Structural sensitivity analysis

Efficient and accurate sensitivity analysis is an essential tool for gradient-based design op-

timization. Inaccuracies in the gradient may lead to poor optimizer performance or even

failure and an inefficient gradient evaluation method will result in long computational times.


In this section, I present the sensitivity methods used for structural design optimization and

evaluate the parallel performance of the implementation. An examination of the accuracy

of the implementation is left until a discussion of accuracy of the aerostructural adjoint

implementation in Chapter 5.

The goal of sensitivity analysis methods is to obtain the gradient, or total derivative,

∇xf ∈ Rnf×nx of a vector of functions of interest, f(u,x) ∈ Rnf . These functions are a

combination of the objective and constraints in a structural optimization problem. The

functions f(u,x) depend smoothly on the state variables u ∈ Rn and the design variables

x ∈ Rnx . Furthermore, the system of residuals that are used to determine the structural

state variables are R(u,x) = 0 ∈ Rn and are also smooth functions of u and x. With these

definitions, the total derivative of f is

∇xf =∂f

∂x− ∂f

∂u

[∂R

∂u

]−1∂R

∂x. (3.34)

The adjoint sensitivity method can be obtained by introducing the matrix of adjoint

variables ψ, such that [∂R

∂u

]Tψ =

∂f

∂u

T

. (3.35)

With the adjoint variables, the total sensitivity (3.34) reduces to the form,

∇xf =∂f

∂x−ψT ∂R

∂x(3.36)

The direct sensitivity method can be obtained by introducing the matrix of state-variable

derivatives φ such that [∂R

∂u

]φ =

∂R

∂x. (3.37)

The total derivative for the direct method may now be written

∇xf =∂f

∂x− ∂f

∂uφ. (3.38)

The relative efficiency of the direct (3.38) and adjoint (3.36) methods depends on the

relative number of functions and design variables. If there are many more design variables

than functions nf nx, then the adjoint method is more efficient. On the other hand, if there

are many more functions than design variables nf nx, then the direct method is more

efficient. Typically, structural optimization problems require both many design variables

and many constraints. These problems cannot be handled efficiently by either the adjoint or

direct gradient evaluation techniques. However, constraint aggregation techniques, such as

the Kreisselmeier–Steinhauser (KS) function [Wrenn, 1989, Poon and Martins, 2007], can be


employed to significantly reduce the number of constraints. These aggregation techniques

are especially important for aerostructural optimization problems where the computational

cost of additional functions is even more severe.

3.4.1 Gradient evaluation performance

The computational cost of evaluating the total derivatives of the functions of interest depends

on many factors including: the number and type of design variables, the number and type

of functions of interest, the element formulation, and the number of processors. The results

presented here focus on the parallel scalability of the gradient-evaluation method, and the

rate of increase in the computational cost with number of functions.


time[s]

32 48 64 80 96

10

20

30

40

50

6070

DV time

adjoint time

adjoint residual product

4thorder

3rdorder

2ndorder

(a) Gradient scaling with processors,

nf = 15

number of functions

time[s]

0 5 10 150

5

10

15

20

25

30

35

DV time

adjoint time

adjoint residual product

4thorder

3rdorder

2ndorder

(b) Gradient scaling with functions, np = 64

Figure 3.12: The scaling performance of the terms required for the adjoint for both processors

and number of functions.

In this study, I use the transonic wing case presented in Section 3.3.3. The functions of

interest are the von Mises failure criteria aggregated over some portion of the wing struc-

ture using the KS function. The design formulation includes 220 thickness variables and 11

geometric design variables consisting of 9 twist, 1 span and 1 chord scaling variable that

modify the entire wing structure. In the present implementation, the computational cost of


computing the total derivative with respect to one geometric design variable is high relative

to material-type design variables, but scales very weakly with increasing numbers of geo-

metric design variables. This is due to the underlying implementation in which the partial

derivatives are first computed with respect to all nodes within the finite-element mesh and

multiplied by the derivative of the nodes with respect to the geometric design variables.

This implementation is very efficient for problems with hundreds or thousands of geometric

design variables that arise in aerostructural shape optimization [Kenway et al., 2012].

One of the purposes of this study is to assess the scaling of the cost of the gradient

evaluation with the number of functions of interest. In order to present a realistic application

for this study, I compute a series of KS functions aggregated over domains formed by splitting

the structure into approximately equally-sized regions. These domains are not selected

randomly, but instead are formed from common structural components such as the skin,

rib and spar panels in the finite-element model. This construction represents a realistic

division of the constraint aggregation that is used in structural optimization [Kennedy and

Martins, 2012].

Figure 3.12 shows an assessment of the computational cost of the terms required to

compute the total derivative. These terms are: the cost of computing the partial derivative

of the function of interest with respect to the design variables, ∂f/∂x, labeled “DV time”,

the time to compute right-hand-side and solve for the adjoint vectors (3.35), and the time

to compute the product of the adjoint vector with the derivative of the residual with respect

to the design variables, ψT∂R/∂x. Figure 3.12a shows the scaling of the terms required

for the total derivative with respect to the number of processors for the case with nf = 15

KS functions. The time to compute the product of the adjoint with the derivative of the

residuals scales ideally to plotting precision. The other terms, however, do not scale as

well. The computation of the adjoint does not scale ideally primarily due to the fact that

the solution times for the direct Schur method do not scale ideally, as was discussed in the

previous section, see Figure 3.11b. The partial derivative of the function with respect to the

design variables does not scale ideally because the KS domains are highly correlated with

the domain decomposition. The domains are not split evenly amongst the processors, but

instead tend to end up on a few processors, with the remaining processors receiving no part

of the domain. This leads to poor load balancing and poor parallel performance.

From Figure 3.12 it is clear that the computational cost of the gradient scales strongly

with element order. On 64 processors with nf = 15, the 4th, 3rd and 2nd order gradients took

approximately 100, 36 and 22 seconds respectively. Therefore, there is a tradeoff between

solution accuracy obtained with higher-order elements and additional gradient-evaluation


costs for design optimization. Based on this analysis, and the frequent appearance of stress

concentrations in practical shell design problems that limit the asymptotic convergence rate,

I usually use 3rd order elements, which provide better stress resolution capability at no

additional analysis cost, with a slightly higher computation cost for the gradient-evaluation

compared to 2nd order elements.

3.5 Conclusions

In this chapter, I have presented the components of a framework for the parallel analysis and

gradient-based design optimization of thin shell structures. These components have been im-

plemented in a unified framework, called the Toolkit for the Analysis of Composite Structures

(TACS). In particular, I have presented a higher-order shell element formulation that enables

the accurate determination of the stress and strain distribution in thin shell structures. In

order to solve the linear systems resulting from the finite-element discretization, I presented

the additive Schwarz preconditioner, approximate Schur preconditioner and the direct Schur

method and I demonstrated that each of these methods exhibit good parallel scalability for

a plane stress problem, when used in conjunction with GCROT, F-GCROT or GMRES. Of

these methods, I demonstrated that the direct Schur approach required the least computa-

tional time to solve the resulting linear system. I also applied these solvers to a structural

model of a transonic wing case. For this case, the inexact preconditioners were inadequate

and only the direct Schur method provided a sufficiently powerful preconditioner. For the

transonic transport wing, I compared three ordering algorithms: AMD, ND and AMD-OD.

I demonstrated that the AMD-OD ordering was the fastest for problems with 24, 32, or 48

processors, while the ND ordering was fastest for the 64 or 96 processor cases. Furthermore,

I demonstrated that the solution times do not vary significantly with increasing element

order. Finally, I presented the implementation of a parallel gradient-evaluation technique

for the determination of derivatives of functions of interest. I examined the scalability of

the adjoint gradient-evaluation method with increasing processor count, increasing numbers

of functions and increasing element order. I demonstrated that the gradient computational

times increase significantly for higher-order elements. Therefore, there is a tradeoff between

higher-accuracy solutions obtained by increasing the element order, and computational time

for gradient-based optimization. When combined, the methods presented in this chapter en-

able efficient, parallel design optimization of complex thin shell structures used for aerospace

applications.

Chapter 4

Laminate parametrization

The parametrization of composite structures for design optimization is a challenging prob-

lem. Realistic structural design optimization problems require numerous manufacturing

constraints, which often include a restriction to select amongst a discrete set of available

ply angles. This discrete problem is not, in its most natural form, amenable to gradient-

based optimization. However, methods for nonlinear mixed-integer programming are almost

inevitably computationally expensive, especially for large numbers of design variables or

large design spaces. In this chapter, I present a laminate parametrization technique that

takes into account the discrete nature of some of the ply angle variables. In order to avoid

solving a large, nonlinear, mixed-integer program, I use a relaxation approach where the

original discrete problem is transformed into a continuous analogue of the original problem.

Gradient-based optimization can then be used to obtain solutions to the modified prob-

lem. To illustrate the effectiveness of the proposed laminate parametrization technique, I

present results from a series of structural optimization problems including design for min-

imum compliance, design for maximum critical end shortening and design problems with

strength criteria.

This chapter is structured as follows. In Section 4.1, I review the relevant literature

on laminate parametrization techniques. In Section 4.2, I describe the proposed laminate

parametrization technique and present a method for predicting material failure within the

context of the laminate parametrization. In Section 4.3, I describe additional manufacturing

constraints that may be required for certain laminate design problems. In Section 4.4, I

present a series of optimization results, including a compliance minimization study and a

stiffened panel buckling optimization. Finally, in Section 4.5, I present an application of

the proposed laminate parametrization technique to the lamination sequence design of a

composite wing-box. Most of the material in this chapter was published in Kennedy and

77

Chapter 4. Laminate parametrization 78

Martins [2012].

4.1 Literature review

Laminate parametrization techniques generally fall into two categories: direct parametriza-

tions that provide an explicit description of the physical laminate, and indirect parametriza-

tions in which intermediate variables are employed and the lamination sequence is only

available in a post-processing calculation. There are difficulties with using either of these

approaches. Direct techniques often introduce many local minima in the design space, while

indirect methods make it difficult to impose manufacturing constraints on the physical con-

struction of the laminate.

The use of ply angle variables and the integer number of plies is the most direct parametriza-

tion of a lamination sequence. However, this type of parametrization suffers from several

drawbacks. First, the combination of the ply-angle variables and the integral number of plies

necessitates mixed-integer programming techniques [Haftka and Gurdal, 1992]. Second, it

is well known that the parametrization using ply-angle variables, for fixed number of plies,

introduces many local minima [Stegmann and Lund, 2005]. Nevertheless, researchers have

developed various techniques to address these issues. For instance, Bruyneel and Fleury

[2002] and Bruyneel [2006] developed an effective, gradient-base optimization approach for

composite structures parametrized with ply angles, for stiffness, strength and weight design

criteria. However, these parametrizations still do not address manufacturing constraints that

limit available ply angles to a discrete set.

The constitutive matrices in classical lamination theory (CLT) and in first-order shear

deformation theory (FSDT) can be expressed in terms of the material invariants and 12

integrals of trigonometric functions of the ply-angle distribution through the thickness of the

laminate. When these integrals are treated as variables, rather than explicit functions of the

lamination sequence, they are referred to as the lamination parameters [Tsai and Pagano,

1968]. Due to the relationship between these integrals, not all combinations of the lamina-

tion parameters represent physically realizable laminates. As a result, constraints must be

imposed to restrict the values of the parameters to a physically-realizable domain [Hammer

et al., 1997]. An expression for the full feasible space of lamination parameters is not known

explicitly, so often a subset of the variables are used for design.

Lamination parameters have often been used as a parametrization for stiffness and buck-

ling design. Fukunaga and Vanderplaats [1991] performed buckling optimization of cylindri-

cal shells with symmetric, orthotropic laminates using two in-plane and two flexural lami-


nation parameters. Fukunaga and Vanderplaats also solved the inverse problem to obtain

the explicit lamination sequences. Later, Fukunaga and Sekine [1992] performed stiffness

design of laminates and obtained explicit expressions for the feasible space of symmetric

laminates. Miki and Sugiyama [1993] used lamination parameters for the compliance and

buckling design of symmetric, orthotropic laminates. Hammer et al. [1997] presented an

extensive theoretical development of the mathematical properties of lamination parameters

and used them for compliance design of symmetric laminates subject to single and multiple

loading conditions. Liu et al. [2004] designed simply supported symmetric plates for buckling

using lamination parameters. They imposed a constraint on the number of plies at 0o, ±45o

and 90o and mapped these constraints into a hexagonal region in the lamination parameter

space.

Other authors have extended the use of the lamination parameters beyond stiffness and

buckling design applications. Foldager et al. [1998], used lamination parameters to avoid

local minima while performing compliance minimization using ply angle design variables.

IJsselmuiden et al. [2008] performed strength-based design studies using lamination parame-

ters by incorporating the Tsai–Wu failure criteria [Jones, 1996] into the lamination parameter

space to obtain a conservative failure envelope.

While lamination parameter based parametrizations have been used effectively in many

applications, one of the primary disadvantages of this approach is that it does not provide a

direct description of the laminate construction. This makes it difficult to impose constraints

on ply angles that may be required due to manufacturing considerations. Furthermore,

lamination parameters, by themselves, do not provide a lamination sequence and therefore

can only be viewed as an intermediate design result.

Often, for manufacturing reasons, the ply angles available to the designer are restricted to

a discrete set of options such as 0o, ±45o and 90o. With this restriction, the laminate sequence

design problem becomes a mixed-integer programming problem. Various authors have used

either mixed-integer programming techniques or genetic algorithms (GAs) to solve laminate

stacking sequence problems poised with a discrete set of ply angles. Haftka and Walsh [1992]

formulated the buckling-load maximization of a simply supported plate, with and without

ply contiguity constraints, as a linear integer programming problem and obtained global

optimum designs using a branch and bound algorithm. Le Riche and Haftka [1993] performed

buckling-load maximization of a simply-supported plate with strength and ply contiguity

constraints using a GA. Later, Liu et al. [2000b] performed buckling-load maximization for a

simply supported plate with a constraint on the number of plies at each available angle using

a permutation GA. More recently, Adams et al. [2004] used a GA for a realistic composite


wing-box design problem with a thick guide laminate and blended plies.

The main advantage of using GAs for laminate design problems, is that they have the

ability to work with integer variables directly. Furthermore, GAs are more likely to get close

to the global optimum, regardless of whether the underlying design space is multi-modal

or discontinuous. However, GAs often require several orders of magnitude more function

evaluations than gradient-based approaches, especially for large design spaces. This property

of GAs is especially problematic when employing high-fidelity computational methods that

require significant computational time for a single analysis.

Discrete material optimization (DMO) approaches can be used as either multi-material

or laminate sequence parametrizations. DMO was first proposed by Stegmann and Lund

[2005] based on the work of Sigmund and Torquato [1997]. In the DMO approach, applied

to laminate design, the stiffness contribution from every discrete ply-angle, in each layer

is multiplied by a weighting function. Instead of using a linear interpolant, a SIMP-type

penalization is employed such that the stiffness-to-weight ratio of intermediate designs are

less favorable. Stegmann and Lund [2005] and later Lund [2009] applied the DMO approach

to the compliance and buckling optimization of composite shells.

Other authors have extended SIMP and DMO-type approaches. Bruyneel [2011] de-

veloped an approach, similar to DMO, for selection amongst a discrete set of four plies

using bilinear shape function weights. This approach, called the shape function with penal-

ization (SFP) parametrization, reduces the number of design variables compared to DMO

approaches. Bruyneel et al. [2011] extended the SFP approach to material selection amongst

different numbers of plies by using different interpolation functions. Using a different ap-

proach, Hvejsel et al. [2011] developed a technique for laminate parametrization that, in a

similar manner to DMO, uses a weighted sum of contributions to the stiffness. In a depar-

ture from the DMO approach, they employed an exact, quadratic concave penalty constraint

function, first used by Borrvall and Petersson [2001], to force the design towards a discrete

solution. They demonstrated their approach on a series of compliance minimization prob-

lems.

One of the main advantages of DMO and DMO-type parametrizations, is that they can be

used with gradient-based optimization techniques. As a result, DMO parametrizations can

be used on very large design problems for which gradient-free methods, such as GAs, would

be ineffective. However, DMO and SIMP-type approaches may only produce a local optimum

solution [Stolpe and Svanberg, 2001b,a]. Furthermore, DMO and DMO-type approaches may

fail to converge to a fully discrete design, especially for objectives other than compliance and

it may be difficult to assess the merits of an intermediate solution.


The laminate parametrization I present below is a direct parametrization that provides

an explicit description of the lamination sequence. The proposed approach is a continuous

regularization of a discrete mixed-integer laminate formulation. Similar to the DMO ap-

proach of Hvejsel et al. [2011], I interpolate between a discrete set of possible angles using

a linear combination of material stiffnesses. In a departure from previous work, I add an

exact `1 penalty function to the objective function to force the design towards a discrete

solution. The `1 penalty function is not differentiable, so I propose an elastic programming

approach that produces the effect of the `1 penalty function in a differentiable manner within

the optimization problem. Simplifications to the penalization can be made if certain linear

constraints are satisfied exactly at all optimization iterations. Compared to DMO-type ap-

proaches, the present approach is more effective at obtaining fully converged designs due to

the exact `1 peanlization. In a departure from previous papers on DMO-type methods, I

also introduce additional complementarity constraints on the ply angles that may arise due

to manufacturing considerations. These complementarity constraints are handled through

a regularization technique proposed by Scholtes [2001]. I apply the proposed approach to

several structural optimization problems of increasing complexity, including a compliance

minimization problem, a stiffened-panel design for buckling and a wing-box sizing and lam-

ination sequence design study.

4.2 The proposed laminate parametrization

In the following laminate parametrization technique, I consider a structure that is split into

a series of M design segments. Each design segment is composed of a single laminate with N

layers, where in each layer, the ply angles must be selected from a discrete set of K allowable

angles, Θ = θ1, θ2, . . . , θK. For ease of presentation, the number of layers and number of

allowable ply angles is fixed to N and K for all segments. This restriction, however, is not

required, and in general, the number of plies and number of available ply angles may vary

between design segments.

Each design segment of the structure is modeled using first-order shear deformation

theory (FSDT), where the in-plane, bending-stretching coupling, bending and transverse

shear constitutive matrices are: A(i), B(i), D(i), A(i)s . Note that the superscript i indexes

the ith design segment, where i = 1, . . . ,M .

In the following description, I first outline the proposed laminate parametrization using

a mixed-integer formulation and then proceed to relax the discrete problem to a continuous

formulation. In the proposed parametrization technique, the constitutive matrices, A(i), B(i),


D(i), A(i)s , are expressed in terms of a series of discrete ply-identity variables ξijk ∈ 0, 1:

A(i) =N∑j=1

(hij+1 − hij)K∑k=1

ξijkQ(θk), B(i) =N∑j=1

1

2(h2

ij+1 − h2ij)

K∑k=1

ξijkQ(θk),

D(i) =N∑j=1

1

3(h3

ij+1 − h3ij)

K∑k=1

ξijkQ(θk), As(i) = κ

N∑j=1


ξijkQs(θk),

(4.1)

where there are N plies in the laminate, Q(θ) and Qs(θ) are the laminae in-plane and shear

stiffnesses in the global coordinate system [Jones, 1996], and hij are the through-thickness

coordinate of the jth layer-interface in the ith design segment.

An active ply-identity variable, ξijk = 1, indicates that the kth ply angle, θk, in the jth

layer of the ith design segment has been selected. To avoid selecting multiple ply angles in

the same layer, it is necessary to impose the following constraint:

K∑k=1

ξijk = 1, i = 1, . . . , N, j = 1, . . . ,M. (4.2)

Note that this discrete formulation is identical to the mixed-integer approach of Haftka and

Walsh [1992]. Equation (4.2) ensures that one and only one ply is active in each layer, thus

ξijp = 1 for some p, while ξijk = 0 for k 6= p.

The number of possible designs increases rapidly as the number of ply angles, layers and

design segments increase. Evaluating all possible designs quickly becomes computationally

intractable as there are KMN possible combinations.

Instead of using the discrete variables ξijk ∈ 0, 1, I relax the mixed-integer problem

and use continuous variables, written as: xijk ∈ [0, 1]. The continuous design variables, xijk,

are called the ply selection variables, and continuous designs that satisfy xijk ∈ 0, 1 are

called 0-1 points. The stiffness can now be expressed in terms of the continuous ply selection

variables as follows:

A(i) =N∑j=1


xPijkQ(θk), B(i) =N∑j=1

1

2(h2

ij+1 − h2ij)

K∑k=1

xPijkQ(θk),

D(i) =N∑j=1

1

3(h3

ij+1 − h3ij)

K∑k=1

xPijkQ(θk), As(i) = κ

N∑j=1


xPijkQs(θk),

(4.3)

where xijk are continuous over the interval [0, 1]. Note that the SIMP parameter P has

been introduced as an exponent on the continuous ply identity variables. The purpose of

the parameter P is to penalize the stiffness of intermediate designs such that 0-1 points have

more favorable stiffness-to-weight ratios. Often, a continuation approach is employed where


a series of optimization problems are solved for increasing values of P [James et al., 2009,

2008]. However, a 0-1 solution is not guaranteed in general when using SIMP penalization,

even for large values of the parameter P [Stolpe and Svanberg, 2001b,a].

As in the mixed-integer formulation, the following linear constraint is imposed on the

continuous ply-angle selection variables:

K∑k=1

xijk = 1, i = 1, . . . ,M, j = 1, . . . , N. (4.4)

This constraint ensures that the weights are a partition of unity and that the design variables

may be used to obtain a reasonable interpolation of the material properties. In the discrete

case, this constraint is sufficient to ensure that a single material is active. However, in

the continuous case, this constraint only forces the design variables to remain on a plane

intersecting the coordinate axes at unity.

In the design problem all the design variables are collected into the design vector x ∈RMNK and all the linear constraints (4.4) for each design patch and each layer are assembled

into the following matrix expression:

Awx = e, (4.5)

where Aw ∈ RMN×MNK is a matrix and all the entries in the vector e ∈ RMN are unity.

In the proposed approach, the SIMP penalization is augmented with an exact penalization

technique. In order to force the design towards a 0-1 solution, I introduce the following

additional constraint:

K∑k=1

x2ijk = 1, i = 1, . . . ,M, j = 1, . . . , N. (4.6)

The conditions that the design variables remain on the interval, xijk ∈ [0, 1], sum to unity,

and remain on the unit (K − 1)-sphere, are sufficient to ensure that only one ply selection

variable, xijk, is active in each layer. In fact, the upper limit on the design variables xijk

is redundant and may be dropped. These criteria are shown graphically in Figure 4.1, for

K = 3, as the intersection of a 2-sphere and a plane for x1, x2, x3 ≥ 0.

For ease of presentation, the spherical constraints for all layers in all design segments,

are collected into a single vector constraint written as follows:

cs(x)− e = 0, (4.7)

where cs(x) ∈ RMN and e ∈ RMN .


x1

x2

x3

x21 + x22 + x23 = 1

x1 + x2 + x3 = 1

Figure 4.1: An illustration of the spherical constraint, forcing a selection of a single ply angle

variable for each layer. The constraint generalizes to arbitrary dimensions beyond K = 3.

If the objective of interest is f(x), and any additional design constraints are written as

h(x) ≥ 0, the design optimization problem, with the constraints (4.5) and (4.7), is:

minimize f(x)

w.r.t. x ≥ 0

s.t. h(x) ≥ 0

cs(x)− e = 0

Awx− e = 0

(4.8)

The difficulty with this problem is that the spherical constraints (4.7) are highly nonlinear

and introduce many local minima. In order to control this effect, I relax the spherical

constraint (4.7) and introduce it through an exact `1 penalty function with penalty parameter

γ. The objective of this modified problem is f(x)+γ||cs(x)−e||1 where || · ||1 is the `1 norm.

However, this modified objective is not differentiable. Instead, I use an elastic programming

technique [Gill et al., 2005], that creates the effect of the `1 norm in a differentiable manner by

adding additional slack variables to the optimization problem. Using the elastic programming

approach, I introduce the vectors of slack variables s+, s− ∈ RMN such that

cs(x)− e = s+ − s−, (4.9)

where s+, s− ≥ 0. The slack variables s+ and s−, represent the positive and negative con-

straint violation of Equation (4.7).


The modified optimization problem then becomes:

minimize f(x) + γeT (s+ + s−)

w.r.t. x, s+, s− ≥ 0

s.t. h(x) ≥ 0

cs(x)− e = s+ − s−

Awx− e = 0

(4.10)

where the parameter γ > 0 is a penalty parameter. For a feasible problem, with a sufficiently

large, but finite value of γ, Problem (4.10) admits solutions, x∗, s∗+ = s∗− = 0, that are also

solutions to Problem (4.8). However, as γ → 0, Problem (4.10) admits solutions that are not

solutions to Problem (4.8) and do not satisfy the 0-1 criteria. To address this problem, I solve

the `1 penalized optimization problem (4.10) for increasing values of penalty parameter. For

small γ, this will allow greater freedom in exploring the design space, but with increasing γ,

the infeasibility measure ||cs(x)− e||1 will decrease.

A further simplification of Problem (4.10) can be achieved when the summation con-

straints (4.5) are satisfied exactly at every iteration. Starting from Equation (4.5), the sum

of the squared design variables must be less than one, i.e.

1 =

(K∑k=1

xijk

)2

≥K∑k=1

x2ijk.

As a result, when the linear constraint (4.5) is satisfied exactly, the constraint violation of

Equation (4.7) is negative, i.e. cs(x) − e ≤ 0. Therefore, the values of the slacks at the

solution are:

s∗+ = 0,

s∗− = e− cs(x∗).

This result can also be observed geometrically. Whenever the design lies on the plane, the

distance from the plane to the sphere is strictly positive, unless it is at a 0-1 point when the

normal distance is precisely zero, see Figure 4.1. Using this result, the optimization problem

Opt(γ), can now be written as follows:

minimize f(x) + γeT (e− cs(x))

w.r.t. x ≥ 0

s.t. h(x) ≥ 0

Awx ≡ e

(4.11)


where the final constraint is written as Awx ≡ e to indicate that it is satisfied at every

iteration.

I use a continuation approach and solve Opt(γn) for a sequence of increasing γn, starting

each subsequent optimization problem from the previous solution. In this work, I use the

sequential quadratic optimization code SNOPT [Gill et al., 2005], through the Python-based

wrapper in the optimization package pyOpt [Perez et al., 2012]. SNOPT is designed to

satisfy all the linear constraints exactly at every iteration.

4.2.1 Failure prediction

The prediction of structural failure is a critical aspect of preliminary structural design. In

this work, I employ a single conservative failure envelope that can be used to ensure that no

material occurs within the laminate. Note that it is also possible to construct a layer-wise

failure envelope within the context of this parametrization [Kennedy and Martins, 2012].

In this work, I use the Tsai–Wu failure criterion [Jones, 1996] to test if the lamina stresses

are within the failure envelope, written as

F (σ) ≤ 1. (4.12)

In practice, any failure criterion that takes the form of Equation (4.12) could be used.

To construct the overall laminate failure envelope, I apply the failure criteria at all

angles θk ∈ Θ, at the upper and lower surfaces of the laminate. Instead of applying each of

these criteria independently, I aggregate them into a single function using a Kreisselmeier–

Steinhauser (KS) aggregation technique [Wrenn, 1989]. This provides a conservative failure

envelope, but does not account for the variation of ply angles within the laminate. This

conservative failure envelope can be written as follows:

F(i)KS = KS(F (σ(p)), ρ) ≤ 1, (4.13)

where the aggregation takes place over the range p = 1, . . . , 2K. Here, σ(2k−1) and σ(2k)

are the laminae stresses at the angle θk on the top and bottom surfaces respectively. The

function KS( · , ρ) is the KS aggregation function with parameter ρ. Equation (4.13) pro-

vides a conservative failure envelope in the sense that when the laminate stresses are within

the envelope, all laminae within the layup are within the failure envelope represented by

Equation (4.12).


4.3 Adjacency constraints

In this section, I introduce a new constraint formulation that is designed to enforce additional

manufacturing requirements on the stacking sequence. This constraint restricts the allowable

ply angles in one layer, based on which ply angle is active in an adjacent layer. Since these

constraints are always imposed between adjacent layers, I call them adjacency constraints.

It may be necessary to impose adjacency constraints on a lamination sequence to prevent

large changes in the ply angles between layers, or to prevent repeated layers at the same ply

angle. The implementation of this type of adjacency constraint is described below.

Without loss of generality, consider first layer, j = 1, of the design segments, i = 1 and

i = 2, with design variables x11k and x21p respectively. If the design variable x11k is active

at the solution, then the purpose of the adjacency constraint is to restrict available choices

in the next design segment to some reduced set of options. Given that the kth-ply is active,

the set of design options that cannot be used in the adjacent segment is given by the set of

design variable indices Ik. Using the set Ik, this type of adjacency constraint can be imposed

as follows:

x11kx21p ≤ 0, k = 1, . . . , K,

p ∈ Ik.(4.14)

This type of constraint, in combination with the condition xijk ≥ 0, is known as a comple-

mentarity constraint [Scheel and Scholtes, 2000, Coulibaly and Orban, 2012]. The less-than

condition is used to conform to a standard complementarity constraint formulation. Un-

fortunately, complementarity constraints violate conventional constraint qualifications such

as the Mangasarian–Fromovitz constraint qualification (MFCQ) or the linear independence

constraint qualification (LICQ) [Nocedal and Wright, 1999, chap. 12]. As a result, these

types of constraints do not admit Lagrange multipliers at the solution and gradient-based

optimizers may encounter difficulties [Scheel and Scholtes, 2000, Scholtes, 2001, Coulibaly

and Orban, 2012].

Instead of using the complementarity constraint (4.14) directly, I use a regularization of

the constraint due to Scholtes [2001]. In this regularization technique, the original comple-

mentarity constraint is perturbed in the following manner:

x11kx21p ≤ τ, k = 1, . . . , K,

p ∈ Ik,(4.15)

for τ > 0. In the approach of Scholtes [2001], a series of optimization problems are then solved

for decreasing values of τ using a conventional SQP optimizer, starting each new problem


from the previous solution. This series of perturbed problems converges to a solution of the

original problem with some conditions on the linear independence of the constraint gradients

excluding the complementarity constraints [Scholtes, 2001].

In this work, I consider three options for the set of indices Ik. In the first case, the index

set Ik is constructed so that the plies may only shift L selections between adjacent layers.

In this case, the index set, Ik, is defined as follows:

Ik = 1, 2, . . . , K \ k − L, . . . , k + L. (4.16)

In the second case, the design selections are allowed to wrap around. For instance, if the

ply selection variable x111 is active, then x211 through x21L, as well as, x21N through x21(N−L)

could be active as well. In this case the index set is defined as:

Ik =

k ≤ L k + L+ 1, . . . , K + k − L− 1k ≥ K − L k −K + L+ 1, . . . , k − L− 1otherwise 1, 2, . . . , K \ k − L, . . . , k + L

. (4.17)

The final option is that once a design is active in one layer, it may not be active in the

second layer. In this case, the index set is defined as follows:

Ik = k. (4.18)

The total number of adjacency constraints can be reduced by combining groups of the

constraints (4.15) into a single equivalent constraint. In this work, I use the equivalent

constraint:

x11k

∑p∈Ik

x21p ≤ τ, k = 1, . . . , K. (4.19)

For ease of presentation, all of the grouped adjacency constraints (4.19) are written in the

following form:

d(x) ≤ τ, (4.20)

where d ∈ Rna , where na is the number of adjacency constraints in the form of Equa-

tion (4.19).

The original optimization problem (4.11) with the additional adjacency constraints as

Opt′(γ, τ), can be written as follows:

min f(x) + γeT (e− cs(x))

w.r.t. x ≥ 0

s.t. h(x) ≥ 0

d(x) ≤ τ

Awx ≡ e

(4.21)


The optimization problem Opt′(γn, τn) must be solved for a sequence γn, τn, with non-

decreasing γn and non-increasing τn.

4.3.1 Avoiding intermediate designs

The penalization approach presented in Section 4.2 ensures that for sufficiently large γ,

solutions to the optimization problem with the full set of spherical constraints (4.8), are also

solutions to the optimization problem (4.11). However, the converse is not true. That is,

local optima of the modified `1 penalty problem (4.11) may not be solutions to the original

problem (4.8), even for large values of γ. For optimization problems with certain constraints,

the sequence of solutions to Opt(γn) may converge to a solution for which ||cs(x∗n)−e||1 6= 0

even for large γn. This is a local minima since any feasible, 0-1 point, xf , with ||cs(xf )−e||1 =

0, has an objective value, f(xf ), lower than the penalized objective, f(x∗n)+γn||cs(x∗n)−e||1,

for a sufficiently large value of γn, even if f(x∗n) < f(xf ).

In practice, I have found that Opt(γn) may fail to converge to a 0-1 solution for problems

in which additional constraints are imposed on the ply selection variables. These additional

constraints impose conditions such that any feasible path away from the local solution yields

a higher value of the penalized objective. As a result, the continuation sequence does not

move away from the local minima and the solution does not proceed to a 0-1 point. To obtain

a 0-1 solution, in these cases, I impose an additional constraint which forces the optimum

away from the local minima. For each ply, I add the following complementarity constraint:

K−1∑k=1

xijk

K∑p=k+1

xijp ≤ τ, i = 1, . . . ,M, j = 1, . . . , N.

Note that this constraint is only imposed after a local minima has been detected. This

constraint ensures that as τ decreases, only a single ply selection variable will be non-zero

in each layer. However, by itself, this constraint does not force the design towards a 0-1

solution. For ease of presentation, I collect these constraints for each ply, in every design

segment, into the following vector of constraints:

g(x) ≤ τ, (4.22)

where g(x) ∈ RMN . As τ → 0, points at which more than one ply selection variable is

non-zero will become infeasible. While it would be possible to include this constraint for all

optimization problems, I have found that optimization problems with the constraint (4.22)

tend to require more function and gradient evaluations. As a result, I only include this

complementarity constraint once a local minima with ||cs(x∗n)− e||1 6= 0 is detected.


Stiffness [GPa] Poisson’s ratio Strength [MPa] Density [kg/m3]

E11 164.0 ν12 0.34 Xt 2410 ρ 1580

E22 8.3 Xc 1040

G12 21.0 Yt 73.0

G13 21.0 Yc 173.0

G23 12.0 S 183.0

Ply thickness tp = 0.125 mm

Table 4.1: Representative IM7/3501-6 stiffness and strength properties.

4.4 Structural optimization studies

In the following section, I present a series of structural optimization problems that demon-

strate the proposed laminate parametrization method. These examples involve the design

of composite structures for compliance and buckling. The representative composite material

properties used for all examples in this section are listed in Table 4.1.

This section is structured as follows: In Section 4.4.1, I present results for compliance

minimization of a square composite plate with and without adjacency constraints. In Sec-

tion 4.4.2, I present results of a critical buckling load maximization problem. In all cases,

the finite-element analysis is performed using the Toolkit for the Analysis of Composite

Structures (TACS), described in more detail in Chapter 3. The efficient sensitivity analy-

sis capabilities in TACS are extremely important, since the design problems presented here

involve hundreds of constraints and thousands of design variables. In particular, I use both

the adjoint sensitivity, and eigenvalue sensitivity capabilities in TACS.

In all cases presented here, I do not use SIMP penalization and therefore set P = 1.

In all optimization problems I employ an optimality and feasibility tolerance 10−6. Unless

otherwise noted, all cases presented here converge to a 0-1 solution such that the infeasibility

of the spherical constraint ||cs(x)− e||1 is less than 10−10.

4.4.1 Compliance minimization of a square composite plate

In this section, I consider the compliance minimization of a fully clamped plate that is

subjected to a uniform surface pressure. The number of layers in the laminated plate is

fixed at 8 and the laminate is parametrized with the lamination parametrization technique

discussed in Section 4.2. The plate is sub-divided into 9 × 9 design segments, each with 8


plies. The lamination angles are restricted to the values 0o, ±45o and 90o, resulting in 4

ply selection variables per layer. As a result, there are 2592 ply selection variables with 648

linear weight constrains (4.4).

The plate is 900 × 900 mm and is subjected to a 1 kPa pressure load, positive out of

the page. This puts the bottom layer of the laminate in compression and the top layer of

the laminate in tension at the middle of the plate. Each of the 9 × 9 design patches are

modeled using 3 × 3, 3rd order MITC9 shell elements [Bathe and Dvorkin, 1986, Bucalem

and Bathe, 1993]. The finite-element model contains 725 elements, 3025 nodes and just over

18 000 structural degrees of freedom.

I solve this compliance minimization problem with and without the adjacency con-

straints introduced in Section 4.3. Here I use the second formulation of the adjacency

constraints (4.17), with L = 1 such that the ply angles are only permitted to change by

45o between constrained plies. The adjacency constraints are applied between the plies in

adjoining design segments along the coordinate directions of the plate, but not along the

diagonal. This scheme requires 1152 adjacency constraints.

The compliance minimization problem, denoted CompOpt(γ, τ), is formulated as follows:

minimize α1

2uTKu + γeT (e− cs(x))

w.r.t. x ≥ 0

governed by Ku = f

s.t. d(x) ≤ τ

Awx ≡ e

(4.23)

where K is the stiffness matrix, f is the consistent force vector and α is a scaling parameter,

set to α = 1/20 000, such that the scale function takes on values close to unity.

I solve a sequence of problems CompOpt(γn, τn), starting each new problem from the

solution of the previous iteration. The sequence of penalty parameter values is, γ1 = 0, and

γn = 2n−210−5 for n ≥ 2, while the sequence of regularization parameters is τn = 1/2(0.9)n−1.

Figure 4.2 shows the continuation convergence history, and function and gradient eval-

uations required to solve the compliance minimization problem without the adjacency con-

straints. Note that the infeasibility here is the violation of the spherical constraint (4.6)

measured using the `1 norm, i.e. ||cs(x)− e||1. The compliance minimization problem con-

verges to a 0-1 point within 9 continuation iterations with a final compliance value of 19635.

The main computational cost is incurred in the first two continuation iterations, while the

remaining continuation optimizations are less expensive.


Iteration

Compliance

Infeasibility

1 2 3 4 5 6 7 8 919475

19500

19525

19550

19575

19600

19625

19650

0

50

100

150

200

Compliance

Infeasibility

Iteration

Evaluations

1 2 3 4 5 6 7 8 90

50

100

150

200

250

300

350

Function evaluations

Gradient evaluations

Figure 4.2: The convergence history and function evaluations for the plate compliance problem

with no adjacency constraints. Note that here the infeasibility is measured as ||cs(x∗n)− e||1.

Iteration

Compliance

Infeasibility

1 2 3 4 5 6 7 8 9 1019500

19600

19700

19800

19900

20000

20100

0

50

100

150

200

Compliance

Infeasibility

Iteration

Evaluations

1 2 3 4 5 6 7 8 9 100

100

200

300

400

500

600

Function evaluations


Figure 4.3: The convergence history and function evaluations for the plate compliance problem

with adjacency constraints. Note that here the infeasibility is measured as ||cs(x∗n)− e||1.

Figure 4.3 shows the convergence history, and function and gradient evaluations required

to solve the compliance minimization problem with the adjacency constraints. The compli-

ance problem with adjacency constraints requires 10 continuation iterations and converges

to a 0-1 solution with a final compliance value of 20005. As before, the first two optimiza-

tions are the most computationally expensive, while subsequent optimizations require fewer

objective and gradient evaluations.

Figure 4.4 shows the optimal ply angles for the compliance minimization problem while

Figure 4.5 shows the results with adjacency constraints. Note that the composite plies

are ordered from the bottom to the top of the laminate. Both solutions share some similar

characteristics in the outer layers, where the center ply angles form roughly concentric circles

while the boundary plies are oriented towards the center of the plate. The middle plies of

the two results differ significantly. In both cases, the laminate is non-symmetric through the

thickness.


Figure 4.4: The compliance minimization results for a 900 × 900 mm, 8-ply laminate. The first

row contains layers 1 through 4, while the second row contains layers 5 through 8.

4.4.2 Buckling optimization of a stiffened panel

In this section, I present the results from a series of buckling optimizations of a stiffened

panel using various design constraints. The geometry of the stiffened panel is shown in

Figure 4.6. The panel consists of four equally spaced stiffeners aligned along the x-direction.

The panel is subjected to a prescribed end shortening in the x-direction such that u = −∆,

at x = Lx, and u = 0 at x = 0. The displacements along the y = 0 and y = Ly edges of the

skin are simply supported, while the stiffeners are permitted to elongate in the z-direction

at the ends x = 0 and x = Lx.

The stiffened panel is modeled using a finite-element mesh consisting of 15 840, 3rd order

MITC9 shell elements: 120 along the length, 128 in the transverse direction and 5 through

the depth of each stiffener. The finite-element model contains just over 383 000 degrees

of freedom. The linearized buckling eigenvalue problem is solved on 16 processors using

the parallel capabilities of TACS. The buckling calculation consists of two steps. The first

step is to determine the initial solution path up, due to forces caused by the prescribed

end-shortening fp:

Kup = fp.

Once the solution path up is calculated, the second step is to solve the following linearized

buckling eigenvalue analysis to determine the critical end-shortening, ∆cr, at the lowest


Figure 4.5: The compliance minimization results for a 900×900, 8-ply laminate with the adjacency

constraints. The first row contains layers 1 through 4, while the second row contains layers 5 through

8.

buckling load:

Ku + ∆crG(up)u = 0. (4.24)

Here G(up) is the geometric stiffness matrix which is a function of the initial load path.

The sensitivities of the eigenvalues d∆cr/dx can be determined if the derivatives of the

stiffness matrix and the geometric stiffness matrices are known [Seyranian et al., 1994]. The

most computationally expensive operation during the computation of d∆cr/dx is the calcula-

tion of the derivative of geometric stiffness matrix with respect to the design variables which

requires a contribution from the load-path computation. The derivative of the geometric

X

Y

Z

Ly = 440 mm

Lx = 450 mm

hs = 20 mm

b = 110 mmwb = 35 mm

Figure 4.6: The geometry of the buckling optimization problem formulation.


stiffness matrix can be found as follows:

dG

dx=∂G

∂x+∂G

∂up· dupdx

,

=∂G

∂x+∂G

∂up·K−1 ∂

∂x[fp −Kup] ,

(4.25)

where the operator (·) is used to denote a tensor-vector inner product.

In this buckling problem, I assume that the geometry of the panel and the number of

plies at 0o, ±45o, and 90o are fixed. This problem could arise during the buckling design

of a stiffened panel where stiffness and strength requirements dictate the geometry and ply

content of the panel. The objective is to maximize the critical end-shortening of the panel

by varying the lamination stacking sequence subject to various constraints on the sequence

of ply angles. The thicknesses of the skin, stiffener-base and stiffener are fixed at 24, 30 and

20 plies, respectively. The number of plies in the skin and the stiffener at 0o, 45o, −45o and

90o are 8, 6, 6 and 4, and 10, 4, 4 and 2 respectively. The outer six plies on both sides of

the stiffener form the bottom 6 plies of the stiffener pad. An additional 8 plies are added in

the middle of the stiffener. Note that the laminates in the skin and stiffener are balanced,

while the laminate of the stiffener-base may be non-symmetric.

To obtain laminates with the prescribed number of plies, I impose the following linear

constraint on the ply selection variables:

N∑j=1

xijk = pik, i = 1, . . . , 3, k = 1, . . . , 4, (4.26)

where pik are the number of plies in component i at ply angle θk.

Matrix-cracking can occur in laminates when several contiguous plies are at the same

angle [Haftka and Walsh, 1992]. To obtain laminate sequences that do not contain more

than four repeated plies, I use the following complementarity constraint:

p+5∏j=p

xijk ≤ τ, i = 1, . . . , 3 p = 1, . . . , N − 5. (4.27)

This constraint ensures that over a five-ply range, no more than four identical plies can be

active.

In the following study, I examine four different lamination stacking sequence problems:

Case A Non-symmetric skin, symmetric stiffener

Case B Symmetric skin and stiffener


Case A Case B Case C Case D

Design variables

Skin ply identity 96 48 96 48

Stiffener ply identity 40 40 40 40

Total 136 88 136 88

Constraints

Contiguity constraint (cp(x) ≤ τ) – – 120 88

Local minima constraint (g(x) ≤ τ) – 22 – –

Ply content (Bx = p) 8 8 8 8

Linear weights (Awx ≡ e) 34 22 34 22

Total 42 52 162 118

Table 4.2: Design problem summary for the buckling optimization studies

Case C Non-symmetric skin, symmetric stiffener and no more than four contiguous plies

at the same angle

Case D Symmetric skin and stiffener and no more than four contiguous plies at the same

angle

Each of these optimization problems can be expressed in the following formulation which

I denote BucklingOpt(γ, τ):

maximize ∆cr − γeT (e− cs(x))

w.r.t. x ≥ 0

s.t. cp(x) ≤ τ governed by Kup = fp

g(x) ≤ τ Ku + ∆crG(up)u = 0

Bx = p

Awx ≡ e

(4.28)

where Bx = p are the ply constraints (4.26) and cp(x) ≤ τ are the contiguous ply con-

straints (4.27). Table 4.2 summarizes the design problems for the four buckling optimization

cases.

Note that for Case B, I have added the complementarity constraints, g(x) ≤ τ , from

Equation (4.22) for avoiding local minima. I have found that, due to symmetry and the ply


content constraints, the ±45o layers converge to a local minima with equal weights of 1/2.

Without this additional constraint, Case B does not converge to a 0-1 point, even for large

γ.

∆cr

A: 1.1036 mm B: 1.1010 mm C: 1.0943 mm D: 1.0914 mm

1 2 3 1 2 3 1 2 3 1 2 3

−45o0o

45o

90o1 2

3

Figure 4.7: The optimal ply angle sequences for the buckling optimization problems. Each

solution shows the skin, stiffener-base and skin layups respectively.

In all cases the following sequences of penalty parameters are employed γ1 = 0, γn =

2n−210−5 for n ≥ 2, with a regularization parameter sequence of τn = (1/2)(0.9)n−1.

The lamination sequences for all cases are shown in Figure 4.7, for the skin, stiffener-base

and stiffener laminates respectively. The non-symmetric skin design, Case A, converges to a

slightly better result than the symmetric skin design, Case B. Likewise, the non-symmetric

skin design with ply-contiguity constraints, Case C, converges to a slightly better design than

the symmetric skin design with ply-contiguity constraints, Case D. In both the symmetric

and non-symmetric designs, the redistribution of ply angles results in about a 1% reduction

in the critical end-shortening.

For the skin layup of both Case A and Case B, the 0o plies are placed in the middle, 90o


Iteration

∆cr[m

m]

0 1 2 3 4 5 6 7 8 9 10 11 12 13 141.09

1.092

1.094

1.096

1.098

1.1

1.102

1.104

1.106

Case A

Case B

Case C

Case D

(a) Objective: ∆cr

Iteration

Infeasibility

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140

2

4

6

8

10

Case A

Case B

Case C

Case D

(b) Infeasibility: ||cs(x∗n)− e||1

Figure 4.8: The convergence history of the objective ∆cr, and the infeasibility as measured by

||cs(x∗n)− e||1 for Cases A, B, C and D.

Iteration

Functionevaluations

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140

50

100

150

200

250

300

Case A

Case B

Case C

Case D

Figure 4.9: The number of function evaluations for the continuation iterations for Cases A, B, C

and D.

plies are placed on the exterior, and ±45o placed in between. The difference between Case

A and Case B is that for Case A the 0o plies are offset from the middle and the arrangement

of the ±45o plies is altered.

Case C and Case D also converge to solutions similar to Cases A and B. However, the

ply contiguity constraint forces Case C and Case D to include additional −45o plies in the

middle of the stiffener and skin to break up the large segment of 0o plies in the original

designs. These requirements have a small, negative impact on the buckling performance.

Figure 4.8a shows the continuation history of the objective for all cases. All cases converge

to a 0-1 design within 10 to 14 continuation iterations. Cases A and C and Cases B and

D arrive at the same design after the first continuation iteration. For Cases C and D, with

the ply-contiguity constraints, the objective drops significantly between the first and second


continuation iterations, while for Cases A and B, without the ply-contiguity constraints, the

objective only decreases near the end of the continuation iterations. Note that only the final

objective value represents a physically realizable laminate. All prior continuation iterations

represent intermediate designs.

Figure 4.8b shows the continuation history of the infeasibility of the spherical constraints,

as measured by ||cs(x)−e||1. Note that the infeasibility falls below 10−10 on the final iteration

for all cases. The infeasibility for Cases A and B do not change between the first and eighth

iteration at which point they both decrease rapidly. The infeasibility for Cases C and D,

with ply-contiguity constraints, decreases more slowly. This behavior is due, in part, to the

ply contiguity constraints.

Figure 4.9 shows the number of function evaluations required for each continuation it-

eration for all cases. The number of gradient evaluations and overall optimization cost is

approximately proportional to the number of functions evaluations and is not shown here.

Case A requires a total of 265 function evaluations and 72 gradient evaluations, Case B

requires a total of 368 function evaluations and 94 gradient evaluations, Case C requires a

total of 821 function evaluations and 218 gradient evaluations, and Case D requires a total of

575 function evaluations and 127 gradient evaluations. Clearly the optimizations for Cases

C and D require significantly more function and gradient evaluations than Cases A and B,

where the main additional cost is incurred in the first three continuation steps. On the other

hand, Cases A and B are less computationally expensive and require far fewer function and

gradient evaluations.

4.5 Wing-box optimization

In this section, I present an application of the laminate parametrization presented in Sec-

tion 4.2, to the optimization of a wing-box, subject to aerodynamic loads. In this design

problem, I consider both local buckling and failure constraints using a global-local analysis.

In the global-local analysis, the global model is a finite-element model with smeared stiff-

eners, while the local panel models consist of finite-strip models of the flat, stiffened-panels

with discrete stiffeners. The finite-strip models are used to calculate an allowable buckling-

free envelope and are also used to compute the equivalent smeared stiffness of the panels for

the global finite-element model. The global finite-element model is used to enforce failure

constraints and determine the average loads on the local panels that are used within the

buckling calculations.

Other authors have performed design studies for composite wing-boxes using global-


local analysis and design methods. Liu et al. [2000a] performed a two level, global-local

optimization of a composite wing with unstiffened panels. In their approach, the global

problem was used to size the panels and determine the number of layers at specified angles,

while the local problem was used to maximize the local buckling loads. Later, Liu and

Haftka [2004] performed an equivalent single-level optimization using lamination parameters

to validate the two-level design approach.

In this study, the sizing and layup sequence design of a wing-box are split into two sepa-

rate steps. First, using continuous lamination parameters, the structural weight is minimized

subject to material failure and local buckling constraints. Second, with fixed structural thick-

nesses, a load-factor applied to the force vector is maximized subject to failure and buckling,

by varying the lamination stacking sequence with the proposed laminate parametrization.

The sizing problem determines a structure that satisfies the failure and buckling criteria

with a unit load factor, but does not satisfy ply-contiguity constraints, or continuity of plies

between adjacent structural segments. The purpose of the second design problem is to ob-

tain a lamination sequence that satisfies the ply-contiguity constraints and continuity of the

laminate across adjacent structural segments, without a minimal reduction in the predicted

load-carrying capability of the structure. The highest possible load factor obtained from the

second problem is unity.

The remainder of this section is structured as follows. In Section 4.5.1, I describe the

geometry of the wing-box structure and the design loads. In Section 4.5.2, I present results

from the lamination parameter-based mass minimization problem. Finally, in Section 4.5.3,

I present the results of the laminate stacking sequence optimization.

4.5.1 Geometry, loads and analysis

The geometry of the wing in this study is roughly based on a Boeing 777-200, similar to

the geometry used in the problems in Section 3.3.3. The wing has a 60.9 m span with a

13.6 m root chord and a tip chord of 2.09 m. The wing crank is located at 30% of the

semi-span. The wing structure consists of two spars, 44 chord-wise ribs and top and bottom

skins stretching between the front and rear spars. The front spar and rear spars are located

at 10% and 70% chord offset from the leading edge. I only model the structural box and

omit any leading and trailing edge structure. The geometry of the wing-box structure can

be seen in Figure 4.11.

To simplify the study, only two design loads are used to size the structure: a 2.5g ma-

neuver and a -1g maneuver load, both at full fuel loads. The mass of the aircraft is set


to 300 000 kg at the maneuver conditions. The aerodynamic loads are calculated at the

maneuver conditions using TriPan, a parallel three-dimensional panel code [Kennedy and

Martins, 2010]. The loads are transferred to the global structural model using a consistent

and conservative load transfer scheme based on the work of Brown [1997].

The global finite-element model of the wing consists of 67 584, 3rd order, MITC9 shell

elements, with 266 852 nodes and just over 1.6 million degrees of freedom. This mesh is

constructed with 30 elements chordwise and 14 elements spanwise along each panel with 12

elements through the depth of each rib and spar.

The finite-strip panel models consist of 4 repeating stiffener-bay segments. The finite-

strip models of the stiffened-panels are used to compute the overall axial and shear panel

buckling modes using a sin-series expansion in the axial direction and a cubic polynomial in

the transverse direction [Plank and Wittrick, 1974, Akhras et al., 1993]. The local stiffened

panel models are used to predict both the critical axial load, Nx,cr, and the critical shear

load, Nxy,cr. The global finite-element model, in turn, is used to compute the average panel

loads, Nx and Nxy. Following Stroud and Agranoff [1976], these results are used to form an

approximate buckling-free envelope as follows:

Nx

Nx,cr

+N2xy

N2xy,cr

≤ 1, (4.29)

for each panel. Each of the local panel buckling analyses are computationally inexpensive,

relative to the global finite-element analysis. However, many individual panel analyses are

required to obtain the buckling envelopes for all panels within the structure. As a result,

the panel analyses are distributed in parallel across all the structural processes. Instead

of designing each stiffened panel in the wing-box independently, adjacent panels are linked

together in pairs of two. This design-linking reduces the number of design variables and the

number of panel analyses required.

4.5.2 Preliminary sizing using lamination parameters

To obtain a preliminary structural design, I minimize the mass of the wing-box, subject

to failure and local buckling constraints with lamination parameters. In this study, I use

two in-plane and three flexural lamination parameters for each composite component in the

structure. This formulation enables continuous changes in the structural thickness, while

omitting the details of the laminate sequence design. In this design problem, the outer

mold line of the wing is fixed. However, certain structure-specific geometric variables are

included in the design formulation. These geometric design variables are: the stiffener pitch,


stiffener-pad width and stiffener height. The wing-box sizing optimization problem includes

both failure and buckling constraints. In order to apply failure constraints while using a

lamination parameter-based formulation, I use the failure envelope (4.13) with the set of

angles, Θ = −45o, 0o, 45o, 90o. This approach provides a conservative failure envelope

without a detailed layer-wise analysis.

There are a total of 1014 design variables in the sizing problem. There are 880 design

variables associated with the top and bottom wing skins: 20 design variables for each skin

panels including the stiffener height, the base width, and 5 lamination parameters and 1

thickness variable for each of the skin, stiffener-base and stiffener. There are 2 design vari-

ables for the top stiffener pitch and bottom stiffener pitch, which are linked across all top

and bottom wing panels. Finally, there are 132 thickness variables: a thickness for each

segment of the front and rear spar, and a thickness for each rib.

There are a total of 456 nonlinear constraints in the sizing problem: 396 lamination

parameter feasibility constraints for each skin, stiffener-base and stiffener, in each panel

for the top and bottom skins; and 15 KS buckling envelope constraints and 15 KS failure

envelope constraints at the 2.5g and -1g maneuver conditions respectively.

The sizing optimization problem can be written as follows:

minimize m(y)

w.r.t. y

governed by Kum = fm m = 1, 2

s.t. KS (FKS(σ), 30), 50) ≤ 1

KS

(Nx

Nx,cr

+N2xy

N2xy,cr

, 50

)≤ 1

h(y) ≤ 1

(4.30)

where y are the geometric, thickness, and lamination parameter design variables, m(y) is

the mass of the wing-box, h(y) ≤ 1 are the lamination parameter feasibility constraints, and

m = 1 and m = 2 correspond to the 2.5g and -1g maneuver conditions respectively.

I solve the sizing optimization problem on 64 processors in approximately 9 hours of wall

time, with 709 function evaluations and 501 gradient evaluations. Of the 60 KS failure and

buckling constraints, 43 are active at the solution. Table 4.3 shows the mass breakdown of

the solution. The top skins are significantly heavier than the bottom skins due, in part, to the

buckling constraints and the lower strength of the composite under compression than under

tension. Additional load cases, and consideration of the engine and landing-gear installation,

would modify these results significantly.


Component Mass [kg] Percentage

Top skin 5377 50.7

Bottom skin 2885 27.2

Ribs 593.3 5.6

Spars 1746 16.5

Total 10601

Table 4.3: Mass component breakdown for the lamination-parameter-based sizing of the wing-box

structure. These quantities are based on the total weight of the entire wing, not just the semi-span.

spanwise station

numberofplies

0 5 10 15 20 25 30 35 400

25

50

75

100

125

150

175

top skin plies

top stiffener plies

bottom skin plies

bottom stiffener plies

(a) Number of plies

spanwise station

dim

ension[cm]

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

top stiffener height

top base width

bottom stiffener height

bottom base width

(b) Stiffener dimensions

Figure 4.10: The number of plies for the top and bottom skin and stiffener, and the top and

bottom the stiffener heights and stiffener base-widths.

Figure 4.10a shows the number of plies on the top and bottom skins and stiffeners for

each spanwise station out the wing. The top skin and stiffener thicknesses are significantly

larger than the bottom skin and stiffener thicknesses. The largest top skin and stiffener

thicknesses occur at the wing crank. Away from the wing-crank, the skin and stiffener

thickness increase gradually towards the wing crank and decrease gradually out towards the

wing tip. The distribution of the stiffener heights and stiffener base widths are shown in

Figure 4.10b. The stiffener heights for the top skin are largest at the root, decrease near the

wing crank, and increase to a local maximum at between the 30th and 35th spanwise station.

The bottom stiffener heights increase gradually towards the wing crank, jump suddenly at

the wing-crank, and then decrease towards the wing tip. The changes in the design that occur

at the wing crank are due to the increase in running loads in the top and bottom stiffened


Figure 4.11: An illustration of the wing-box analysis for the 2.5g maneuver load. The figure

shows the in-plane resultant Nx out the span of the wing and the finite-element discretization with

3rd order MITC9 shell elements.

wing structure. Almost everywhere else in the wing, buckling and strength constraints are

simultaneously active, while at the wing crank, only the strength constraints are active. The

changes in stiffener dimensions and skin thickness at the wing crank save structural weight,

while meeting the failure constraint.

Figure 4.11 shows the force resultant out the span of the wing for the displaced structural

solution for the mass-minimization problem, under the 2.5g maneuver load. Figure 4.11 also

illustrates the global-local analysis, where the panel-level buckling loads are calculated using

the finite-strip analysis.

4.5.3 Lamination sequence optimization

In this section, I present the results of lamination sequence design optimization study using

the laminate parametrization method presented above. I consider two optimization problems.

In the first problem, the thickness distribution, stiffener height distribution and stiffener base

width distribution are fixed based on the values determined from the mass-minimization

results presented in the previous section. In the second case, the thickness distributions are

fixed, but the stiffener height, stiffener base-width and stiffener pitch are allowed to vary. In

this second formulation, a mass constraint is also imposed such that the mass of the structure

does not change.


In the lamination sequence design problem, I link groups of plies together and apply the

lamination parametrization to these ply groups instead of the individual plies themselves.

For modeling purposes, the lamination sequences must be balanced and symmetric. To

enforce this condition, I group the plies as follows: 0o2, ±45o and 90o

2, and construct the

sequence symmetrically about the middle of the laminate. In addition, I apply a contiguity

constraint such that no more than four plies may be contiguous. In order to obtain the initial

thickness distribution, I round up the thickness distribution to the nearest even multiple of

the ply thickness.

In this design, I use one lamination sequence for each of the following components: the

upper skin, the upper skin stiffeners, the lower skin, and the lower skin stiffeners. Changes in

thickness are accomplished by adding or removing plies from the outer-most portion of the

laminate, symmetrically on both sides. This is designed to model a situation in which a layer

in a single ply could be extended over the entire wing-box skin or stiffener. More realistic

ply-blending schemes should be investigated, but this simple blending scheme provides a

starting point for future work.

The objective of this optimization problem is to maximize the load factor λ, subject to

failure and buckling constraints. The wing-box optimization problem can be expressed in

the following form, denoted WingBoxOpt(γ, τ):

maximize λ− γeT (e− cs(x))

w.r.t. λ,x,y ≥ 0

governed by Kum = λfm m = 1, 2

s.t. KS

(Nx

Nx,cr

+N2xy

N2xy,cr

, 50

)≤ 1 m(y) = mtarget

KS (FKS(σ), 30), 50) ≤ 1 cp(x) ≤ τ

Awx ≡ e

(4.31)

where y are the stiffener dimensions, m(y) is the mass, and mtarget is the initial mass of the

wing-box.

There are a total of 526 design variables in this optimization problem. There are 101 ply

groups, 33 for the top skin, 40 for the top stiffener, 13 for the bottom skin and 15 for the

bottom stiffener. These 101 groups result in 303 design variables. There are 222 geometric

design variables, the stiffener height and stiffener base width for 44 wing panels, the two

stiffener pitch variables for the top and bottom wing skin, and the 132 thicknesses variables

for the front spar, rear spar and ribs. Finally, there is the load factor, λ, which is treated as

a design variable.


Iteration

Loadfactor(λ)

Infeasibility

1 2 3 4 5 6 7 8 9 10 11 12 130.86

0.87

0.88

0.89

0.9

0.91

0.92

0

5

10

15

20

Load factor (λ)

Infeasibility

(a) With stiffener dimensions

Iteration

Loadfactor(λ)

Infeasibility

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.63

0.65

0.67

0.69

0.71

0.73

0.75

0

10

20

30

40

Load factor (λ)

Infeasibility

(b) Without stiffener dimensions

Figure 4.12: The continuation history of the load factor, λ, and the infeasibility ||cs(x∗n) − e||1for the wing-box optimization.

Iteration

Evaluations

1 2 3 4 5 6 7 8 9 10 11 12 130

50

100

150

200Function evaluations


(a) With stiffener dimensions

Iteration

Evaluations

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

50

100

150

200

250

300Function evaluations


(b) Without stiffener dimensions

Figure 4.13: The number of function evaluations and gradient evaluations required for the wing-

box optimization.

As before, I solve a sequence of optimization problems, WingBoxOpt(γn, τn), with an

increasing sequence of penalty parameters, γ1 = 0, γn = 2n−210−5 and decreasing sequence

of regularization parameters, τn = 0.5(0.9)n−1. I solve the optimization WingBoxOpt(γ, τ)

in parallel on 64 processors using TACS. Each function evaluation requires approximately

26 seconds of wall time, and each gradient evaluation, requires approximately 45 seconds of

wall time.

Figure 4.12 shows the continuation history of the load factor λn and the infeasibility

||cs(x∗n)− e||1, for the optimization problems with and without the stiffener variables. Fig-

ure 4.13 shows the number of function and gradient evaluations required for each optimization

within the continuation sequence. The lamination sequence optimization problem without

stiffener variables requires approximately 11 hours of wall time, with 1004 function evalua-


tions and 323 gradient evaluations. The solution converges to a 0-1 point with λ∗ = 0.6313

where only one of buckling constraints for the -1g maneuver condition is active at the solu-

tion, out of the 60 KS constraints. The optimization problem with stiffener variables requires

roughly 13 hours of wall time, with 885 function evaluations and 543 gradient evaluations.

The solution converges to a 0-1 point with λ∗ = 0.8682 where 32 out of the 60 KS constraints

are active at the solution. The redistribution of the structural mass permitted in the second

problem enables a higher load factor to be achieved without a change in the structural mass.

Figure 4.14 shows the top half of the symmetric lamination sequences for both optimiza-

tion problems, for the top skin, top stiffener, bottom skin and bottom stiffener respectively.

The changes in skin and stiffener thickness illustrated in Figure 4.10 are obtained by remov-

ing plies symmetrically from the top and bottom of the laminate. Note that the inner layers

of the bottom skin and stiffener exhibit the same lamination sequence, for both the design

cases. This is due to the fact that the optimization problem without stiffener variables is

essentially dictated by a single buckling constraint that is active on the lower skin. This con-

straint effectively determines the entire lamination sequence. For the problem with stiffener

design variables, the lamination sequence for the top skin and top stiffener consist predomi-

nantly of 0o and ±45o plies. The outer layers of the stiffener contain more 90o plies, however,

these plies are only used for a small portion of the thickest part of the wing-skin near the

wing-crank. The bottom skin is mainly composed of 90o and ±45o plies, while the bottom

stiffener is composed of mostly 0o and ±45o plies. These results illustrate that the proposed

laminate parametrization can be used successfully for large, complex wing-box design prob-

lems. However, without variables to modify thickness and structural sizing variables, results

may be sub-optimal. Further investigation should be performed in the future into techniques

that combine the laminate stacking sequence problem and thickness variables.

4.6 Conclusions

In this chapter, I have presented a novel laminate parametrization technique that can be

used to determine the laminate stacking sequence of a layered composite structure. In this

approach, the stiffness and failure criterion are expressed in terms of ply selection variables.

Instead of using discrete variables in the optimization problem, which leads to a nonlinear

mixed-integer formulation, I use a continuous relaxation of the discrete problem and impose

additional spherical constraints so that the solution to the continuous problem must be

a 0-1 solution. Instead of introducing these constraints directly into the problem, I add

them through an exact `1-penalty function so that solutions to the relaxed problem are also


solutions to the penalized problem, for sufficiently large values of the penalty parameter

γ. Additional simplifications can be achieved if the set of linear constraints on the ply-

identity variables are satisfied exactly at every iteration in the optimization problem. This

approach can be used as an independent penalization, or as an additional penalization to

discrete material optimization (DMO) parametrizations that use a SIMP-approach. I applied

the proposed parametrization technique to a series of structural optimization problems of

increasing complexity, including a wing-box lamination sequence optimization. These results

demonstrate that the proposed parametrization method is an effective approach to obtain

0-1 designs for large, complex lamination sequence design problems.


Top Bottom

Skin Stiffener Skin Stiffener

1A 2A 3A 4A 5A 6A 1B 2B

With stiffener geometry

Without stiffener geometry

Top skin and stiffener laminates

1A2A3A

4A 5A 6A

Bottom skin and stiffener laminates1B

2B

−45o0o

45o

90o

Figure 4.14: Ply angle sequences for the wing-box optimization problem. Only the top half of

the symmetric laminate is shown.

Chapter 5

Aerostructural analysis and design

optimization

Aerostructural design optimization using high-fidelity models is a computationally intensive

multidisciplinary design optimization problem. Many authors have focused on using high-

fidelity aerodynamic analysis in the aerostructural problem, while using considerably smaller

finite-element models. This discrepancy is often justified since the primary area of interest in

these studies is the aerodynamic performance of the aerostructural system. Several authors

have devised optimization techniques specifically designed to take advantage of this imbal-

ance [Chittick and Martins, 2008, 2009, Kennedy et al., 2008]. However, it is also important

to examine the consequences of larger and more sophisticated structural analyses, and the

implications these more costly structural models have on both aerostructural solution algo-

rithms and the aerostructural optimization problem. Examining these consequences is an

important research problem, as unconventional aircraft design concepts and the increasing

use of composites and advanced composite systems place additional computational demands

on both the structural and aerodynamic analyses.

In this chapter, I present an aerostructural analysis and design optimization framework

that is designed to be efficient when both the aerodynamic and structural disciplines require

significant computational resources and time. This situation leads to aerostructural solution

and sensitivity methods that separate the computational resources of the disciplines. In the

present work, the aerodynamic analysis uses a parallel, three-dimensional panel code, Tri-

Pan, coupled to the finite-element code TACS, described in greater detail in Chapter 3. The

inter-disciplinary coupling is handled using a parallel implementation of a consistent and

conservative load and displacement transfer technique. While TriPan is a medium fidelity

aerodynamic analysis tool, it does provide accurate induced drag, lift, and pressure distri-

110

Chapter 5. Aerostructural analysis and design optimization 111

bution results at low to moderate Mach numbers. Furthermore, TriPan requires significant

computational time and resources for large aerodynamic meshes. The framework presented

here has been successfully extended and integrated with a high-fidelity aerodynamic anal-

ysis tool, the Stanford University multi-block (SUmb) solver [Kenway et al., 2012]. The

work presented here should be regarded as a first step towards fully integrated high-fidelity

multidisciplinary analysis and optimization, with special attention paid to the structural

problem.

The remainder of this chapter is organized as follows: In Section 5.1, I present a re-

view of significant contributions to the literature in the areas of aerostructural analysis and

design optimization. In Section 5.2, I outline the components of the present aerostruc-

tural design optimization framework, including the aerodynamic analysis performed using

the three-dimensional panel code TriPan, the structural analysis performed using TACS,

the load and displacement transfer approach, and the geometric parametrization technique.

In Section 5.3, I describe the aerostructural solution methods that are used to solve the

coupled aerostructural system. In Section 5.4, I present the coupled aerostructural adjoint

sensitivity method. Finally, I present results for various aerostructural optimization studies

in Section 5.5.

5.1 Review of aerostructural optimization

Many authors have developed methods for aerostructural analysis and design optimization.

Reuther et al. [1999] developed an aerostructural analysis and optimization framework that

coupled a linear finite-element structural model to a finite-volume Euler CFD solver. They

obtained a coupled solution to the aerostructural system using a pseudo-time marching

scheme with periodic updates of the displaced shape. Their structural model consisted of

either solid, three-dimensional elements, to represent a wind tunnel wing model, or shell and

beam elements to represent a stiffened aircraft wing. Following the work of Brown [1997],

they developed a systematic scheme to pass loads and displacements across the aircraft outer

mould line (OML). Martins et al. [2005] developed both an adjoint and direct sensitivity

formulation for the coupled aerostructural system, and implemented a solution method for

the coupled adjoint system based on a block Gauss–Seidel technique. Martins et al. [2004]

applied this framework to the optimization of a supersonic business jet.

Maute et al. [2001] performed an aerostructural analysis that coupled the an Euler CFD

solver to a linear finite-element structural model. They employed a mesh movement strategy

based on a spring analogy and a load and displacement transfer technique following the


earlier work of Maman and Farhat [1995] and Farhat et al. [1996]. Maute et al. used a

nonlinear block Gauss–Seidel method with relaxation for the solution of the coupled nonlinear

equations. Furthermore, they presented formulations of both the adjoint and direct methods

for computing the sensitivities of the coupled aerostructural system. These linear equations

were solved using a block Gauss–Seidel method that mirrors the method of solution for the

coupled system. Two types of structural models were employed: an equivalent flat plate

composed of either a single isotropic material or multiple composite layers and a full shell

and stiffener model composed of shell and beam elements.

Later, Maute and Allen [2004], using a similar aerostructural solution and sensitivity

techniques as Maute et al. [2001], developed an aerostructural optimization problem in which

the internal structure of the wing box was parametrized using a single isotropic material with

penalization (SIMP) approach. Maute and Allen used the SIMP method to determine the

topology of the optimal structure.

In order to improve the robustness and efficiency of the methods developed by Maute

et al. [2001], Barcelos et al. [2006] developed a class of Newton–Krylov–Schur methods for

solving the coupled nonlinear fluid-structure-mesh movement problem. In their approach,

a globalized Newton’s method is used to solve the coupled system. At each iteration, the

linearized system is solved using a Schur complement approach. They found that their

technique is more robust and efficient than the original Gauss–Seidel method presented by

Maute et al. [2001]. More recently, Barcelos and Maute [2008] presented an aerostructural

solution technique coupling a Navier–Stokes solver, including a turbulence model, to a lin-

ear finite-element structural analysis and a mesh movement strategy using a linear spring

analogy.

Aerostructural analysis techniques are a specialization of more general fluid-structure

interaction solution methods. A significant amount of research has focused on this field.

For instance, Felippa et al. [2001] performed an extensive review of solution techniques for

coupled nonlinear problems using partitioned solvers. Kim et al. [2003] developed a solution

procedure for coupled multi-physics problems using a multi-level Newton’s method. They

applied their approach to a coupled fluid-structure interaction problem, noting the impor-

tance of using an accurate linearization of the coupling terms. In two papers, Biros and

Ghattas [2005a,b] presented a Lagrange–Newton–Krylov–Schur approach to the simultane-

ous solution of PDE-constrained optimization problems. They applied their approach to a

design problem using the incompressible Navier–Stokes equations. Heil et al. [2008] solved

a time-dependent fluid-structure interaction problem by applying Newton’s method to a

second-order backward difference discretization of the coupled system. The resulting equa-


tions were solved using a fully-coupled approach, without neglecting coupling terms in the

linearization, with both direct and iterative solvers. They demonstrated that fully-coupled

methods are competitive with loosely-coupled methods that do not use coupling terms in

the linearization, even in cases of relatively weak coupling.

5.1.1 Load and displacement transfer schemes

One of the primary tasks in aerostructural analysis is to develop a scheme to couple the

aerodynamic and structural disciplines into a single analysis. Many authors have devised load

and displacement transfer techniques and a brief overview of some important contributions

is covered below.

Maman and Farhat [1995] developed a method for the direct transfer of loads and displace-

ments between fluid and structural meshes where the boundaries of the two domains may

not be coincident everywhere. In their scheme, pressure from the fluid mesh is transferred

to a projected point on the structural mesh, where a local normal is defined. For displace-

ment transfer, structural displacements are projected back onto the fluid mesh. From the

projected points, local interpolations are used to determine the values of the quantities of

interest. In a second paper, Farhat et al. [1996] devised two methods for load and displace-

ment transfer for transient problems: a method based on consistent interpolation between

coincident surfaces, and a second, more general method based on displacement and load

transfer between discrete surfaces.

Brown [1997] focused on the development of load and displacement transfer schemes

where the structural and fluid model are non-conforming. Brown used displacement interpo-

lation functions that are either a continuous extension of the finite-element shape functions,

or a rigid attachment to the nodal degrees of freedom. Brown constructed the load transfer

from the fluid to structural model using the principle of virtual work. This technique is

consistent (i.e. the sum of the forces on the fluid model is equal to the sum of the nodal

forces on the structure), and is also conservative (i.e. the work done on the fluid model in

moving through the displacements defined by the structure, is equal to the work performed

on the structure).

In two recent papers, Allen and Rendall [2007] and Rendall and Allen [2008] developed

interpolation and mesh movement schemes that employ radial basis functions (RBFs). In

these methods, the structural and aerodynamic models are embedded in an RBF volume.

Structural displacements are transferred through the volume using RBF displacement in-

terpolation. Load transfer is performed using the principle of virtual work. Consistency is


Level Process group Parallelism

1st Optimization Multiple flight conditions and load cases run concurrently

2nd Aerostructural Interdisciplinary coupling and coordination

3rd Discipline Parallel discipline-level analysis

Table 5.1: Levels of parallelism and process groups within the aerostructural optimization frame-

work.

achieved by ensuring that rigid body modes are preserved within the RBF interpolation.

5.1.2 The proposed framework

Here, I present an aerostructural analysis and optimization framework that utilizes three lev-

els of parallelism within a multidisciplinary feasible MDO framework [Cramer et al., 1994].

These levels of parallelism are summarized in Table 5.1. In the first level of parallelism,

different flight conditions are analyzed concurrently. These analyses are independent and

can be performed in an embarrassingly parallel manner. Sophisticated aerodynamic and

structural analysis tools used for large, high-fidelity analysis often have stringent memory

and performance requirements. In order to handle these requirements, I develop techniques

in which the aerodynamic and structural processes are split into non-overlapping process

groups. As a result of this division, the second level of parallelism in the framework is

coordination of the aerodynamic and structural, discipline-level tasks that are required to

perform an aerostructural solution or sensitivity calculation. The last level of parallelism is

at the discipline-level. At this level, efficient single-disciplinary codes may be run on indepen-

dent process groups. I use these three levels to perform efficient, parallel multidisciplinary

analysis and optimization. Efficient multidisciplinary analysis and optimization must exploit

all three levels to attain efficient use of computational resources.

5.2 Aerostructural analysis components

The following subsections outline the relevant details of the aerodynamic analysis, load and

displacement transfer technique, structural analysis and geometric parametrization. These

constitute the independent components of the aerostructural analysis.


5.2.1 TriPan: An aerodynamic panel code

Within the present aerostructural framework, the aerodynamic analysis is performed using

TriPan, an unstructured three-dimensional panel code for calculating the aerodynamic char-

acteristics of inviscid, external lifting flows governed by the Prandtl–Glauert equation [Katz

and Plotkin, 1991]. TriPan uses constant source and doublet singularity elements distributed

over the surface of a body discretized with quadrilateral and triangular panels [Katz and

Plotkin, 1991, Hess and Smith, 1967]. Aerodynamic forces and moments are calculated

using surface pressure integration. The induced drag is calculated using a far-field wake

integration scheme that provides a more accurate drag estimate than surface pressure in-

tegration [Smith, 1996]. The discretized set of boundary conditions governing the doublet

strengths are represented by the vector of aerodynamic residuals,

RA(w,u) = 0, (5.1)

where w is a vector of doublet strengths and u is a vector of the structural displacements.

I solve the linear system represented by Equation (5.1) using the parallel linear algebra

routines in PETSc [Balay et al., 2004, 1997]. A dense matrix storage format is used to

store the aerodynamic influence coefficients for each panel. The matrix is split between

processors such that the rows of the matrix are stored in contiguous segments on different

processors. The ownership range of each segment is determined by performing a domain

decomposition of the surface mesh using METIS [Karypis and Kumar, 1998] to determine

the surface-panel processor assignment. For efficiency reasons, each processor has a local

copy of the entire mesh. The surface mesh is relatively small, so copying the entire mesh

is not too costly in terms of memory when compared to the dense aerodynamic influence

coefficient matrix. I have also implemented an adjoint sensitivity method within TriPan that

may be used to compute the sensitivities of common functions of interest for gradient-based

design optimization.

I solve the linear system of equations represented by Equation (5.1) or the correspond-

ing adjoint system using GMRES(30) [Saad and Schultz, 1986] with a block-Jacobi ILU(0)

preconditioner. The preconditioner is assembled by considering only those panels that are

within a predetermined physical radius of the current panel centroid. The rationale for this

construction is that the closest panels have the strongest effect on a given panel. The effect

of a higher ILU preconditioner level of fill can be achieved by selecting a larger radius.

Figure 5.1 shows the timing results for both an aerodynamic analysis and sensitivity

evaluation for a TriPan model of a straight wing with aspect ratio 10, composed of NACA0012



time[s]

2 4 6 8 10 12 14 16

20

120

220

320

420

520 solution time

adjoint set up time

total sensitivity time

Figure 5.1: Timing results for analysis and sensitivity evaluation using TriPan with a 14 400

surface panel mesh

airfoils. The TriPan mesh contains 14 400 surface panels and 9600 wake panels. These timing

results demonstrate the excellent scaling behavior of both the aerodynamic and sensitivity

analysis in TriPan. Note that the sensitivity evaluation is considerably more expensive than

the aerodynamic analysis itself. This is due to the complexity of the expressions that are

required to evaluate the derivatives of the panel influence coefficients with respect to the

panel surface coordinates.

Figure 5.2 shows a verification between TriPan and the Stanford University multi-block

(SUmb) code [van der Weide et al., 2006] for the ONERA M6 wing at a Mach number of

0.5 and angle of attack of 3.06o. Here, SUmb is used to solve the Euler equations on a mesh

with 3.14 million cells with an off wall spacing of 5× 10−4. The plot shows the sectional Cp

distribution at the stations η = 0.2 and η = 0.8 where η is the span-wise location normalized

by semi-span. The results demonstrate excellent agreement apart from a small discrepancy

at the trailing edge of the wing.

5.2.2 Load and displacement transfer

I have implemented a parallel load and displacement transfer scheme that follows the work

of Brown [1997]. In this approach, the displacements are extrapolated to the aerodynamic

surface mesh through the use of rigid links. These links connect the nodes on the aerodynamic

surface to the closest point on the structure. The nodal displacement on the aerodynamic


X [m]

Cp

0.2 0.3 0.4 0.5 0.6 0.7 0.8

1

0.5

0

0.5

TriPan

SUmb

(a) η = 0.2

X [m]

Cp

0.6 0.7 0.8 0.9 1

1

0.5

0

0.5

TriPan

SUmb

(b) η = 0.8

Figure 5.2: Verification of TriPan against SUmb for the ONERA M6 at M = 0.5 and α = 3.06.

surface may be written in terms of the displacements and rotations defined by an element

as follows:

uA = uS + φS × r, (5.2)

where uA and uS are the displacements of the aerodynamic and structural points, respec-

tively, φS are the rotations at the structural surface and r is the vector of minimum distance

connecting the points in the structure to the points on the aerodynamic surface. Note that

φS may be either the rotations used in the element formulation, as in the case of shell or

beam elements, or the rotation field defined in terms of the x, y, z displacements, as in the

case of three-dimensional elements. Equation (5.2) defines the displacements at the aerody-

namic surface in terms of displacements in the structure. The method of virtual work can

be used to determine a consistent and conservative set of nodal forces and moments acting

on the structure. The virtual work of the aerodynamic pressure forces is

δ W =

∫SA

pn · δuA dS,

=

∫SA

pn · δuS − pn · (r× δφS) dS,

= F(u,w)T δu,

(5.3)

where p is the surface pressure, n is the normal defined on the aerodynamic surface mesh and

F(u,w) are the set of aerodynamic loads on the structural nodes. Note that the integration

in Equation (5.3) is performed on the aerodynamic surface.


The load and displacement transfer technique of extrapolating the displacements to the

aerodynamic surface mesh using Equation (5.2) and computing the set of equivalent nodal

loads using Equation (5.3) is very flexible, since these equations can be used for any arbitrary

combination of aerodynamic surface mesh and structural mesh. Even problems where the

structure lies outside the aerodynamic surface can be connected in this manner. However,

there are two important issues that may arise with this scheme:

1. Adjacent points on the aerodynamic surface may cross when the structure experiences

large rapidly varying rotations. No strain energy is associated with the deformation

of the aerodynamic surface. Therefore, there is not necessarily any structural stiffness

that counteracts the movement of adjacent surface points towards one another.

2. Large point moments may be produced when the structure lies far away from points on

the aerodynamic surface. These large moments are required to maintain consistency

and conservativeness. Usually, this indicates that the model is inadequate and addi-

tional structural elements should be added to the mesh to more accurately represent

the transfer of loads through the physical structure.

For many aerostructural problems where the meshes are close together over most of the

domain, neither of these issues arise. These issues do not affect the results presented in this

work.

5.2.3 Structural analysis

The structural analysis is performed using the finite-element code TACS, discussed in detail

in Chapter 3. However, there are some additional modeling complexities associated with

aerostructural loads. The residuals of the structural governing equations are:

RS(w,u) = RS(u)− F(w,u), (5.4)

where u is a vector of displacements and rotations, RS(u) are the residuals due to conserva-

tive forces and internal strain energy, and F(w,u) are the follower forces due to aerodynamic

loads (5.3).

The Jacobian of the structural residuals involves two terms: the tangent stiffness matrix,

K, and the derivative of the consistent force vector with respect to the structural displace-

ments:∂RS

∂u= K− ∂F

∂u. (5.5)


The second term in this expression, ∂F/∂u, has significant implications for the parallel

implementation of aerostructural solution methods. While the tangent stiffness matrix, K,

is distributed over the group of structural processors, the derivative of the consistent force

vector with respect to the displacements, ∂F/∂u, must be computed using operations that

involve all aerostructural processors. This is due to the load-transfer scheme where the

displacements are extrapolated from the structural processors using Equation (5.2), while

the integral in Equation (5.3) is computed on the aerodynamic processors. As a result,

calculations involving the Jacobian of the structural system require the synchronization of

all processors and cannot be performed concurrently with aerodynamic calculations. It is

often advantageous to ignore the contribution of the follower forces, by excluding the term

∂F/∂u during the analysis. However, for sensitivity calculations this term cannot be ignored

due to its significant contribution to the adjoint system.

5.2.4 Geometry parametrization

The geometric parametrization of the aerodynamic surfaces and structural surfaces and

volumes, including all internal structure, is a key component of aerostructural design op-

timization. Here, I use a CAD-free approach to manipulate the underlying discipline-level

meshes in a continuous and differentiable manner that is well-suited for aerostructural de-

sign optimization problems. The geometric parametrization uses a free-form deformation

(FFD) [Sederberg and Parry, 1986] approach that defines a modification or deformation of

the initial geometry. In the FFD approach, the mesh points for each discipline are em-

bedded in a parametric volume. The control points that define the parametric volume are

then manipulated to modify the embedded mesh points to obtain smooth changes to the

discipline-level meshes. The disadvantage of the FFD approach is that the initial source

geometry representation and the final geometry representation are not the same. However,

the FFD approach is very flexible and can be applied to any mesh without knowledge of the

underlying geometric representation. Furthermore, the FFD approach can be used to obtain

efficient and accurate derivatives of the mesh points with respect to the geometric design vari-

ables. Obtaining these derivatives efficiently and accurately is crucial for multidisciplinary

gradient computation, and is difficult to achieve with CAD-based approaches.

In the following section, I outline a systematic way to manipulate the FFD control points

to obtain geometry changes for an aircraft wing. In particular, I include changes to the

local twist angle, span, chord, thickness-to-chord ratio, dihedral and sweep. In this work,

I use three-dimensional B-spline volumes as the FFD volumes. However, the control point


manipulation scheme presented here could be extended to other parametric volumes, such as

radial basis function volumes. In the proposed scheme, geometric modifications are applied

to the initial set of FFD control points, pijk ∈ R3, to obtain the final set of control points,

Pijk ∈ R3, where all coordinates are given in a global Cartesian reference frame. Chord,

span and thickness-to-chord ratio are modified through an anisotropic scaling of the geometry

along different directions, while twist, dihedral and sweep changes are applied in a consistent

manner that avoids self-intersecting surfaces for large changes to sweep and dihedral and

moderate changes to twist.

To apply these changes in a consistent manner, I employ a series of unit vectors that

define a span-wise direction, ts, a chord-wise direction, tc, and a vertical direction, tv. In

addition, I also employ a series of reference points, rn ∈ R3, for n = 1, . . . , N , connected

by line segments. The geometry modification is divided into two steps: first, the geometry

changes are applied to the reference points, second the location of the initial FFD points,

pijk, relative to the initial reference line segments is used to determine the final position of

the FFD control points. The geometric variables are split into two groups: those given for

each span-wise segment, and those given at each span-wise station. The geometric variables

given for each segment consist of the scaling along the span-wise direction, sn, dihedral, Γn,

and sweep, Λn, while the geometric variables given for each span-wise station consist of the

twist, θn, chord-wise scaling, cn, and vertical scaling, vn.

The following rotation matrix is used extensively in the proposed FFD manipulation

scheme:

C(a, ϕ) = cosϕI + (1− cosϕ)aaT − sinϕa×,

where a ∈ R3 is a unit vector such that aTa = 1, and ϕ is the angle of rotation about the unit

vector a [Hughes, 2004]. Note that this rotation matrix is defined such that the components

of the transformed vector are expressed in the transformed reference frame.

In the proposed scheme, the geometric changes are first applied to the reference line

segments. The difference between adjacent reference line points is denoted, an = rn+1 − rn.

The reference line segment is modified in the following manner: first, the dihedral is applied,

followed by a sweep modification and finally by a span scaling operation. These operations

can be written as follows:

An = snC(b,Λn)TC(tc,Γn)Tan,

where b = C(tc,Γn)T tv is the vertical direction vector rotated through the dihedral angle.

The final reference point locations, Rn, are determined from applying the following update:

Rn+1 = Rn + An, (5.6)


with R1 = r1, for n = 1, . . . , N − 1.

The twist axis, tθ, which defines the axis about which the twist rotation is applied, is

determined by projecting the segment direction, Ak, onto the plane defined by the span axis,

ts, and vertical axis, tv, as follows:

tθ =(tst

Ts + tvt

Tv )An

||(tstTs + tvtTv )An||2. (5.7)

To obtain the final geometry, the vertical axis and the chord axis are scaled and rotated

based on the values of the twist, dihedral, chord and vertical scaling. In the final geometry,

the modified vertical and chord axes are denoted vn and cn, respectively. These vectors are

defined for each segment as follows:

c1 = c1C(ts, θ1)T tc, v1 = v1c1C(ts, θ1)T tv,

cn = cnC(tθ, θn)T tc, vn = vncnC(tθ, θn)TC(tc, Γn)tv,

where, Γn = 1/2 (Γn + Γn+1), ΓN = ΓN .

After the final reference line locations and the transformed chord and vertical axes, cn

and vn, have been calculated, the final FFD control point locations are determined based

on the values of the following projections:

us =tTs

tTs an(pijk − rn),

uc = tTc (pijk − rn − usak),uv = tTv (pijk − rn − usak),

where us is the projection onto the span direction, uc is the projection onto the chord direction

and uv is the projection onto the vertical direction. If 0 ≤ us < 1, then the following update

is applied:

Pijk = Rn + usAn + uc((1− us)cn + uscn+1) + uv((1− us)vn + usvn+1). (5.8)

If us < 0 or us ≥ 1, then Pijk is unmodified by the segment. Figure 5.3 shows the FFD

volume points and reference points and line segments for an initial straight wing, and a

modification of geometry to a swept C-wing with taper and a crank.

5.3 Aerostructural solution methods

The aerostructural residuals are the concatenation of the aerodynamic and structural resid-

uals, represented by:

R(q,x) =

[RA(w,u,x)

RS(w,u,x)

]= 0, (5.9)


(a) Initial FFD points (b) Final FFD points

(c) Initial shape (d) Final shape

Figure 5.3: A geometry modification from an initial straight wing to a swept C-wing with taper

and a crank.

where RA and RS are the aerodynamic and structural residuals, w and u are the aerodynamic

and structural state variables and x is a vector of design variables. Often, x will be omitted

for brevity. Occasionally it will be convenient to combine the unknown state variables into

a single vector, qT = [wT ,uT ].

During the solution procedure, a point is considered converged when the relative tolerance

of both residuals are reduced below a specified tolerance, typically εr = 10−8, such that

||RA(w(n),u(n))||2 < εr||RA(w(0),u(0))||2,||RS(w(n),u(n))||2 < εr||RS(w(0),u(0))||2.

(5.10)

I apply the stopping criterion to each discipline separately rather than the system of

aerostructural residuals as a whole to avoid situations where the initial residual of one disci-

pline is significantly larger than the initial residual of the other. In addition, this condition

is easier to apply when the disciplines are split across groups of processors.

Other authors have concentrated on aerostructural analysis techniques that are suitable

for solving the coupled system when the aerodynamic and structural residuals are distributed


across the same set of processors, or the structural residuals are on every processor [Martins

et al., 2004, Barcelos and Maute, 2008]. Here, I focus on the situation where either memory

or performance requirements dictate that the aerodynamic and structural solvers be split

between groups of processors. This requires an additional level of parallelism reflected in the

solution algorithm.

In the following sections I present two variants of an approximate Newton–Krylov solution

algorithm. A more thorough investigation of other aerostructural solution algorithms is

presented in Kennedy and Martins [2010] where it is demonstrated that the approximate

Newton–Krylov approach is faster and more robust than either the block-Jacobi or Gauss–

Seidel methods.

5.3.1 Approximate Newton–Krylov methods

Newton’s method applied to Equation (5.9) results in the following linear system of equations

for the update ∆q(n),∂R

∂q∆q(n) = −R(q(n)). (5.11)

Newton’s method converges quadratically provided the starting point is sufficiently close to

the solution and the Jacobian remains non-singular. In order to achieve convergence from

points far from the solution, Newton’s method is often globalized with some strategy to

ensure progress is made towards the solution until a suitable starting point is found. I have

found, however, that globalization is not necessary for the aerostructural system (5.9).

In the Newton–Krylov approach, the linearized system (5.11) is solved inexactly using

a Krylov subspace method. The advantage of Krylov subspace methods is that they can

be constructed using only vector operations, matrix-vector products and a preconditioner

operator. In order to obtain an aerostructural solution method that is independent of the

details of the discipline-level solvers, I use a preconditioner based on generic discipline-level

preconditioners. Furthermore, I have implemented the off-diagonal matrix-vector products

and transpose matrix-vector products using a product-rule implementation [Kennedy and

Martins, 2010].

Many authors have used Jacobian-free Newton–Krylov methods where the matrix-vector

products are calculated using a finite-difference calculation. These techniques have been

applied successfully to many nonlinear problems [Knoll and Keyes, 2004]. However, for this

analysis, the formation of the aerodynamic residuals in TriPan is almost as costly as calcu-

lating the exact aerodynamic Jacobian, therefore, I have employed an exact computation of

the block-diagonal components of the aerostructural Jacobian.


Solving Equation (5.11) inexactly is typically more computationally efficient than finding

an accurate solution. Here, I solve the Newton update to a tolerance of εnk = 10−3, and

apply the stopping criterion:

||R(q(n)) +∂R

∂q∆q(n)||2 < εnk||R(q(n))||2, (5.12)

with the update q(n+1) = q(n)+∆q(n). I use the following stopping criterion, for the Newton–

Krylov methods:

||R(q(n))||2 < εrmin(||RA(q(0))||2, ||RS(q(0))||2). (5.13)

The treatment of the off-diagonal blocks requires a more detailed discussion. Since, a

Krylov approach is used to solve the linearized system of equations, only matrix-vector prod-

ucts are required for the linearization of the coupling terms. In the present implementation,

I compute the matrix-vector product [∂RA/∂u] v as the product of two terms:

∂RA

∂uv =

∂RA

∂Xs

∂Xs

∂uv, (5.14)

where Xs are the aerodynamic nodal locations, and v is an arbitrary vector. Note that the

first term, ∂RA/∂Xs, is computed on the aerodynamic processors, while the second term

∂Xs/∂u is computed on the structural processors.

The first term in Equation (5.14), ∂RA/∂Xs, is a dense matrix that is fully populated

due to the nature of the panel method. This term is also required for the adjoint system of

equations. The formation of ∂RA/∂Xs is a time consuming operation, typically requiring

20 to 50 times the computational time of forming the aerodynamic residuals. As a result,

I do not recalculate this term at every iteration. Instead, I recompute ∂RA/∂Xs when

mod(n,m) = 0 for m = 10. This high value of m is chosen since I have found that the

Newton updates computed in this manner are effective, and that for the best computational

performance, ∂RA/∂Xs should be updated only if the aerostructural problem is difficult to

solve. This modification renders this algorithm an approximate Newton method since the

exact Jacobian is formed only every mth iteration. I label this variant of the approximate

Newton–Krylov method ANK 1.

I have found that the ANK 1 method is robust, but that the high-cost of forming

∂RA/∂Xs makes ANK 1 significantly slower than the nonlinear block Gauss–Seidel ap-

proach. In order to achieve better computational performance, I have implemented a method

to approximate the term ∂RA/∂Xs within the aerodynamic code. This approximation is

constructed by computing the contributions from panels only within a given radius of one


another, in a manner analogous to the technique used to form the aerodynamic precondi-

tioner. This reduces the computational cost of this term significantly. I label this second

variant of the approximate Newton–Krylov method ANK 2.

The matrix-vector products with the other off-diagonal block [∂RS/∂w] s, are calculated

using the following series of operations:

∂RS

∂ws =

∂RS

∂fA

∂fA∂w

s = − ∂F

∂fA

∂fA∂w

s, (5.15)

where fA is a vector of the integrated forces over the aerodynamic surface. Note that the

term ∂fA/∂w is formed on the aerodynamic processors, while the term ∂RS/∂fA is computed

on the structural processors. The product of these terms is implemented using a matrix-

free approach in which the matrix components are computed and discarded each time a

matrix-vector product is computed.

At each iteration, I approximately solve the Newton update (5.11), using preconditioned

F-GMRES(60) [Saad, 1993]. The preconditioner is based on a single application of block

Jacobi. In this approach, each discipline applies a block-preconditioner to its own set of

equations. I have found that an effective approach is to use GMRES(5) for the aerodynamic

processors, and one application of the structural preconditioner — usually the direct Schur

method discussed in Section 3.3.2. This set of preconditioning options is set to approximately

balance the time required for the aerodynamic and structural preconditioning operations for

typical problem sizes.

5.3.2 Aerostructural analysis performance

In this section, I examine the parallel performance of the aerostructural solution algorithms

described in the previous section. The results presented here are based on calculations

performed on the GPC at SciNet [Loken et al., 2010] (see Section 3.3.3 for a more detailed

description of SciNet’s configuration.)

I assess the parallel solution performance of the ANK 1 and ANK 2 algorithms for an

aerostructural system consisting of a finite-element structural model that contains 1.6 million

degrees of freedom, and an aerodynamic model that contains 14 440 surface panels and 18 000

wake panels. Figure 5.4 shows the solution times for test cases run with 16, 24, 32, 48, and 64

processors for a stiff structure with a span of b = 60.9 m, and a flexible structure with a span

of b = 81.0 m. For all cases the processors were divided evenly amongst the structural and

aerodynamic process groups. Both the ANK 1 and ANK 2 algorithms exhibit good parallel

scalability. The ANK 2 method performs roughly 1.5 times faster than the ANK 1 method for



solutiontime[s]

16 24 32 40 48 56 64

200

400

600

800

1000

1200

1400ideal

ANK 1

ANK 2

(a) Stiff structure

number of processorssolutiontime[s]

16 24 32 40 48 56 64

200

400

600

800

1000

1200

1400ideal

ANK 1

ANK 2

(b) Flexible structure

Figure 5.4: The solution times for an aerostructural problem with 1.6 million structural degrees

of freedom and 14 400 aerodynamic surface panels for a stiff and a flexible structure

both the stiff and the flexible structure. Furthermore, both methods are robust to changes in

the structure since the flexible structure requires only roughly 1.15 times the computational

time as the stiff structure. These results demonstrate that the approximate Newton–Krylov

method is a robust method for solving the coupled aerostructural system (5.9).

5.4 Aerostructural gradient evaluation

Efficient gradient-based optimization requires the accurate and efficient evaluation of gra-

dients of functions of interest. In the aerostructural optimization problem, there are typi-

cally far fewer objective and constraint functions than there are design variables, therefore,

an adjoint implementation of the sensitivity equations is appropriate. I have developed

an aerostructural adjoint that is based entirely on analytic derivatives, without the use of

finite-difference computations.

The coupled aerostructural adjoint equations can be written in the following form:

∂R

∂q

T

ψ =∂f

∂q

T

, (5.16)

where ψ is the adjoint vector and f(q,x) is either an aerodynamic or structural function of


interest. The total derivative is determined using the additional computation:

∇xf =∂f

∂x−ψT ∂R

∂x. (5.17)

I have implemented the adjoint sensitivity method for aerodynamic lift, drag and moments,

as well as the Kreisselmeier–Steinhauser (KS) function of the structural failure criteria and

buckling envelope.

I use a Krylov method to solve the linear coupled aerostructural adjoint equations (5.16).

In the Krylov approach, the matrix-vector products are computed using the exact Jacobian-

transpose of the coupled aerostructural system. One iteration of a transpose block Jacobi

iteration is used as the preconditioner, with similar settings to those used in the Newton–

Krylov solution method. The adjoint equations are solved to a relative tolerance of εrA =

10−10.

Once the adjoint vector ψ has been determined, the total sensitivities must be computed

using Equation (5.17). This calculation requires the partial derivative of the residuals with

respect to the design variables. I have implemented the aerostructural adjoint equations with

geometric, structural and angle of attack design variables. The geometric design variables

are the most costly to compute, requiring the following calculations:

ψTA

∂RA

∂x= ψT

A

∂RA

∂Xs

∂Xs

∂x, (5.18)

ψTS

∂RS

∂x= ψT

S

∂RS

∂x−ψT

S

∂F

∂fA

∂fA∂Xs

∂Xs

∂x. (5.19)

These computations must again take into consideration the distribution of the disciplines

across multiple process groups. Equation (5.18) involves terms only on the aerodynamic pro-

cesses, while Equation (5.19) involves terms on both structural and aerodynamic processes.

As a result, this operation requires significantly more communication.

5.4.1 Sensitivity performance study

Figure 5.5 shows the results of a sensitivity parallel scalability test run on 16, 24, 32, 48,

and 64 processors. The aerostructural model is the same as that described in Section 5.3.2.

The timing results are shown for both the Trefftz plane drag function and the KS failure

function, and the time required to compute ∂RA/∂Xs and other terms required for the

adjoint sensitivity computations labeled “adjoint set up” time. Note that these adjoint terms

can be reused for each gradient evaluation at the given values of the state variables. Thus,

the adjoint set up time can be amortized over the different gradient evaluations. Figure 5.5



time[s]

16 24 32 40 48 56 64

100

200

300

400

500

600

700 ideal

adjoint set up

drag

KS

(a) Stiff structure

number of processorstime[s]

16 24 32 40 48 56 64

100

200

300

400

500

600

700 ideal

adjoint set up

drag

KS

(b) Flexible structure

Figure 5.5: The computational times for setting up the adjoint, and solution and total derivative

times for the Trefftz plane drag and KS failure function.

shows that the set up time and adjoint solve times scale well over a range of processors.

Furthermore, there is almost no difference in solution or set up times for the stiff and flexible

structures.

5.4.2 Sensitivity accuracy study

I have verified the accuracy of the adjoint sensitivity implementation for aerodynamic and

structural functions of interest using the complex-step method [Squire and Trapp, 1998,

Martins et al., 2003]. In this approach, complex arithmetic is used throughout the entire

code and the total derivative is calculated as follows:

df

dxi=

Im(f(x + ihei))

h+O(h2), (5.20)

where i =√−1, h is a step size, and ei is the ith Cartesian basis vector. The advantage of

this formula (5.20) is that it does not suffer from subtractive cancellation. As a result, very

small step sizes may be used, yielding gradients accurate to machine precision. I have used

h = 10−30 for all results presented here.

Here, I compare the complex-step calculations to the adjoint implementation for a small


Component

Absolute

derivativevalue

Relativeerror

0 2 4 6 8 10 12

102

101

100

101

102

103

1012

1011

1010

109

108

|∇KS|

|∇Lift|

KS rel. err.

Lift rel. err.

Figure 5.6: Aerostructural sensitivity comparison for lift normalized by the dynamic pressure and

the KS function with the the complex step method.

aerostructural problem with 566 surface panels and 1956 structural degrees of freedom. A

small case is chosen to allow rapid testing of all aerodynamic and structural functions.

The first five design variables are aerodynamic twist variables, the next eight are structural

thicknesses and final design variable is the angle of attack. Figure (5.6) shows the absolute

derivative value and relative error for the lift, normalized by the dynamic pressure, and the

KS function. The results demonstrate that the relative error of any gradient component is

less than 10−7. These results also show the large difference in magnitudes between different

gradient components of the same function. These large differences are a common charac-

teristic of aerostructural design problems, since certain design variables may only have an

indirect influence on a function of interest, e.g. thickness design variables and lift.

5.5 Aerostructural optimization studies

It has long been understood that there is a fundamental trade-off between induced drag

and aircraft weight. This trade-off has been examined by many authors. For instance, Jones

[1950] presented an analysis of wings with minimum induced drag for fixed lift and fixed root

bending moment. Jones found that a 15% reduction in the induced drag could be obtained

by increasing the span 15% while keeping the root bending moment fixed. Later, Jones and

Lasinski [1980] presented an analysis of nonplanar lifting surfaces using an integrated bending

moment constraint. Jones and Lasinski found that winglets and wing tip extensions provided


approximately equal reduction in the induced drag for a given integrated bending moment

constraint. More recently, Ning and Kroo [2010] performed an analysis and optimization

of wings with various wing tip devices. They used a calibrated weight model that included

an integrated bending moment calculation and a historical weight correlation to predict the

relative changes in weight of different designs from a baseline configuration.

Other studies have used structural analysis techniques to obtain the flying shape of the

wing, and to size a portion of the aircraft structure and thus predict a partial wing-weight. In

one of the earliest examples of aerostructural optimization, Haftka [1977] compared the trade-

off between structural weight and induced drag for both composite and isotropic wings of a

fighter aircraft. Haftka obtained the displaced shape through an iterative procedure and used

stress constraints to size the aircraft wing skins. More recently, Jansen et al. [2010] presented

optimizations of various nonplanar configurations using a gradient-free optimization method.

They used a calibrated lifting line method to predict both induced and viscous drag, and

analyzed all configurations in the displaced, flying shape.

In this work, I examine aerostructural induced drag minimization of a transonic transport

wing. Induced drag, or drag due to lift, is a portion of the drag that can be attributed to the

generation of lift [Kroo, 2000]. The induced drag, Di, can be computed using the following

formula:

Di =L2

πqb2e, (5.21)

where L is the lift, q is the dynamic pressure, b is the span, and e is the span efficiency

factor. In a steady, level flight condition the lift generated by the aircraft is equal to the

total aircraft weight, L = W . As a result, there is a trade-off between decreasing the induced

drag by increasing the span, b, leading to higher overall structural weight, and decreasing

the structural weight of the wing leading to overall lower aircraft weight. Based on a simple

analysis presented by Kroo [2000], this trade-off occurs when the wing-weight is roughly 1/3

of the aircraft weight. Typical transport aircraft wings constitute closer to 10% of maximum

takeoff weight (MTOW), and are far from the region where this trade-off becomes active.

Instead, more practical aircraft design objectives, such as fuel burn, place a larger emphasis

on structural weight than an analysis of Equation (5.21) would suggest.

In order to focus on the trade-off between induced drag and wing weight, and obtain

designs that have a reasonable span, I include a target mass constraint within the aerostruc-

tural optimization problem. The target mass is set as a fraction of the structural mass of a

wing obtained from a structural optimization under fixed aerodynamic loads. The results of

aerostructural optimizations for different target masses are examined below. In addition, in


order to constrain the viscous drag, I impose a wetted area constraint such that the initial

and final wetted areas must be equal. This constraint does not account for higher-order vis-

cous drag effects due to variation in slenderness ratio or Reynolds number, but does provide

a useful starting point for comparison purposes.

The geometry for all cases presented below is based on a Boeing 777-200 aircraft wing

with an initial span of 60.9 m, a taper ratio of 0.2, and a root chord of 13.2 m. The wing

crank is set at 30% of the semi-span. The initial wing is a linear loft of RAE2822 airfoil

sections, without twist or dihedral. The wetted surface area of the wing is 837.47 m2. The

wing structure consists of 44 equally spaced chord-wise ribs, with front and rear spars located

at 10% and 70% chord offset from the leading edge, respectively. The leading and trailing

edge structure is not modeled. The aerodynamic mesh consists of 4200 surface mesh panels:

60 span-wise panels and 70 chord-wise panels, with 100 stream-wise wake panels for each

trailing edge panel. The structural model consists of 29 216 3rd order, MITC9 shell elements

with a total of 114 556 nodes and just over 687 000 degrees of freedom.

The aerostructural optimization problem consists of an on-design cruise flight condition,

and two maneuver conditions used to size the wing-box: a 2.5g maneuver condition and a

-1g maneuver condition. The cruise condition is calculated at a Mach number of 0.84 at an

altitude of 10 688 m, or 35 000 ft. The maneuver conditions are calculated at an altitude

of 4000 m or approximately 13 100 ft, at a Mach number of 0.9 in order to represent a dive

condition at low altitude. Standard atmospheric conditions are used at these altitudes. The

maneuver flight conditions at this low altitude and high Mach number are possible without

stalling the wing. Within the context of the optimization problem, the angles of attack at

these flight conditions are considered design variables. In all cases, the MTOW of the aircraft

is 297 550 kg. The cruise condition is calculated at mcruise = MTOW − 1/2mfuel, where

mfuel = 171 175 kg, while the maneuver conditions are calculated at MTOW. No inertial

relief from the fuel or self-weight of the structure is included in the calculations presented

here, but should be considered in future work. Note that the Mach numbers at the cruise and

maneuver flight conditions are beyond the range of validity of the Prandlt–Glauert equation,

and future work should consider the use of at least Euler CFD methods. However, using

lower Mach numbers, within a range suitable for the panel method, would lead to high

cruise and maneuver CL values, which would result in unrealistically large sectional pitching

moments at the maneuver conditions.


Span extension Raked wing tip Winglet

Span scaling 1 1 2

Chord scaling 1 1 1

Vertical scaling 10 10 10

Twist 9 9 9

Sweep 1

Structural parametrization 1014 1014 1014

Panel length 44 44 44

Angle of attack 3 3 3

Total 1082 1083 1083

Table 5.2: Summary of the design variables in the aerostrucutral induced drag study

5.5.1 Design parametrization

The variables in the aerostructural design parametrizations are summarized in Table 5.2. I

consider three different geometric parametrizations in this study, one parametrization for a

span extension, a raked wing tip and a winglet, respectively. For each of these parametriza-

tions, there are 10 reference point locations positioned from the root to the tip at the trailing

edge of the wing. The first 3 reference points are positioned uniformly from the wing root

to the wing crank, while the remaining sections are positioned uniformly from the wing

crank to the wing tip. In all cases, the chord scaling variables are linked such that cn = c1,

for n = 2, . . . , 10. For the span extension and raked wing tip parametrizations, the span

scaling variables are linked such that sn = s1 for n = 2, . . . , 9, while for the winglet, an

additional winglet span scaling variable is added such that sn = s1, for n = 2, . . . , 8, with

s9 set as an addition design variable. All parametrizations also use the vertical scaling vari-

ables, 0.75 ≤ vn ≤ 1.25. Since the initial airfoil section has a t/c ratio of approximately

12%, these bounds ensure that the t/c ratio varies between 9% and 15%. These bounds

are imposed to maintain reasonable t/c ratios in the absence of viscous and compressibility

effects that would penalize large t/c values. A series of linear constraints are imposed on

the vertical scaling variables such that the variables v1, v3, v10, are independent, while all

remaining vertical scaling variables are interpolated linearly between these values. Finally,

all parametrizations use the twist design variables, θn, with the root-twist fixed, θ1 = 0, and


Nonlinear constraints Linear constraints

Wetted area 1 t/c linearity 7

Mass constraint 1 Twist linearity 6

Lift constraint 3 Thickness variation 126

2.5g maneuver KS 5 Stiffener height variation 42

-1g maneuver KS 5 Spar variation 86

Lamination parameter feasibility 396 Stiffener dimension 132

Geometric compatibility 44

Box constraints 27

Total 482 399

Table 5.3: Summary of the constraints in the aerostrucutral induced drag study

with a set of linear constraints such that θ3, θ6 and θ10 are used as independent variables,

and all remaining twists are interpolated linearly between these values. For the raked wing

tip case, an additional sweep variable is added to the tip such that 10o ≤ Λ9 ≤ 30o. Finally,

for the winglet case, the dihedral of the last section is set such that Γ9 = 85o.

The structural design parametrization used in this aerostructural optimization study

is identical to the parametrization presented in Section 4.5.2. I use panel-level structure-

specific geometric design variables: the stiffener height and stiffener base width as design

variables, in addition to 5 lamination parameters and one thickness variable for the stiffener,

stiffener base and skin. This results in 20 design variables for each independent panel in

the structure. The designs of adjacent panels are linked in groups of two to reduce the

number of structural design variables. As before, this parametrization has a total of 1014

structural design variables. Unlike the formulation presented in Section 4.5.2, however, the

geometry is not fixed and as a result, the length of the stiffened panels vary. To maintain

geometric compatibility, these panel lengths are added as design variables, and additional

geometric compatibility constraints are added such that the physical panel length, and the

design variable length are equal at a feasible solution.

5.5.2 Constraint formulation

The constraints in the aerostructural design problem are summarized in Table 5.3. There

are a total of 881 constraints in the aerostructural problem: 482 nonlinear dense and sparse


constraints, and 399 sparse linear constraints. All constraints requiring an adjoint solution

are dense, in addition to the wetted area and mass constraints. All other constraints are

treated as sparse. The KS constraints at the 2.5g and -1g maneuver flight conditions consist

of 3 KS failure functions: one aggregated over each of the top skins, bottom skins, and spars

and ribs, and 2 KS buckling functions: one aggregated over each of the top and bottom

skins. In all cases, I use an aggregation parameter of ρ = 50. The lamination parameter

feasibility constraints enforce the feasibility of the lamination parameters, and the geometric

compatibility constraints ensure that the panel lengths correspond to the physical lengths of

the panels. Finally, the box constraints are imposed for all design problems on the control

point locations, but are inactive for all but the span-constrained results.

The linear constraints consist of the constraints to impose the piecewise linearity of the

twist and t/c distributions. The thickness variation, stiffener height variation and spar

variation constraints ensure that the change in thickness and spar height do not exceed

5 mm, or 1 cm between adjacent panels, respectively. Finally, a series of linear constraints are

imposed on the spar height and stiffener width to ensure that they remain within reasonable

bounds.

5.5.3 Summary of the proposed studies

The aerostructural optimization studies presented in the following sections can be written

in the following manner:

minimize Di(q1,x)

w.r.t. x

governed by R(qj,x) = 0 j = 1, 2, 3

s.t.m(x)

mfixed

= β KS (FKS(σ), 30), 50) ≤ 1 j = 2, 3

Lj(qj,x)

qjSref

=njm(x)g

qjSref

+njmfixedg

qjSref

KS

(Nx

Nx,cr

+N2xy

N2xy,cr

, 50

)≤ 1 j = 2, 3

Swet

Swet init

= 1 h(x) ≤ 1

where x are the design variables listed in Table 5.2, β = m/mfixed is the fixed mass fraction,

nj is the load factor, and qj = 1/2ρjV2j is the dynamic pressure. Here, j indexes the

flight condition, where j = 1 corresponds to level flight, n1 = 1, j = 2 corresponds to the

2.5g maneuver condition, n2 = 2.5, and j = 3 corresponds to the -1g maneuver condition,

n3 = −1. Note that the KS constraints for material failure and buckling are applied only


at the 2.5g and -1g maneuver conditions. Finally, h(x) represents the remainder of the

constraints listed in Table 5.3.

Span extension, raked wing tip and winglet designs

I solve the aerostructural induced drag minimization problem with the design formulation

presented in Section 5.5.1 and the constraint formulation presented in Section 5.5.2, for

mass fractions m/mref = 1, 0.95, 0.9, 0.85, and 0.8. Here, mref = 12117 kg is the mass of the

wing obtained from a structural mass minimization problem under fixed aerodynamic loads.

The purpose of these studies is to examine the effect of aeroelastic load alleviation and the

trade-off between larger spans resulting in lower induced drag and structural weight.

Span extension without twist variables

In order to investigate the effect of jig twist and aeroelastic coupling, I also solve the

aerostructural induced drag minimization problem for fixed mass fractions m/mref = 1,

0.95, 0.9, 0.85, and 0.8, without twist variables. The purpose of this study is to examine the

extent to which aeroelastic tailoring can be used to obtain a structurally-favourable, inboard

lift distribution. Furthermore, this study assesses whether raked wingtips on an untwisted

wing can lead to further load-alleviation at the tip.

Span constrained designs

The span of aircraft wings may be limited due to gate constraints or operational require-

ments. When span constraints are present, non-planar configurations, such as winglets,

provide additional induced drag reduction potential [Kroo, 2000, Jansen et al., 2010]. How-

ever, structural weight penalties must be considered with each new design. In order to

explore these possibilities, I also solve the aerostructural induced drag minimization prob-

lem presented above, for mass fractions m/mref = 1, 0.95, and 0.9, with a span constraint of

70 m, imposed on the reference point locations.

5.5.4 Aerostructural induced drag minimization results

In this section, I present the results from the aerostructural optimization studies. All results

here were run using 72 processors on SciNet: 24 processors for each flight condition, with

16 processors for the aerodynamic analysis, and 8 processors for the structural analysis. I

found that starting the aerostructural optimization problem from a good initial design is


essential in order to achieve convergence within a reasonable number of design iterations. In

the present work, I use two types of starting points. For cases with m/mref = 1, 0.9, and 0.8,

I start the optimization with structural design variables set to the minimum mass structural

solution and the aerostructural lift constraints satisfied using the angles of attack, while for

cases with m/mref = 0.95 and 0.85, I start the optimization from the m/mref = 0.9 design.

For all cases presented here, I use the sequential quadratic optimization code SNOPT [Gill

et al., 2005], through the Python-based wrapper in the pyOpt [Perez et al., 2012] optimiza-

tion package. In SNOPT, I use a feasibility tolerance of 10−5 and an optimality tolerance

of 10−4. Note that these tolerances correspond to the maximum scaled optimality gap and

the scaled maximum constraint violation, respectively [Gill et al., 2005]. All cases presented

here converge to the full feasibility tolerance. However, not all of the cases converge to the

full optimality tolerance. The cases that have not fully converged are only accepted if they

satisfy the feasibility tolerance, have converged to below 5 × 10−4, and have exceeded 600

major iterations, which corresponds roughly to 3 days of wall time. I have found that it is

difficult to achieve optimality tolerances for any aerostructural induced drag minimization

problem tighter than 10−4. However, Hicken and Zingg [2010b] found it necessary to converge

aerodynamic-only induced drag minimization problems to tolerances as tight as 10−7. This

incomplete convergence makes it difficult to assess whether two designs are in fact distinct,

or are two points converging towards the same design. Aerodynamic-only optimizations in

TriPan can achieve convergence tolerance as low as 10−7 [Kennedy and Martins, 2010], there-

fore it must be concluded that the nature of the aerostructural induced drag minimization

problem is causing the convergence issues.

Figure 5.7 shows a summary of all the aerostructural optimization studies. Figure 5.7a

shows the normalized drag, D/Dref, as a function of m/mref, where Dref is the aerostructural

drag of the reference wing at the cruise condition with the structure obtained from the struc-

tural mass-minimization problem. Note that the drag for all results decreases for increasing

mass fractions. Further drag reduction is achieved when twist variables are added. These

additional variables give extra freedom to alleviate loading at the maneuver conditions, and

thus enable further span extension. In all cases, the wing rake angle goes to minimum bound

Λ9 = 10o and there is only a small difference between the raked wing tip and the span exten-

sion results. These results demonstrate that from the baseline, structurally optimized wing,

it is possible to achieve either a 43% induced drag reduction with no weight penalty, a 28%

induced drag reduction and 10% wing weight reduction, or a 20% wing weight reduction with

a 5% induced drag penalty. These induced drag improvements are mainly due to increased

spans.


m/mref

D/D

ref

0.8 0.85 0.9 0.95 1

0.6

0.7

0.8

0.9

1

1.1

1.2 Span extension

Raked wing tip

Winglet

No twist variables

With twist

Span constrained

(a) Drag

m/mref

b/b

ref

0.8 0.85 0.9 0.95 1

0.9

1

1.1

1.2

1.3

1.4 Span extension

Raked wing tip

Winglet

No twist variables

Span constrained

With twist

(b) Span

m/mref

e

0.8 0.85 0.9 0.95 1

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1 Span extension

Raked wing tip

Winglet

No twist variables

With twist

Span constrained

(c) Span efficiency

m/mref

mcomponent/m

ref

0.8 0.85 0.9 0.95 1

0.2

0.25

0.3

0.35

0.4

0.45

0.5 Span extension

Raked wing tip

Winglet

Top skin

Bottom skin

Ribs and spars

(d) Mass fraction with twist variables

Figure 5.7: Summary of the results from all the induced drag minimization studies

Figure 5.7b shows the wing span as a function of the target mass fraction, m/mref. The

spans of all designs increase for increasing mass fraction. The span of the wings without twist

variables are slightly larger than the spans with twist variables for m/mref = 0.8 and 0.85.

The designs with twist at these points, however, have lower induced drag due to higher span

efficiency factors. Figure 5.7c shows the span efficiency factor for all cases for increasing

m/mref. The winglet designs exhibit the highest span efficiencies, with the exception of


the span constrained case at m/mref = 0.9. The designs without twist have the lowest

span efficiencies, while the span efficiencies of the designs with twist increase gradually for

increasing m/mref.

Figure 5.7d shows the component masses of the top skins and stiffeners, bottom skins

and stiffeners, and the spars ands ribs as a fraction of the reference mass, for increasing

target mass fraction, m/mref. In Figure 5.7d, only results for the optimizations with twist

variables are shown. The mass of the top skins and stiffeners increases the most rapidly

with increasing m/mref due to the buckling constraints. The mass of the bottom skins and

stiffeners also increase with m/mref, but at a slower rate. The mass of the spars and ribs

decreases for increasing m/mref, due to the decreasing chord lengths and higher-aspect ratios

of the larger-span designs. The smaller chord shortens the ribs, making them lighter. In

practice, aircraft employ fixed rib spacings and the number of ribs would increase for larger

spans. Furthermore, the fixed rib spacing would modify the buckling constraints for the

upper skin panels. These effects are not captured in the present model.

(a) Span extension (b) Raked wing tip (c) Winglet

Figure 5.8: The planforms from the span extension, raked wing tip and winglet parametrizations.

Figure 5.8 shows a comparison between the planforms of the span extension, raked wing

tip and winglet designs. In all cases the span extension designs have the largest wing spans,

followed by the raked wing tip designs, and finally the winglet designs, which have the

smallest spans. For all winglet designs, the winglet height is at the lower bound. From


Figure 5.7a, it is clear that the span extension results have the lowest drag, due primarily

to their larger spans. For the cases presented here, any additional load aleviation provided

by the raked wing tip does not enable lower induced drag at the cruise condition.

spanwise direction [m]

t/c

0 10 20 30 400

0.05

0.1

0.15

m/mref

= 0.8

m/mref

= 0.85

m/mref

= 0.9

m/mref

= 0.95

m/mref

= 1

Figure 5.9: The t/c distribution for the span extension results

Figure 5.9 shows the t/c ratio for the span extensions results for m/mref = 1, 0.95, 0.9,

0.85 and 0.8, respectively. Note that the sharp drop in t/c at the wing tip in all designs is due

to the pinched wing tip geometry employed in this study. All designs exhibit significantly

higher t/c ratios near the wing root, reaching the upper bound at the wing root in all cases

except for the m/mref = 0.8 design. The higher t/c ratios near the wing root enable larger

spans to be achieved. However, as the wing span decreases and the chord increases, the

physical wing thickness becomes larger. For the m/mref = 0.8 case, when the upper bound

on the t/c ratio at the root is not active, the extra spar and rib mass required to increase

the t/c ratio is instead used to increase the span further.

Figure 5.10 shows the thickness and stiffener height distributions for the initial structure,

and thickness and stiffener height distributions for the span extension results with twist

variables for m/mref = 1, 0.9, and 0.8. The raked wingtip cases with twist variables exhibit

similar thickness and stiffener height distributions. Figure 5.10a shows the initial distribu-

tions obtained from the mass minimization with fixed aerodynamic loads. The design has

significantly large thicknesses at the wing crank to compensate for the large torque generated

by the span-wise bending moment. Figure 5.10b, Figure 5.10c, and Figure 5.10d show the

thickness and stiffener height distributions for the aerostructural results for m/mref = 1,

0.9, and 0.8, respectively. For these cases, the thicknesses at the wing break are smaller,


and do not vary as rapidly as the mass-minimization result. This behaviour is due to the

addition of the vertical scaling variables within the aerostructural problem. These additional

variables enable a smoother distribution of the thicknesses and stiffener heights through the

wing crank region, by adjusting the wing thickness through the vertical scaling variables.

The redistribution of the thickness and stiffener heights is most noticeable for the bottom

skin, where the aerostructural distributions are completely smooth through the wing crank

region.

Figure 5.11 shows the span-wise lift distribution and aerostructural and jig twist distri-

butions for the 3 flight conditions for the span extension case with m/mref = 1, 0.9, and

0.8. Note that positive twist is defined as a nose-up rotation that generates more lift. For

each flight condition, the span-wise lift distribution is normalized by the dynamic pressure,

q, which is equivalent to c`c, where c` is the section lift coefficient, and c is the chord. For all

cases there is considerable aerostructural twist, especially at the 2.5g maneuver condition.

While the jig twist distribution changes for the different designs, the aerostructural twist for

the 2.5g maneuver condition remains roughly the same for all cases. For the m/mref = 1,

and 0.9 cases, the aerostructural deflection results in a negative change in twist. This is due

to the small positive lift generated at the tip for the -1g maneuver conditions, resulting in

a negative torque. There is significant wash-out at the 2.5g maneuver conditions, resulting

in greater in-board loading of the wings. These aerostructural lift distributions are more

structurally-favourable, enabling larger spans.

5.6 Conclusions

In this chapter, I have described the implementation of a parallel, aerostructural analysis

and design optimization framework that couples a three-dimensional panel method, TriPan,

to the finite-element code TACS. In particular, I described in detailed the consistent and

conservative load and displacement transfer technique, the geometric manipulation scheme

based on free-form deformation volumes, and the parallel aerostructural analysis and adjoint-

based gradient-evaluation methods. I have demonstrated that the FFD approach can enable

large shape changes for exploratory gradient-based optimization, and that the approximate

Newton–Krylov solution algorithm, ANK 2, that makes use of an approximate Jacobian with

periodic updates, is robust and 1.5 times faster than the ANK 1 method. Furthermore, I

have demonstrated that using a Krylov approach to solve the coupled aerostructural adjoint

system is robust and nearly independent of the flexibility of the structural model. Finally, I

have applied this framework to a detailed aerostructural induced-drag minimization study.


Within the context of this study, I have demonstrated that it is possible to achieve either a

43% induced drag reduction with no weight penalty, a 28% induced drag reduction and 10%

wing weight reduction, or a 20% wing weight reduction with a 5% induced drag penalty from a

baseline wing obtained from a structural mass-minimization problem with fixed aerodynamic

loads.

Chapter5.

Aerost

ructuralanaly

sisand

desig

noptim

ization

142

spanwise station

thickness[m

m]

stiffenerheight[cm]

0 5 10 15 20 25 30 35 400

5

10

15

0

2

4

6

8

10top skin thickness

top stiff height

bottom skin thickness

bottom stiff height

(a) Initial thickness distribution

spanwise station

thickness[m

m]

stiffenerheight[cm]

0 5 10 15 20 25 30 35 400

5

10

15

0

2

4

6

8


top stiff height


bottom stiff height

(b) m/mref = 1

spanwise station

thickness[m

m]

stiffenerheight[cm]

0 5 10 15 20 25 30 35 400

5

10

15

0

2

4

6

8


top stiff height


bottom stiff height

(c) m/mref = 0.9

spanwise stationthickness[m

m]

stiffenerheight[cm]

0 5 10 15 20 25 30 35 400

5

10

15

0

2

4

6

8


top stiff height


bottom stiff height

(d) m/mref = 0.8

Figure 5.10: The thickness and stiffener height distributions for the initial structure and the span extension results with twist variables for

m/mref = 1, 0.9, and 0.8, respectively.

Chapter5.

Aerost

ructuralanaly

sisand

desig

noptim

ization

143

η

twist[degrees]

0 0.2 0.4 0.6 0.8 1

5

0

5

10

15jig

cruise

2.5g

1g

η

clc

0 0.2 0.4 0.6 0.8 1

2

0

2

4

6 cruise

2.5g

1g

(a) m/mref = 1

η

twist[degrees]

0 0.2 0.4 0.6 0.8 1

5

0

5

10

15jig

cruise

2.5g

1g

η

clc

0 0.2 0.4 0.6 0.8 1

2

0

2

4

6 cruise

2.5g

1g

(b) m/mref = 0.9

η

twist[degrees]

0 0.2 0.4 0.6 0.8 1

5

0

5

10

15jig

cruise

2.5g

1g

η

clc

0 0.2 0.4 0.6 0.8 1

2

0

2

4

6 cruise

2.5g

1g

(c) m/mref = 0.8

Figure 5.11: The twist and clc distributions for the span extension parametrization, for m/mref = 1, 0.9, and 0.8, respectively. Note that

these spans correspond to b/bref = 1.33, 1.20, and 0.993, respectively.

Chapter 6

Contributions, conclusions and future

work

In this chapter, I outline the main contributions from my thesis, summarize the main con-

clusions from each chapter and discuss future extensions of the work presented herein.

6.1 Contributions and conclusions

6.1.1 Homogenization-based beam theory

In Chapter 2, I presented a novel beam theory designed to accurately determine the through-

thickness stress and strain distribution in isotropic and composite beams. The proposed

beam theory includes several novel aspects, including the stress and strain representation

in the beam as a linear combination of the fundamental state solutions, and the use of ho-

mogenized stress, strain, and displacement quantities. This homogenization-based approach

enables a rigorous mathematical treatment with assumptions that can be tested against

three-dimensional solutions. The proposed beam theory contains a self-consistent proce-

dure to determine the shear strain correction matrix, and pressure corrections based on the

fundamental states. While some fundamental state solutions can be obtained for simple

geometries, it is difficult to obtain these solutions for realistic cross-sections with complex

geometries. To address this issue, I developed a finite-element approach to determine the fun-

damental state solutions for arbitrary cross-sections. I demonstrated that this approach can

be used to obtain highly accurate, three-dimensional stress and strain distributions in up to

three orders of magnitude less computational time when compared to full three-dimensional

calculations. The accuracy of the stress and strain distributions as well as the computational

144

Chapter 6. Contributions, conclusions and future work 145

efficiency, make this theory a powerful tool for analysis and design.

6.1.2 Structural analysis and design optimization

In Chapter 3, I presented a detailed description of the analysis of thin composite shell

structures. I presented high-order shell element formulations using both a displacement-

based approach and a mixed interpolation of tensorial component (MITC) formulation that

is not susceptible to shear and membrane locking [Dvorkin and Bathe, 1984, Bathe et al.,

2000]. Next, I presented parallel solution methods used to solve the large, sparse, linear

systems resulting from the finite-element discretization of thin shell structures. I outlined

the implementation of the direct Schur method and demonstrated its parallel scalability on

a series of extremely large finite-element problems. I demonstrated that the factorization

time for the direct Schur method is nearly independent of element order. I also discussed

the implementation of a new matrix ordering scheme, which I called AMD-OD. While ND

exhibited the most consistent parallel scalability, the AMD-OD scheme outperformed both

AMD and ND for the 24, 32 and 48 processor cases. Finally, I presented the sensitivity

analysis methods used to compute the derivatives of objectives of interest. The high-order

elements prove to be effective for analysis, yielding the most accurate solutions for a fixed

computational cost. On the other hand, the computational cost of computing the derivatives

of these higher-order elements increases dramatically with element order. Therefore, there

is a trade-off between accuracy of the solution and the computational cost of the gradients

for design problems.

6.1.3 Laminate parametrization

In Chapter 4, I presented a laminate parametrization technique for laminated composite

structures. This parametrization takes into account the discrete nature of the ply-angle vari-

ables that often arise due to manufacturing constraints. Frequently, these ply parametriza-

tion problems are solved with gradient-free approaches [Haftka and Walsh, 1992, Le Riche

and Haftka, 1993, Adams et al., 2004], however, this parametrization results in a continuous

formulation that is amenable to gradient-based design optimization. The parametrization

I developed, uses an exact penalty function to ensure that there are no intermediate plies

in the final design. I also present additional constraints that can be imposed to enforce

other manufacturing requirements such as a restriction on the number of contiguous plies

at the same angle [Haftka and Walsh, 1992], or that adjacent ply angles be restricted to a

reduced set of values. I demonstrated this ply parametrization on a series of problems: the


compliance minimization of a plate, the critical end-shortening maximization of a stiffened

panel, and a layup determination study of a composite wing.

6.1.4 Aerostructural analysis and design optimization

In Chapter 5, I presented an aerostructural optimization framework. This framework in-

cluded a consistent conservative load and displacement transfer scheme, a geometric parametriza-

tion based on free-form deformation volumes, an approximate Newton–Krylov solution al-

gorithm and a gradient-evaluation technique based on the adjoint method. I demonstrated

that an approximate Newton–Krylov method with periodic approximate Jacobian updates,

is fast, robust and exhibits good parallel scalability. In addition, I demonstrated that the

coupled adjoint system can be solved effectively using a Krylov subspace method. I applied

the aerostructural optimization framework to an aerostructural induced-drag minimization

study. I demonstrated the trade-offs between structural wing weight and induced drag. In

particular, the results show that it is possible to achieve a 43% induced drag reduction with

no weight penalty, a 28% induced drag reduction with a 10% wing weight reduction, or a

20% wing weight reduction with a 5% induced drag penalty from a baseline wing obtained

from a structural mass-minimization problem with fixed aerodynamic loads.

6.2 Future work

Dynamic analysis of beams The homogenization-based beam theory should be extended

to include dynamics. The extension to non-uniform through-thickness density distri-

butions would be particularly challenging, but may be important for the analysis of

sandwich structures.

Beam section design optimization The homogenization-based beam theory, in conjunc-

tion with the finite-element sectional analysis, provides a powerful tool for stress and

strain prediction. These analysis tools should be used to perform design optimization

of composite and isotropic cross-sections.

Extensions to curved beams The entire beam theory rests on the assumption that the

fundamental states are far-field solutions. Introducing curvature violates this assump-

tion. An extension of the beam theory to curved beams should be developed to account

for curvature, or to assess the errors in the analysis of curved beams.


Extension to nonlinear analysis The homogenization-based beam theory also relies on

the assumption that there are no material nonlinearities within the beam. For deep

beams, where large strains may occur, modeling material nonlinearity could be of

practical value.

Investigation of iso-geometric analysis for design The iso-geometric technique applied

to shell analysis has been a rapidly developing area of research [Hughes et al., 2005]. A

detailed assessment of the application of this analysis technique for structural design

optimization should be performed.

Simultaneous thickness and sequence design The laminate parametrization presented

in this thesis focused on laminate sequence design with a fixed-thickness distribution.

Better results could be obtained if the thickness-distribution and laminate sequence

design were performed simultaneously. The proposed laminate parametrization should

be extended to enable thickness changes.

Blended laminate designs The proposed laminate parametrization scheme should also

be extended to include a simultaneous sequence and blending design. In this combined

problem, the lamination sequence and the order of ply-additions and ply-removals could

be performed simultaneously.

Geometric parametrization While the free-form deformation approach presented in this

thesis was effective for the aerostructural optimization problems addressed herein, fur-

ther investigation of geometric parametrizations is warranted. In particular, the geo-

metric parametrization of the displaced shape should be considered, and methods for

moving the internal structure should be investigated.

Panel post-buckling analysis The analysis in this thesis has relied on a global-local ap-

proach to buckling analysis, where the local panels are examined for buckling using a

linearized buckling analysis with the local loads from a linear global analysis. These

local analyses should be extended to include possible post-buckling behaviour. This

will be important for the blended-wing-body (BWB) center fuselage sections where

large in-plane loads from wing bending, and pressure loads will act simultaneously.

The correct sizing of these structures will require post-buckling analysis and design.

Nonlinear structural kinematics Nonlinear structural kinematics should be used to ob-

tain the displaced shape of the wing. This will have important effects for large displace-


ments where a linear analysis will over-predict the displaced area due to the linearized

rotations.

Additional load conditions Additional load conditions must be considered within the

aerostructural design optimization framework to obtain realistic structural sizing re-

sults. These additional load cases include, but are not limited to, gust loads, landing

loads and inertial loads.

6.3 Epilogue

In the coming years, new aircraft designs will continue to employ increasing amounts of com-

posites and other advanced materials. Optimization techniques will play an ever-increasing

role in determining how best to use these materials. In this thesis I have endeavored to

develop tools and techniques that will facilitate the optimal usage of composite materials in

future aircraft. I have developed a novel beam theory to enable better analysis of thick com-

posite beams. I have examined parallel solution strategies for large finite-element problems

that may be used to better understand the behaviour of stresses in composite and isotropic

structures. I have developed a parametrization method that incorporates many manufac-

turing constraints that conventional methods do not include. Finally, in an integration of

these contributions, I have developed a parallel aerostructural analysis and optimization

framework that considers the multidisciplinary nature of aircraft design.

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Appendix A

Shell element tests

In this appendix, I present a series of examples of the displacement-based and MITC-based

shell element formulations presented in Section 3.1. Figure A.1 and Figure A.2 show accu-

racy studies for the displacement and MITC-based shell elements for a fully clamped plate

subject to a uniform pressure load, and a cylinder subject to a sinusoidally varying pressure

distribution, p(x, θ) = p0 sin(x/L) sin(θ), respectively. In both test cases, the computations

are performed on distorted finite-element meshes that are not aligned with the coordinate

lines of the natural shell parametrizations. These distorted meshes make for a more diffi-

cult accuracy test. The MITC-based shell elements of all orders perform better than their

displacement-based counterparts. In particular, the low-order MITC shell elements exhibit

far better behaviour than the low-order displacement-based elements.

Figure A.3a shows a comparison of the prediction of the snap-through behaviour of a

partial cylinder subject to a central point load. Note that Horrigmoe and Bergan [1978]

used a coarse mesh due to the limited memory available on computers at the time. I have

run results with the original mesh and a mesh with nine-times as many degrees of freedom

with 4th order MITC shell elements. The original and more refined snap-through behaviour

exhibit similar trends, with the more-refined mesh predicting a smaller snap-through load.

Figure A.3b shows a comparison of the critical pressure loads obtained from the 4th order

MITC shell elements and those calculations performed by Sobel [1964] using the Donnell

equations. The relative difference between the two predictions is also plotted in Figure A.3b.

These results demonstrate that the discrepancy between the two models is between 2% and

4%.

162

Appendix A. Shell element tests 163

∆x [mm]

|ww

h|

5 10 15 2010

4

103

102

101

100

101

102

2ndorder

3rdorder

4thorder

Ideal

(a) Displacement-based shell element

∆x [mm]

|ww

h|

5 10 15 2010

4

103

102

101

100

101

102

2ndorder

3rdorder

4thorder

Ideal

(b) MITC-based shell element

Figure A.1: Accuracy study for a fully clamped plate subject to a uniform pressure on a dis-

torted finite-element mesh. The error is measured using |w − wh| =∫

Ωw − whdΩ. The low-order

displacement based elements exhibit poor convergence behavior, while the MITC elements of all

order perform well.

∆x [mm]

|ww

h|

1 2 3 4 5 610

2

101

100

101

102

103

104

2ndorder

3rdorder

4thorder

Ideal

(a) Displacement-based shell element

∆x [mm]

|ww

h|

1 2 3 4 5 610

2

101

100

101

102

103

104

2ndorder

3rdorder

4thorder

Ideal

(b) MITC-based shell element

Figure A.2: Accuracy study for a cylinder subject to a distributed pressure load, on a distorted

finite-element mesh. The error is measured using |w − wh| =∫

Ωw − whdΩ, where w is the radial

displacement. The low order displacement-based elements perform poorly, while the MITC elements

of all orders perform well.

Appendix A. Shell element tests 164

central deflection

load

fact

or

-30-25-20-15-10-50

-0.4

-0.2

0

0.2

0.4

0.6

Current refinedCurrent coarseHorrigmoe

(a) Horrigmoe snap-through test

L/R

1000

p crR

/Eh

%di

ffer

ence

1 1.5 2 2.5 3 3.5 4

0.4

0.6

0.8

1

1.2

1.4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

computedSobel (Donnell eqns)% difference

(b) Critical pressure load from Sobel

Figure A.3: The load-displacement history for a snap-through of a partial cylinder with 4th order

MITC shell elements, and a comparison of the critical buckling pressure calculated by Sobel [1964]

and that predicted by the present fourth order MITC finite-element formulation.

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Aerostructural analysis and design optimization of ... Aerostructural analysis and design optimization of composite aircraft Graeme James Kennedy Doctor of Philosophy Graduate Department