Composite Optimization of Gearbox Casing used in LMP1

race car under service and FIA regulation load cases

Diogo André Baltazar Gonçalves Domingues

Thesis to obtain the Master of Science Degree in

Aerospace Engineering

Supervisors: Prof. Nuno Miguel Rosa Pereira Silvestre

Prof. José Firmino Aguilar Madeira

Examination Comitee:

Chairperson: Prof. Fernando José Parracho Lau

Supervisor: Prof. Nuno Miguel Rosa Pereira Silvestre

Member of the Committee: Prof. José Arnaldo Pereira Leite Miranda Guedes

November 2015

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Aos meus pais, que tanto me apoiaram

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Acknowledgments

First I would like to express my gratitude to the two institutions that made the realization of this

project possible: the Scientific Area of Applied Mechanics and Aerospace Engineering at Insituto Superior

Técnico – University of Lisbon and the engineering company Optimal Structural Solutions.

Furthermore, I would like to thank guiding teachers Nuno Silvestre and José Aguilar Madeira for

the helpful insights into some of the more technical aspects of this project and for the time dedicated and

interest demonstrated when visiting Optimal Structural Solutions for a project meeting.

I would like to issue a big thanks to the technical supervisor for this project and Engineering

Director at Optimal Structural Solutions engineer António Reis for giving me the opportunity to develop this

thesis in close collaboration with the company, allowing me to accomplish my ambition of working on the

final Aerospace Engineering project at a corporate level.

Additionally, I would like to give a special thank you to Catarina Vicente and André Coelho,

engineers at Optimal Structural Solutions, for their technical support along the course of the project and for

the motivation they transmitted.

Finally I want to thank my family and close friends for encouraging me throughout the duration of

this project.

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Resumo

Este trabalho consiste no processo de optimização do casing da caixa de velocidades usada num

carro de competição da categoria LMP1 do Campeonato Mundial de Endurance. Para tal, são realizadas

optimizações topográficas e compósitas com o objectivo principal de redução de peso da estrutura,

mantendo a sua rigidez e respeitando o critério de falha.

Os estudos são realizadas recorrendo a um software comercial de análise estrutural de elementos

finitos, acopolado a um solver que utiliza avançados algoritmos de optimização.

A estrutura em análise é sujeita a casos de carga de serviço, projectados de forma a replicar

os carregamentos a que o casing está sujeito em condições normais de corrida, e ainda a casos de

carga de homologação delineados pela organização reguladora da competição, a Fédération

Internationale de l’Automobile (FIA).

Esta dissertação foi elaborada em colaboração com o Instituto Superior Técnico e com a

empresa de engenharia Optimal Structural Solutions, que desempenhou um papel importante nos

aspectos mais técnicos do projecto.

Palavras-chave: Optimização topográfica, Optimização de Estruturas Compósitas, Redução de Peso,

Rigidez, Teoria de Falha, OptiStruct

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Abstract

This work consists of the optimization process carried out on the gearbox casing used in an LMP1

race car of the World Endurance Championship. To that effect, the author undertakes topography and

composite optimizations with the objective to reduce the overall weight of the structure, while maintaining

the stiffness of the initial model and satisfying the failure criterion.

The studies are performed using a commercial finite element analysis software, coupled to a

solver that uses advanced optimization algorithms.

The structure under analysis is subjected to service load cases, designed to replicate the loadings

that the gearbox casing must withstand under normal racing conditions, and also to regulation load cases

devised by the sport’s governing body, the Fédération Internationale de l’Automobile (FIA).

This dissertation was developed in collaboration with Insituto Superior Técnico – University of

Lisbon and engineering company Optimal Structural Solutions, who played a very important role in the

most technical aspects of the project.

Keywords: Topography Optimization, Optimization of Composite Structures, Weight Reduction, Stiffness,

Failure Theory, OptiStruct

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Contents

Acknowledgments ................................................................................................................................. v

Resumo................................................................................................................................................. vii

Abstract ................................................................................................................................................. ix

Contents ................................................................................................................................................ xi

List of Tables ...................................................................................................................................... xiv

List of Figures ..................................................................................................................................... xvi

Acronyms ............................................................................................................................................ xix

Nomenclature ...................................................................................................................................... xxi

1. Introduction .................................................................................................................................... 1

1.1. Gearbox Housing ..................................................................................................................... 1

1.2. Objectives ................................................................................................................................ 2

1.3. Literature Review ..................................................................................................................... 2

1.4. Thesis Outline .......................................................................................................................... 3

1.5. MSc in Aerospace Engineering ............................................................................................... 4

2. The Object of Study ....................................................................................................................... 6

2.1. The Gearbox Housing ............................................................................................................. 6

2.2. Dimensions .............................................................................................................................. 8

2.3. FIA Regulations ..................................................................................................................... 10

2.3.1. Gearbox Housing ........................................................................................................... 10

2.3.2. Regulation load cases ................................................................................................... 10

3. Optimization ................................................................................................................................. 14

3.1. Basic Optimization Concepts................................................................................................. 15

3.2. Topography Optimization ...................................................................................................... 16

3.2.1. Variable Generation ....................................................................................................... 17

3.2.2. Topography Optimization Example ............................................................................... 18

3.3. Composite Optimization ........................................................................................................ 19

3.3.1. Stage 1 – Free-size Optimization .................................................................................. 21

3.3.2. Stage 2 – Ply-based sizing Optimization ....................................................................... 24

3.3.3. Stage 3 – Stacking Sequence Optimization .................................................................. 25

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4. Optimization Setup ...................................................................................................................... 27

4.1. Design space and meshing ................................................................................................... 27

4.2. Initial Model............................................................................................................................ 29

4.3. Boundary conditions and load cases ..................................................................................... 30

4.3.1. Regulation load cases ................................................................................................... 31

4.3.2. Service load cases......................................................................................................... 32

5. Optimization Results ................................................................................................................... 36

5.1. Topography Optimization ...................................................................................................... 36

5.2. Composite Optimization ........................................................................................................ 38

5.2.1. Preparatory actions ....................................................................................................... 39

5.2.2. Free-size optimization ................................................................................................... 42

5.2.3. Ply-bundle sizing optimization ....................................................................................... 44

5.2.4. Stacking Sequence Optimization ................................................................................... 47

5.2.5. Composite optimization overview .................................................................................. 49

6. Failure Analysis ........................................................................................................................... 51

6.1. Service load cases ................................................................................................................ 51

6.2. Regulation load cases ........................................................................................................... 52

7. Conclusions and Recommendations for Future Work ............................................................ 59

7.1. Conclusions ........................................................................................................................... 59

7.2. Recommendations for future work......................................................................................... 60

Bibliographic References ................................................................................................................... 62

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List of Tables

Table 4.1 - Mechanical properties of Aluminium ................................................................................... 29

Table 4.2 - Static analysis of initial model under regulation load cases ................................................ 29

Table 4.3 - Service load cases .............................................................................................................. 34

Table 5.1 - Topography optimization results ......................................................................................... 37

Table 5.2 - T1000 and T800 carbon fiber mechanical properties ......................................................... 38

Table 5.3 - Initial model’s compliance values for service load cases .................................................... 39

Table 5.4 - Design space initial laminate ............................................................................................... 40

Table 5.5 - Non-design space initial laminate ....................................................................................... 41

Table 5.6 - Free-size optimization results ............................................................................................. 43

Table 5.7 - Continuous ply-bundle sizing results .................................................................................. 45

Table 5.8 - Discrete ply-bundle sizing results ........................................................................................ 45

Table 5.9 - Stacking sequence optimization results .............................................................................. 47

Table 6.1 - Failure index values for service load cases ........................................................................ 52

Table 6.2 - Failure index values for FIA regulation load cases ............................................................. 52

Table 6.3 - Failure index values for FIA regulation load cases (with reinforcements) .......................... 54

Table 6.4 - Failure index values for FIA regulation load cases (with redesigned reinforcements) ....... 55

Table 6.5 - Failure index values for FIA regulation load cases (with final reinforcements) ................... 56

Table 7.1 - Final results for the stiffness study ...................................................................................... 59

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List of Figures

Figure 1.1 - Gearbox housings used in road cars ................................................................................... 1

Figure 1.2 - Gearbox housings used in racing cars ................................................................................ 2

Figure 2.1 - 2012 Audi R-18 e-tron quattro LMP1 car ............................................................................. 6

Figure 2.2 - Gearbox Housing’s original geometry .................................................................................. 7

Figure 2.3 - Design and non-design spaces............................................................................................ 8

Figure 2.4 - Gearbox housing dimensions (displayed in mm) ................................................................. 8

Figure 2.5 - Gearbox housing dimensions (displayed in mm) ................................................................. 9

Figure 3.1 – General optimization problem ........................................................................................... 16

Figure 3.2 - Topography reinforcements ............................................................................................... 16

Figure 3.3 - Topography design variables ............................................................................................. 17

Figure 3.4 - Element normal method for topography optimization ........................................................ 17

Figure 3.5 - Metallic plate under torsion ................................................................................................ 18

Figure 3.6 - Results of topography optimization of plate under torsion ................................................. 18

Figure 3.7 - Composite Optimization process using Altair’s OptiStruct ................................................ 20

Figure 3.8 - Super-ply concept .............................................................................................................. 21

Figure 3.9 - Super-ply concept .............................................................................................................. 22

Figure 3.10 - Ply-bundles shape editing ................................................................................................ 24

Figure 4.1 - Gearbox housing regions ................................................................................................... 27

Figure 4.2 - 2D Mesh detail ................................................................................................................... 28

Figure 4.3 - 3D Mesh adjustments ........................................................................................................ 28

Figure 4.4 - Single-point constraints (SPC) ........................................................................................... 30

Figure 4.5 - Simulated cover ................................................................................................................. 30

Figure 4.6 – Static side load tests ......................................................................................................... 31

Figure 4.7 - Impact test .......................................................................................................................... 32

Figure 4.8 - Acceleration load case ....................................................................................................... 33

Figure 4.9 - Load center for service load cases .................................................................................... 34

Figure 4.10 - Suspension arms and detailed rigid bodies ..................................................................... 35

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Figure 4.11 - Acceleration and cornering load cases ............................................................................ 35

Figure 5.1 - Mesh adjustment performed in topography optimization ................................................... 36

Figure 5.2 - Reinforcement beads obtained for topography optimization (10mm draw height) ............ 37

Figure 5.3 - Normals of 2D elements .................................................................................................... 39

Figure 5.4 - Material orientation ............................................................................................................ 40

Figure 5.5 - Initial laminates .................................................................................................................. 41

Figure 5.6 - Ply-bundles obtained after free-size optimization .............................................................. 42

Figure 5.7 - Optimized thickness distribution ........................................................................................ 43

Figure 5.8 - Deformed shape of bump load case (multiplied by a scale factor of 100) ......................... 44

Figure 5.9 - Example of ply-editing (ply-bundle 101300) ...................................................................... 44

Figure 5.10 - Manufacturable plies ........................................................................................................ 46

Figure 5.11 - Optimized stacking sequence .......................................................................................... 48

Figure 5.12 - Variation of WCOMP along the composite optimization process .................................... 49

Figure 6.1 - Distribution of composite failure index for FIA regulation load cases ................................ 53

Figure 6.2 - Composite reinforcements ................................................................................................. 53

Figure 6.3 - Redesign of carbon fiber reinforcements ........................................................................... 54

Figure 6.4 - Isolated elements with CFI values greater than (static side load test) .............................. 55

Figure 6.5 - Distribution of composite failure index for impact load case .............................................. 55

Figure 6.6 – Final reinforcements .......................................................................................................... 56

Figure 6.7 – Poor-quality elements with CFI values greater than 1 (impact test) ................................. 57

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Acronyms

CFRP Carbon Fiber Reinforced Polymer

CG Center of Gravity

CFI Composite Failure Index

CAD Computer Aided Design

FIA Fédération Internationale de l’Automobile

FE Finite Elements

FEA Finite Element Analysis

FEM Finite Element Method

TMANUF Manufacturable Thickness

MOGA Multi-Objective Genetic Algorithm

OEM Original Equipment Manufacturers

RIS Rear Impact-absorbing Structure

WCOMP Weighted Compliance

WEC World Endurance Championship

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Nomenclature

𝒈𝒋(𝒙) Constraint response

𝑿𝒏 Design variable

𝑳 Lower bound

𝑵𝑬 Number of finite elements

𝑵𝒑 Number of super-plies

𝒇(𝑿) Objective function

𝑷 Ply percentage

𝝈𝒊 Principal stress

𝑭𝒊 Strength parameter

𝒙𝒊𝒌 Thickness of the 𝑖-th super-ply of the 𝑘-th element

𝑻 Total thickness

𝑼 Upper bound

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1

Chapter 1

1. Introduction

The first chapter covers the introduction of the scope and objectives of this thesis: the study and

optimization of the gearbox housing. In addition, the reader is presented to the objectives for this work,

literature review and thesis structure.

1.1. Gearbox Housing

The gearbox housing is a structural component of a gearbox assembly. It surrounds the

mechanical components of the gearbox (shafts, gears, etc.), providing them mechanical support and

protection from the outside world (against debris, elements) as well as offering a fluid-tight container that

holds the lubricant that bathes those mechanical components.

Depending on the vehicle’s powertrain layout, the gearbox assembly can be placed either at the

front or at the rear of the car. It can also adopt many shapes and sizes, according to the number of gears

that sit inside the gearbox housing and the type of gearbox itself (manual, automatic, etc.). Figure 1.1

shows examples of gearbox housings used in road cars.

a) Front-wheel-drive transaxle gearbox housing b) Rear-wheel-drive longitudinal gearbox housing

Fonte: http://www.gearboxman.co.uk Fonte: http://www.jdmgarageuk.com

Figure 1.1 - Gearbox housings used in road cars

Conventionally, the gearbox housing is made from cast iron or cast aluminium, using methods of

permanent mold casting or shell molding. However, in racing applications (as is the case of the object of

study for this thesis) the gearbox housing can be made out of lighter metals such as magnesium or even

composite materials such as carbon fiber. Figure 1.2 demonstrates examples of gearbox housings used in

racing cars.

In addition to the structural properties mentioned above, in racing applications the gearbox

housing is subjected to service loads that result from cornering, accelerating, braking and bumps and, in

some cases, it might even integrate the rear impact-absorbing structure (RIS) of the car. As a result, its

http://www.gearboxman.co.uk/http://www.jdmgarageuk.com/

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integrity must be evaluated through impact and static loading tests, to ensure that the structure is capable

of withstanding extreme forces without failing.

a) Cast titanium gearbox housing used in Formula 1 b) Composite gearbox housing used in LMP1

Fonte: http://www.ret-monitor.com Fonte: http://.fourtitude.com

Figure 1.2 - Gearbox housings used in racing cars

1.2. Objectives

The main objective of this thesis is to develop an optimized composite version of an existing

metallic gearbox housing, which was used in a 2014 LMP1 Le Mans race car [1]. These cars compete in

what is one of the most demanding forms of motorsport: the World Endurance Championship (WEC) [2].

As a result, engineers and designers always look for the lightest and most efficient solutions [3].

With this in mind, the goal of this work is to significantly reduce the housing’s weight while, at the

same time, maintain its structural stiffness and satisfy the strength criterion. In order to achieve this, the

housing itself is modelled using finite elements (FE) and undergoes several optimization processes,

namely topography and composite optimizations.

The FE modelling and structural optimizations were done in collaboration with engineering

company Optimal Structural Solutions [4]. All the structural calculations and final solutions were obtained

with respect to the regulations set by the sport’s governing body the Fédération Internationale de

l’Automobile (FIA).

1.3. Literature Review

In this section the reader is presented to a brief review of some literary work that was of great

importance to the foundation of the procedures and tools utilized in this project.

Optimization of composites is a process that has seen greater development in recent years. The

papers published by Zhou and Fleury [5-7] are good reviews on the subject. While different forms of

composite materials exist, the predominant usage is composite laminate where thin plies of various

orientations are stacked together to form a shell structure. For shell structures, continuously varying

thickness, termed free-size, provides a meaningful alternative to topology optimization that targets

constant thickness shells (metal structures). Publications on the free-size formulation of shell structures

date back to the 1960s, with the landmark paper of Lucien Schmidt [8] being the first reference. More

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recently, and with the constant increase in finite element analysis (FEA) modelling power, free-size

optimization was further studied by Cervellera, Zhou and Schramm [9] and added to commercial FEA

solvers like OptiStruct. Optimization of simplified composite models was studied as early as 1989 by

Peterson [10-11] and complete composite optimization studies were done in more recent years. A good

example is the work of Radny [12], in which the author studied free-size optimization for composite by

directly applying element wise sizing optimization to each ‘super-ply’ of composite laminates.

An additional type of optimization is performed in this thesis: Topography Optimization. This is a

special class of shape optimization, which can be used to change the sheet metal shapes with various

objectives to the structure, which can involve changes, for example, to its overall stiffness or natural

frequencies. Good reviews of topography optimization can be found in the technical papers of Chakravarty

[13], Krishna [14] and Dai and Ramnath [15].

One of the fundamental tools used in this thesis is the finite element method (FEM), which was

created after the need to solve complex structural problems in both civil and aeronautical engineering. The

first contributions to the study of FEM can be found in the works of Hrennikoff [16] and Courant [17]. While

Hrennikoff presented the lattice analogy, in which a structure was modelled as a framework consisting of a

set of beams, Courant’s approach segmented the domain into a finite number of triangular regions in order

to solve potential differential equations. In the early 1950s, Turner, Clough and Topp [18] and Argyris [19]

published significant developments in the FEM and in 1960 Clough [20] introduced the designation of finite

elements, which has been used since.

1.4. Thesis Outline

This thesis contains seven chapters characterized as follows.

Chapter 2 covers the introduction of the object of study for this project, specifically its shape,

dimensions and positioning in the racing application. Moreover, in chapter 2 the reader is presented to the

regulations tests established by the FIA that the structure must go through to receive certification.

In chapter 3 the reader is introduced to the numerical tools used to perform the structural analysis

process and extensive insights to topography and optimization of composites are also presented, as well

as the fundamental optimization concepts.

Chapter 4 described the introduction of all the pre-processing work that went into the optimization

runs, including the definition of design and non-design spaces, the discretization procedure and the

establishment of boundary conditions and load cases.

In chapter 5 all the results obtained for topography and composite optimizations are presented, as

well as specific preparatory actions that were required for each optimization. An important overview of the

composite optimization process is reported at the end of this chapter.

Chapter 6 covers the failure analysis performed on the final optimized design and showcases the

strategies that were carried out to solve the problems that occurred in the process.

Finally, chapter 7 presents the most relevant results that were obtained from the optimization

method, together with the author’s final conclusions and suggestions for future work.

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1.5. MSc in Aerospace Engineering

The theme of this thesis fits perfectly in the scope of the Master’s degree in Aerospace

Engineering offered by Insituto Superior Técnico – University of Lisbon.

Technical knowledge regarding 3D modelling and the finite element method, obtained in the

courses of Technical Drawing and Mechanical Modelling I and Computational Mechanics, respectively,

was of particular importance in the early stages of this work while editing the preliminary geometry of the

gearbox housing, building the FE model and applying the boundary conditions and load cases.

In the later stages of this thesis, acquired knowledge in Structural Mechanics allowed for the

undertaking of structural analysis of the FE model when subjected to service and regulation load cases,

while the academic experience in Laminated Composite Materials paved the way for the creation of the

composite model of the gearbox housing and further failure analysis.

As the final project of the Master’s degree in Aerospace Engineering, the work performed in this

thesis covers two of the most important stages in the creation of a new and technologically advanced

engineering product: design and testing. After being fully dimensioned and its laminate thoroughly

customized, the final composite model of the gearbox housing revealed total compliancy with the

regulations set by the FIA.

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Chapter 2

2. The Object of Study

In this chapter the reader is presented to the initial shape of the gearbox housing, as well as its

overall dimensions and positioning in the car. Additionally, this chapter also covers the 2014 Procedure for

the approval of safety structures for LMP sports cars, a document established by the FIA stating the tests

required for certification.

2.1. The Gearbox Housing

The gearbox housing used in the LMP1 car was built out of magnesium, a very light metal often

used in motorsport applications (for instance, the wheels on an F1 car). Although a composite housing is

theoretically a better solution, since carbon fiber is lighter and stronger than most metals, its complicated

manufacturing process may deem the composite solution unviable. Given the complex shape of this

gearbox housing, the magnesium solution is a good compromise between performance and manufacturing

efforts. The main objective of this thesis to engineer a composite alternative of this gearbox housing that

can outperform the one used in the real application.

The positioning of the gearbox housing in an LMP1 car can be seen in figure 2.1 a). It shows that

the gearbox housing is located at the very back of the car, between the rear tires. Further back sits the

RIS which is specially designed to withstand the excessive forces that result from a rear impact or side

loads. Given the positioning of the gearbox housing, it is clear that a major part of these forces travels

through its structure. As a result of this, the gearbox housing must perform two important functions:

provide mechanical support for the components that holds within and a fluid-tight container, as well as

deliver enough structural stiffness to the gearbox assembly to prevent damage to its core components in

the event of a strong impact

a) Cutaway (gearbox housing highlighted) b) Monocoque

Fonte: https://garagemonkey.com Fonte: http://img.icpcw.com

Figure 2.1 - 2012 Audi R-18 e-tron quattro LMP1 car

As with any type of motorsport, engineers always try to shed as much weight as possible out of

the car because a lighter car is a faster car. To that end, LMP1 cars are designed with a monocoque

construction, where all the major components, such as suspension arms, engine or aerodynamic parts,

https://garagemonkey.com/

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are bolted directly to the composite safety cell that protects the driver (figure 2.1 b)), without the need for a

tubular frame or a more conventional chassis structure. The gearbox housing itself is bolted to the rear of

the engine which in turn is connected to the rear bulkhead of the monocoque. The gearbox housing

studied in this work also features pickup points for the rear suspension and supports for other components

such as driveshafts.

The gearbox housing studied in this project is shown in the figure 2.2:

a) Front Three-Quarter View b) Back Three-Quarter View

Figure 2.2 - Gearbox Housing’s original geometry

This complex geometry was used in the LMP1 race car and represents the outset for the

optimization work performed in this thesis. As a starting point to this study, and similarly to the real LMP1

application, the housing is modelled as a fully metallic structure, in this case out of aluminium. To that

effect, the geometry of the outside surfaces of the gearbox housing is extracted from the CAD model and

imported to the FEA software. The solid and shell elements used in the initial model are then created into

the inside of the geometry.

This first interpretation of the gearbox housing is subjected to a topography optimization process

in order to determine its optimal shape before performing the composite optimization which will determine

the laminate’s thickness throughout the structure as well as its optimum stacking sequence of plies.

The design space considered for the optimizations is shown in purple in figure 2.3. Finite elements

in this space are modelled as aluminium or carbon fiber shell entities, depending on the optimization

process.

The non-design space is comprised of the two remaining zones. The engine mounting points,

pickup points for the suspension arms and other solid supports are highlighted in light blue and are

modelled in aluminium. The areas displayed in green consist of critical zones of the gearbox housing that

work as reinforcements and are modelled as non-optimizable carbon fiber shell elements in the composite

optimization process.

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a) Front Three-Quarter View b) Back Three-Quarter View

Figure 2.3 - Design and non-design spaces

2.2. Dimensions

Figures 2.4 and 2.5 gives an insight to the gearbox housing’s overall dimensions.

a) Front view

b) Rear view

Figure 2.4 - Gearbox housing dimensions (displayed in mm)

402

25° 25°

394,9

260 230,8

Design space

Solid

supports

2D Non-design

space

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c) Right view

d) Left view

d) Top and bottom views

Figure 2.5 - Gearbox housing dimensions (displayed in mm)

126

363,5

335,6

390,5

316

500,24

521,3

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2.3. FIA Regulations

Before the start of every new WEC season the sport’s governing body, the FIA, releases a new

set of sporting and technical regulations that will dictate the competition’s new rules and define the

working space within which the engineers will design their new cars. In order to be certified to compete,

the prototypes must undergo a series of tests where vital components are subjected to various load cases

and must comply with safety and performance standards.

2.3.1. Gearbox Housing

While there are several technical regulations stated in the FIA’s Technical Regulations for LMP1

Prototypes document [21] regarding the operation of the gearbox system, there are no specific constraints

to the gearbox housing itself, allowing engineers to implement innovative shapes and use better

performing materials. Article 1.26 of document [21] defines the gearbox system and is the first mention of

the gearbox housing (also referred to as gearbox casing or transmission casing):

“Gearbox

A gearbox is defined as all the parts in the drive line which transfer torque from the Power Unit output

shafts to the drive shafts (the drive shafts being defined as those components which transfer drive torque

from the sprung mass to the unsprung mass).

It includes all components whose primary purpose is for the transmission of power or mechanical selection

of gears, bearings associated with these components and the casing in which they are housed.”

Art. 1.26, 2015 Technical Regulations for LMP1 Prototype

Additional articles of [21] state that the car’s rear wing may be rigidly mounted directly to the

transmission casing (art. 3.6.2c), in which case the housing is subjected to the aerodynamic forces

produced by the rear wing and must be able to sustain these loads without failure of the structure, and that

the gearbox casing can be constructed out of carbon fiber (art. 11.4).

2.3.2. Regulation load cases

As previously mentioned in Chapter 2.1, and in article 18.1 of [21], LMP1 cars must have a RIS

fitted just behind the gearbox assembly. This structure must be rigid enough to prevent damage to the

gearbox and other vital components in the event of a crash.

The mechanical tests that this structure must undergo are defined in the FIA’s 2014 Procedure for

the Approval of Safety Structures for LMP Sports Cars document [22] and are, at an initial stage of this

thesis, used to establish the thickness of the housing’s preliminary shape by evaluating stress values that

result from testing these load cases. Later, the final composite design is also subjected to these load

cases in order to evaluate its strength. These tests are detailed below.

According to article 2.5 of [22], the rear impact absorbing structure must be subjected to a static

side load test and an impact test. For these tests, both the rear impact structure and the gearbox housing

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(including other components such as the jacks and rear wing pillars) must be solidly fixed to a vertical

flange that reproduces the rear face of the engine, while the flange itself is also solidly fixed to the ground.

These two strength tests consist of the following:

1. Static side load test

In this test, a constant transverse and horizontal load of 40 kN must be applied to one side of the

impact-absorbing structure using a pad at a point 400 mm behind the rear wheel axis. The center of area

of the pad must pass through the plane mentioned above and the midpoint of the height of the structure at

the relevant section.

After 30 seconds of application, there must be no failure of the structure or of any attachment

between the structure and the gearbox housing. During the test the gearbox housing and the structure

must be solidly fixed to the flange but not in a way that could increase the strength of the attachments

being tested, and the gearbox housing must be blocked laterally through a pad of identical dimensions to

the one used to apply the load, positioned before the junction with the rear absorbing structure. The load

must be applied in less than 3 minutes and maintained during at least 30 seconds.

Acceptance criteria: There must be no failure of the structure or of any attachment between the structure

and the gearbox housing, or of the gearbox housing itself.

2. Impact test

The structure and the gearbox housing must be rigidly fixed to the ground and a solid object,

having a total mass MT and travelling at a velocity of no less than 11 meters/second, will be projected into

it.

The total mass of the trolley consists of the minimum weight of the LMP1 prototype (as per

technical regulations – 870 Kg) and an additional mass of 150 Kg. Given the maximum deceleration

permitted by the acceptance criteria described below, the impact force of this test is estimated at 250 kN.

The object used for this test must be flat, measure 450 mm (+/-3 mm) wide by 550 mm (+/-3 mm)

high and may have a 10 mm radius on all edges. Its lower edge must be at the same level as the car

reference plane (+/-3 mm) and must be so arranged to strike the structure vertically and at 90° to the car’s

centerline.

During the test, the striking object may not pivot in any axis and the crash structure may be

supported in any way, provided that this does not increase the impact resistance of the parts being tested.

Acceptance criteria: The average deceleration of the trolley must not exceed 25 g. It is calculated from the

unfiltered deceleration data, from the instant of impact (T0 defined by electronic contact) to the first instant

the trolley speed is less than 0 m/s (V0). There must be no damage to the clutch bell housing or the

gearbox housings.

12

The implementation of these load cases is explained in detail in Chapter 4. Less extreme load

cases, such as acceleration, bump, cornering and braking are later used to evaluate displacement values

and determine the housing’s stiffness by incorporating these in the topography and composite optimization

processes.

13

14

Chapter 3

3. Optimization

In this chapter the reader is introduced to the tools used in the structural analysis process

performed in this thesis. Insights to topography and composite optimizations, as well as basic optimization

concepts, are also presented.

All the structural analysis work carried out in this project is performed with the software package

HyperWorks® 12 [23].

The initial stage of this work, including the editing of the gearbox housing’s geometry, creation of

the mesh and establishment of the boundary conditions and loadcases is executed in HyperWorks’s pre-

processor: HyperMesh®. HyperMesh is a high-performance finite element pre-processor into which the

initial CAD (Computer Aided Design) model is loaded. It also supports the direct use of existing FE

models.

The structural calculations and optimizations are executed using one of HyperWorks’s solvers:

OptiStruct®. OptiStruct is a modern structural analysis solver that can perform both linear and non-linear

structural problems under static and dynamic loadings. It is based on FE and multi-body dynamics

technology and allows for the development of innovative, lightweight and structurally efficient designs.

OptiStruct is used to analyze and optimize structures for various characteristics such as stiffness,

strength, stability, durability, noise, vibration and harshness. It also has the ability to perform thermal

analysis as well as evaluate kinematics and dynamics.

Its most noticeable downside is the fact that it can only perform single-objective optimizations.

However, multiple desired characteristics can be attributed to the structure by way of optimization

constraints. For instance, the designer/engineer may define minimizing the structure’s compliance as the

optimization objective and set a constraint for its maximum weight, resulting in a lightweight and rigid

design.

OptiStruct makes use of gradient-based algorithms to perform Topology, Topography, Size and

Shape optimizations and is highly compatible with the popular solver NASTRAN, using the latter’s

standard type input syntax and writing results in NASTRAN formats. In this work, the interest is focused on

the characteristics of Topography optimization and optimization of composite structures, which consists of

a three-stage process: Free-sizing, Ply-based Sizing and Stacking Sequence optimizations.

Finally, optimization results are visualized using HyperWorks’s post-processor: HyperView®.

HyperView is a complete post-processing and visualization environment for FEA, computational fluid

dynamics, multi-body system simulation, digital video and engineering data.

15

In the highly competitive world of engineering, it is no longer enough to design a specific

component whose performance of its required function is just acceptable. Engineers and designers always

look for the best combination of effectiveness and efficiency from a particular system. As an example, one

can think of motorsport monocoques, which not only have to be strong enough to protect the driver in the

event of a crash but also must remain lightweight to avoid compromising the car’s performance.

Numerical optimization is a design tool that allows engineers to obtain the desired results in a

timely and economical fashion, since it makes use of computer power to analyze alternative designs

rapidly, thus saving the need to manually evaluate every iteration.

3.1. Basic Optimization Concepts

In order to better understand numerical optimization, one should be introduced to the basic

concepts that regulate and guide its process: design space, design variables, objective function and

design constraints.

The group of independent parameters that are allowed to change while searching for the best

design are called design variables. According to Arora [24] the “(…) set of variables that describe the

system, called design variables (…) are referred to as optimization variables and are regarded as free

because we should be able to assign any value to them.” Additionally, these variables “(…) should be

independent of each other as far as possible (…)” otherwise “(…) their values cannot be specified

independently because there are constraints between them.” (p. 20). As an example, in topography

optimization the design variables are the minimum width, draw height and draw angle of the structural

beads, which consist of spatial variations of mesh elements on the design space. Upper and lower bounds

can be attributed to these variables, defining their range of variation. Defining a maximum beam draw

height in a topography optimization is a good example.

The design space is defined as the domain of the structure that will be subjected to the

optimization process. In this case, this is the region highlighted in purple in figure 2.3.

The objective function is the dependent variable that the optimizer attempts to either minimize or

maximize. Since it is a function of the design variables, changing the values of these variables should

change the value of the objective function. If, for instance, the objective of the optimization process is to

maximize the stiffness of the structure, then the objective function will be linked to the structure’s

compliance and the optimizer will attempt to decrease the value of this parameter to a minimum.

In order for the final design to be acceptable it must comply with certain requirements. These

requirements are function of design variables or system responses and are called design constraints. In

the example referenced above, in addition to the objective to maximize de structure’s stiffness, a design

constraint could be attributed to the overall mass of the structure, setting, for example, a maximum desired

weight. Manufacturing constraints are often set for the optimization of composite structures, such as

manufacturable ply thicknesses and orientations.

Figure 3.1 illustrates the general form of an optimization problem according to Belegundu and

Chandrupatla [25] where the goal is to minimize the objective function.

16

Topography reinforcements (beads)

Figure 3.1 – General optimization problem

In summary, numerical optimization iteratively changes the values of the design variables in the

design space, in order to find the minimum or maximum of the objective function, while satisfying all the

required design constraints. Once the optimization process has converged and all the constraints are

satisfied, a feasible design is achieved and the optimization process is complete.

It is important to acknowledge the high difficulty for a given solver to obtain the absolute best

design for a particular optimization process. This may be due to the existence of a large number of local

optima or non-differentiable objective functions.

3.2. Topography Optimization

Topography optimization is an advanced procedure of shape optimization usually performed on

shell structures. Fine examples of topography optimizations are the works performed on suspension

modeling by Kilian, Zander and Talke [26] and the study of automotive body structures by

Chakravarty [13].

As opposed to topology optimization, where density variables are used, in topography optimization

no material is added to or removed from the structure. Instead, geometrical changes to the shape of the

structure optimize its performance under specific load cases. The structure’s compliance, its natural

frequencies or its moment of inertia are a few examples of responses that can be optimized using

topography optimization.

In the solver OptiStruct, topography

optimization is accomplished by creating, on the

design space, a pattern of shape reinforcements

(also referred to as beads) based on shape

variables generated internally.

These protrusions can increase the stiffness

of the structure by increasing its moment of inertia.

Figure 3.2 - Topography reinforcements

𝑋 =

𝑋1𝑋2…𝑋𝑁

Design

Variables

Design Space

Function: 𝑓(𝑋)

Objective

Minimize 𝑓(𝑋)

Constraints:

Inequality: 𝑔𝑖(𝑋) ≤ 0 𝑖 = 1,… ,𝑚

Equality: ℎ𝑗(𝑋) = 0 𝑗 = 1,… , 𝑙

Bounds: 𝑋𝑈 ≤ 𝑋 ≤ 𝑋𝐿

17

Over a series of iterations, the influence on the structure of a large number of shape variables is

calculated and optimized. These variables are part of the design space and allow the user to create any

reinforcement pattern within the design domain.

Basic topography shape variables (minimum bead width and draw angle) are circular in shape,

and they are laid out across the design domain in a roughly hexagonal distribution.

Figure 3.3 - Topography design variables

Each topography shape variable has a circular central region of diameter equal to the minimum

bead width. Grids within this region are perturbed as a group, which prevents the formation of any

reinforcement bead of less than the minimum bead width. Grids outside of the central circular region of

the topographical variables are perturbed as the average of the variables to which they are nearest. This

results in smooth transitions between neighboring variables. If two adjacent variables are fully perturbed,

all of the nodes between them will be fully perturbed. If one variable is fully perturbed and its neighbor is

unperturbed, the nodes in between will form a smooth slope connecting them at an angle equal to the

draw angle. The spacing of the variables is determined by the minimum bead width and the draw angle in

such a way that no part of the bead reinforcement pattern forms an angle greater than the draw angle.

3.2.1. Variable Generation

There are three methods of automatically generating shape variables for topography optimization:

element normal, draw vector and user-defined. The first two are performed entirely in OptiStruct and

element normal is the one used in this project.

In this method, the normal vectors of the 2D

elements are used to define the vector along which

the shape variables are allowed to change. This

method is especially effective for curved surfaces

and enclosed volumes where the beads are

intended to be drawn normal to the surface.

Figure 3.4 - Element normal method for topography optimization

Draw height

Minimum bead width

Draw

angle

18

3.2.2. Topography Optimization Example

The intention of presenting the following structural study is to give the reader an illustrative

demonstration of the results of topography optimization on a common structural problem.

Figure 3.5 shows a finite element model of a metallic plate being subjected to a torsion load case.

This part is assumed to be formed using a stamping process.

Figure 3.5 - Metallic plate under torsion

The objective of this topography optimization was to minimize the displacement of the node where

the force is applied in the positive z-direction. The topography design variables mentioned above and

optimization parameters were defined and the OptiStruct software was used to determine the optimal

reinforcement patterns.

Figure 3.6 shows the converged solution for the topography optimization.

Although the optimization converged, the overall shape obtained proved difficult to manufacture.

Its main contribution to the final design was to show what kinds of patterns were likely to optimize the

structure, in this case to minimize the displacement at the selected node. A possible pattern suggested by

the converged solution and consisting of channels parallel to a diagonal was implemented in the form of a

pattern grouping constraint at the last iteration. In this example, the diagonal emerging from the node

where the load is applied was selected.

a) Converged solution b) Manufacturable reinforcements

Figure 3.6 - Results of topography optimization of plate under torsion

Constrained in z-direction

F = 100 N

Constrained

in z-direction

Parallel

channels

19

3.3. Composite Optimization

A new era in the production of composite materials began when leading aircraft companies

decided to innovate and start using carbon fiber reinforced materials in the design and manufacture of

composite airframes for their commercial airliners.

Over the years, the manufacturing process of composite materials has evolved and new materials

have been studied and tested, allowing engineers to design and produce composite components with

desired characteristics such as strength, stiffness, weight and various dimensions. A great example of the

progress in the design and manufacturing of composite materials for aerospace application is the recently

launched Boeing 787 Dreamliner (in 2007), which features an airframe comprising of 50% carbon fiber

reinforced plastic and other composites (see [27]).

Composite components also played a major role in the world of motorsport , allowing

engineers to fabricate parts that were both lighter and stiffer, therefore improving the performance

and the safety of their cars. The usage of carbon fiber components in racing cars began with Formula

One in 1981, when engineers at McLaren revealed the MP4/1: the first Formula One car to use a

carbon fiber composite monocoque chassis. Nowadays, carbon fiber is used for structural,

aerodynamic and other body parts in the majority of motorsport disciplines including DTM cars, World

Rally cars, Formula Racing series, GT cars and Endurance prototypes (see [28]).

In the automotive industry, carbon fiber composites were first used in ultra-expensive limited-

series road cars. The first of which was the McLaren F1, released in 1992. Over the last decade,

developments in reducing the production costs of carbon fiber composites has allowed engineers to

slowly implement this technology in mass-produced road cars (see [29]). In 2014, BMW became the

first automotive company to launch a volume production car featuring a passenger cell constructed of

carbon fiber reinforced polymer (CFRP) (see [30]).

Although there are different forms of composite materials, the most commonly used is the

composite laminate where thin plies of various orientations are stacked and bonded together to form a

shell structure.

When it comes to design, composite structures offer unmatched tailoring potential since the

properties of the laminate material can be almost continuously customized throughout the structure.

However, with this amplified design liberty come additional challenges for the design process itself,

leading from concept to design details, and for the software that performs the structural calculations.

In recent years, engineers at Altair Engineering have established a complete framework for

composite optimization (figure 3.7), in a process consisting of three consecutive stages [5-7]:

1. Stage 1 of the composite optimization process is a free-size optimization that focuses on

material distribution in terms of orientation and thickness. This is achieved by allowing the

thickness of each ‘super-ply’ of a unique fiber orientation to change freely throughout the model,

obtaining a thickness contour for each fiber orientation. A concept design of ply layout and

thickness results from the interpretation of the thickness contours.

20

2. Stage 2 is a ply-based sizing optimization in which the interpreted ply-based model is further

optimized under all design constraints (such as manufacturable thicknesses) with discrete design

variables representing the number of individual plies of each ply bundle.

3. Finally, Stage 3 of composite optimization consists of a stacking sequence optimization in

which the final design is refined to satisfy all manufacturing and performance constraints.

It’s important to stress that manufacturing constraints are considered in all three optimization

stages. For instance, important requirements for aerospace or motorsport composite components can be

specified in composite optimization, such as the maximum number of consecutive plies with the same

orientation. This requirement would translate into a percentage requirement during the first two stages of

the optimization process in order to achieve a balanced distribution of fiber orientation that would allow a

feasible stacking sequence in the last stage of the process.

The whole composite optimization method is executed in the commercial software Altair

OptiStruct, which features bespoke FEA modeling techniques and ply layup definitions directly associated

with the optimization process. These make up for a modeling approach that follows the design and

manufacturing language known as Ply-Book, allowing for a more instinctive interface between design and

analysis

OptiStruct’s software package has seen increasing adoption among aerospace and automotive

Original Equipment Manufacturers (OEMs), with singular importance attributed to large to medium size

airliners, in regard to their large economic scale.

The three composite optimization stages presented in figure 3.7 are explained in-depth in the

following chapters.

Figure 3.7 - Composite Optimization process using Altair’s OptiStruct

Stage 1

Stage 3

Stage 2

21

3.3.1. Stage 1 – Free-size Optimization

Topology optimization is widely regarded as a very important method in the design of highly

efficient and innovative structural concepts. Nevertheless, it has been proven in [5] that, while topology is

better suited for the layout of shells of constant thickness, free-size optimization offers an expressive

alternative with its ability to allow for continuous variation of thicknesses throughout the shell structure.

The main principle of composite free-sizing optimization is to generate design concepts that

extract all the capabilities of a composite structure by designing simultaneously the structure and the

material.

Free-size-optimization answers the question of “In which areas of the structure is a specific fiber

orientation required?” To this end, a thickness distribution is generated for each fiber orientation that is

defined by the user in pre-processing, while letting the total thickness of the laminate vary continuously all

through the structure. Simultaneously, an optimum laminate composition is obtained for every finite

element on the mesh.

The concept of ‘super-ply’ is introduced in free-size optimization (represented in Figure 3.8).

These super-plies result from allowing the thickness of every fiber orientation to be dimensioned freely and

represent the total designable thickness for each fiber orientation. Each super-ply results in the creation of

a default number of 4 ply-bundles (illustrated in figure 3.9), since this number of bundles delivers a

satisfying balance between accurate representation for the thickness distribution and the complexity of the

ply tailoring.

A ply-bundle consists of a continuous set of plies of the same shape. Multiple ply shapes can be

determined for each orientation and generated from the free-size optimization phase.

Usually only one super-ply is required for a particular fiber orientation since the majority of shell

structures are designed to sustain in-plane loading locally, while still being able to provide bending

capacity as an assembly. In this case, shell properties are often associated with a smear option, which

counterbalances the effect of stacking sequence. Multiple super-plies and other shell properties options

may also be selected to allow for the selection of stacking sequence.

Figure 3.8 - Super-ply concept

22

Figure 3.9 - Super-ply concept

It is perhaps important to explain the labeling used for the ply-bundles represented in figure 3.9,

taking as an example ply-bundle ‘12300’. The first number (1) represents the first free-size design variable

created (in this case, the only one). All ply-bundles begin with the number 1. The second number in ply-

bundle ‘12300’ represents the second fiber orientation defined in the initial laminate (45°). Finally, the

number 300 identifies the third ply-bundle of the initial super-ply.

For the free-size stage, the optimization problem can be mathematically defined as follows:

Minimize 𝑓(𝑥)

Subject to 𝑔𝑗(𝑥) − 𝑔𝑗𝑈 ≤ 0, 𝑗 = 1,… ,𝑀 (1)

𝑥𝑖𝑘𝐿 ≤ 𝑥𝑖𝑘 ≤ 𝑥𝑖𝑘

𝑈 , 𝑖 = 1,… ,𝑁𝑝 𝑘 = 1,… ,𝑁𝐸

In this problem, 𝑓(𝑥) represents the objective function, 𝑔𝑗(𝑥) and 𝑔𝑗𝑈 represent the 𝑗-th constraint

response and its upper bound, respectively. 𝑀 is the total number of constraints, the number of finite

elements is 𝑁𝐸 and 𝑁𝑝 represents the number of super-plies. 𝑥𝑖𝑘 is the thickness of the 𝑖-th super-ply of

the 𝑘-th element. 𝑥𝑖𝑘𝐿 and 𝑥𝑖𝑘

𝑈 respectively represent the lower and upper bounds for 𝑥𝑖𝑘.

In this first design phase, both the objective function and constraints are linked to a wide range of

responses. Generally, engineers aim to optimize the structural stiffness of the component and therefore

resort to compliance or displacement responses.

Since a composite laminate is traditionally manufactured through a stacking and curing process,

specific manufacturing requirements must be defined in order to reduce unwanted secondary effects that

result from the curing process.

As mentioned before, manufacturing constraints are considered throughout the optimization

process. These constraints are essential for the design of the composite structure and start being

established right at the beginning of the concept design procedure.

In pre-processing, the following five plies

with distinctive fiber orientations are

created:

Free-size optimization generated

4 ply-bundles for each initial super-ply:

23

There are two significant constraints to be considered at the start of the free-size optimization:

number of consecutive plies of the same orientation and the total thickness of the laminate.

Typically for carbon fiber reinforced composites, no more than 3 or 4 consecutive plies of the

same fiber orientation can be stacked consecutively (in order to prevent manufacturing failure during the

curing process), thus ruling out design concepts that feature areas with predominance of single fiber

orientation. Therefore, to achieve a manufacturable design concept, it is possible to either constrain the

percentage (𝑃) of each fiber orientation in the overall thickness or set lower and upper bounds on the

thickness of individual orientations, consequently ensuring that enough alternative ply orientations are

offered to break the succession of plies of the same orientation.

The total thickness (𝑇) of the laminate can be constrained through the setting of lower and upper

bounds (𝑇𝑘𝐿 and 𝑇𝑘

𝑈, respectively) on the free-size design variable.

These two types of manufacturing constraints can be represented mathematically as follows:

Total thickness: 𝑇𝑘𝐿 ≤ ∑ 𝑥𝑖𝑘

𝑁𝑝𝑖=1 ≤ 𝑇𝑘

𝑈 𝑘 = 1,… ,𝑁𝐸

(2)

Ply percentage: 𝑃𝑗𝐿 ≤

𝑥𝑗𝑘

∑ 𝑥𝑖𝑘𝑁𝑝𝑖=1

≤ 𝑃𝑗𝑈

𝑗 = 1,… ,𝑁𝑝 𝑘 = 1,… ,𝑁𝐸

In free-size optimization, it is possible to establish several instances of the constraints mentioned

above, as these can be applied locally through the definition of sets of finite elements. This allows

instituting different constraints in different regions of the structure and still maintaining continuity of plies

throughout the model. This approach is particularly useful when the structure features critical regions such

as bolted areas.

Furthermore, an additional balance constraint ban be defined in which, usually, a pair of

symmetrical fiber orientations are required to compensate each other. For instance, setting the balance

between the number of plies with 45° and -45° fiber orientations eliminates twisting of a plate under

bending along the 0 axis.

Alternatively, free-sizing optimization can also be executed as a zone based process, through the

definition of clusters of finite elements that are subjected to optimization. The purpose of this method is to

turn the design interpretation process more straightforward and improve the manufacturability of the

structure. This approach has the added disadvantage of somewhat reducing the design freedom.

The final result of the free-size optimization phase is the thickness contribution of each fiber

orientation defined in pre-processing. OptiStruct automatically generates ply bundle data that works as a

starting point for phase two of the composite optimization process: Ply-based sizing Optimization.

24

3.3.2. Stage 2 – Ply-based sizing Optimization

The second stage of composite optimization is designated ply-based sizing optimization.

As mentioned before, OptiStruct adopts the native language of Ply-book standards for composite

laminate modeling and manufacturing. In this format, laminates are defined in terms of ply entities and

stacking sequences, making the whole design and optimization process close to the way real laminate

composites are manufactured.

In this second stage of composite optimization, each individual ply thickness is directly selected as

a designable unit, which allows for a simplified definition of design variables since ply continuity across

patches is automatically taken into account.

The starting point for ply-based sizing optimization is the output data from free-size optimization,

which consists of a continuous distribution of thickness for each fiber orientation. The ply bundles are

automatically set up for optimization and ready to be sized to determine the optimum thickness per bundle

per fiber orientation.

Working as a preparatory action for the optimization run, the first step in sizing optimization is to

manually edit the ply-bundles that resulted from free-size, since these present impractical shapes that are

often a considerable challenge to manufacture. This is achieved by adding or removing specific finite

elements from the set that characterizes each ply-bundle. An example of this action is illustrated in figure

3.10.

a) Ply-bundle before shape editing b) Ply-bundle after shape editing

Figure 3.10 - Ply-bundles shape editing

Composite manufacturing constraints defined in the free-sizing phase are automatically carried

over into the ply-based sizing optimization phase.

By defining discrete manufacturable thicknesses (TMANUF) and capturing different level-sets of

the thickness distribution for each fiber orientation, the solver OptiStruct defines the layout of ply-bundles

and forces these to reach thicknesses reflecting a discrete number of physical plies. Therefore, from a ply

bundle sizing optimization, the number of plies required per orientation can be established.

25

The optimization problem solved in ply-based sizing optimization is the same shown previously in

equations (1) and (2), only with discrete thicknesses as design variables. These thicknesses work as unit

ply thickness increments for the ply-bundles that resulted from free-size optimization.

Finally, OptiStruct automatically generates an input file for the final phase of composite

optimization.

3.3.3. Stage 3 – Stacking Sequence Optimization

Stacking sequence optimization is the third and final stage in the composite optimization process.

In this design phase, the composite plies previously generated through ply-bundle sizing are

shuffled to determine the optimal stacking sequence for the given design optimization problem.

Although the design that resulted from sizing optimization contained all ply layout and stacking

details, it is unlikely that specific manufacturing constraints are fully satisfied. As a result, OptiStruct’s

stacking sequence optimization allows the user to define important constraints such as the ones presented

below:

The stacking sequence should not contain any section with more than a given number of

successive plies of same orientation;

The 45° and -45° orientations should be paired together;

The cover and/or core sections should follow a predefined stacking sequence

An efficient proprietary technique is developed to allow the process to evaluate a large number of

stacking combinations from both performance and manufacturability perspectives.

26

27

Chapter 4

4. Optimization Setup

In this chapter the reader is presented to all the pre-processing work that was performed before

the optimization runs, including the identification of both the design and non-design spaces, the meshing

process and the establishment of boundary conditions and load cases.

4.1. Design space and meshing

As is previously mentioned in section 2.1, the gearbox housing’s geometry consists of 3 different

regions: design space, non-design space and solid supports. Although the solid supports (represented in

light blue in figure 4.1) are not subjected to optimization either, they differ from the remaining non-design

space as these supports are modeled as solid elements. Aluminium is attributed to the solid supports in

both topography and composite optimizations.

The remaining non-design space (represented in green in figure 4.1) consists of areas around the

solid supports and other connection points to important gearbox components. Therefore, these areas are

not subjected to optimization, since their shape has to remain unchanged. An additional cause for

incorporating these areas in the non-design space is the fact that the objective of minimizing the

structure’s weight could result in insufficient material being attributed to these areas, potentially weakening

these critical regions and resulting in local failure under service and FIA regulation load cases.

a) Solid supports b) Design regions

Figure 4.1 - Gearbox housing regions

After defining the design and non-design spaces of the gearbox housing’s geometry, the meshing

process was initiated.

28

Using OptiStruct’s ‘automesh’ function, the 2D mesh was created on the initial geometry with

a standard 4mm element size and mixed mesh type, i.e., presenting both 3-node triangular (CTRIA3)

and 4-node quadrilateral (CQUAD4) 2D elements. All 2D elements were edited so that each element’s

normal pointed to the inside of the gearbox housing.

Figure 4.2 - 2D Mesh detail

2D elements referring to the solid supports were converted to CHEXA (3D hexahedral

elements with 8 nodes) and CPENTA (3D triangular prism pentahedral elements with 6 nodes) solid

elements. These solid elements were attributed the PSOLID property of OptiStruct.

The remaining non-design space and design space 2D elements were assigned to the

PSHELL property and attributed a constant 8mm thickness throughout the structure (to be explained

in section 4.1).

Since both two and three-dimensional finite elements coexisted in the same mesh, special

attention was given to the critical areas around the solid supports to guarantee the coincidence of

nodes in close proximity. To this effect, in regions throughout the structure where the 2D non-design

mesh featured a very pronounced curvature, the solid elements were modified to meet the 2D mesh

without a significant loss in element quality.

The meshing process created 41980 shell elements, 32995 solid elements (74975 finite

elements in total) and 86105 nodes. The degree of discretization (associated with the average

element size) used in the model was a compromise between the accuracy of the optimization results

and reasonable computation times. All optimizations presented in chapters 5 and 6 represent

converged solutions. The confirmation of convergence is automatically reported to the user by the

optimization software OptiStruct.

Figure 4.3 - 3D Mesh adjustments

29

4.2. Initial Model

For the initial model of the gearbox casing, all finite elements were modeled in Aluminium, whose

mechanical properties are presented in Table 4.1.

Table 4.1 - Mechanical properties of Aluminium

Material Young’s Modulus [GPa] Poisson’s Ratio Density [kg/m3]

Aluminium 70 0,34 2700

As previously mentioned in section 2.3.2, the regulation load cases are initially tested on

several iterations of the FE model that use different thicknesses for the shell elements. The goal is to

evaluate the resulting stresses and displacements in order to determine the thickness of the shell

elements at the starting point of the optimization process. The most relevant results are shown in

table 4.2.

Table 4.2 - Static analysis of initial model under regulation load cases

Regulation load cases

PSHELL Thickness

[mm]

Mass [Kg]

Side Static (left) Impact Side Static (right)

Max Stress (von Mises)

[MPa]

Max Disp [mm]

Max Stress (von Mises)

[MPa]

Max Disp [mm]

Max Stress (von Mises)

[MPa]

Max Disp [mm]

10 20,39 153,9 0,2076 884,4 1,289 200,2 0,2538

8 17,16 174,4 0,2494 1009,0 1,942 234,9 0,315

6 13,92 272,0 0,3231 1171,0 3,284 295,5 0,4029

The left and right static side load tests identified in table 4.2 are illustrated in section 4.3.1.

An overall shell thickness of 8mm was set for the shell elements (design space and 2D mesh of

non-design space) given the 3,23kg weight saving compared to 10mm shells and the satisfactory

maximum stresses and displacements obtained for a starting point FE model.

While shell thicknesses greater than 10mm resulted in lower values for both maximum stress and

displacement, these renderings of the FE model were deemed too heavy to be accepted as the starting

point for the optimization process. On the other hand, shell thicknesses lower than 8mm resulted in FE

models that, despite being lighter, did not deliver satisfactory strength to the structure under the regulation

load cases.

The starting point for the optimization process was obtained: an interpretation of the gearbox

housing’s FE model featuring 8mm thick shell elements throughout the structure and with a total mass of

17,16kg.

30

4.3. Boundary conditions and load cases

When it is mounted to the LMP1 car, the front of the gearbox housing is fixed directly to the back

of the engine with six bolts. These mountings translate into the constraints that were applied to the FE

model and these are represented in figure 4.4 a).

a) Six mounting points b) SPC detail

Figure 4.4 - Single-point constraints (SPC)

Figure 4.4 b) demonstrates one of the rigid fixing points for the engine in detail. The lines

displayed in orange represent RBE (Rigid Body Element) one-dimensional rigid elements. These are rigid

bodies whose independent degrees-of-freedom are specified at a single grid point and whose dependent

degrees-of-freedom are specified at an arbitrary number of grid points. In other words, all the nodes on the

inner face of the ring-shaped solid support are dependent and are always at the same distance from the

central independent node. Since this is a fixed support, all the degrees-of-freedom of the central node are

null. This approach is applied to all six mounting points.

Additionally, rigid bodies are also used to simulate the presence of a cover on the top of the

gearbox casing. To that end, all nodes referring to the bolt-holes of the top solid support are dependent of

a node that sits in the geometrical center of this support.

Figure 4.5 - Simulated cover

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4.3.1. Regulation load cases

The first load cases to be considered in this thesis are the static side load test and impact test set

by the FIA and previously described in section 2.3.2 of this document. The load cases are resumed below

and represented in figures 4.6 and 4.7.

Static Side Load Test: A constant transverse and horizontal load of 40 kN is applied to one side

of the structure. Left and right side load tests are performed since the gearbox housing in question is not

longitudinally symmetric, thus offering different compliances to both load tests.

Impact Test: Total mass of ≅ 1000 kg, travelling at a velocity of not less than 11 meters/second is

projected into the back of the gearbox housing. Estimated impact force: ≅ 250 kN (based on maximum

allowed deceleration). For the purpose of this work, the impact test was interpreted as a static test where a

maximum dynamic force is applied.

a) Left

b) Right

Figure 4.6 – Static side load tests

F

F

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c) Impact test

Figure 4.7 - Impact test

The engine mounts constraints introduced in section 4.3 are implemented in all three load cases

represented in figure 4.6. Both static side load tests feature side solid supports that are fixed as per

regulation (represented in red). Rigid bodies are modeled to mimic the RIS and thus transfer the load from

the independent node where the force is applied to the dependent nodes. These nodes are at the center

of the bolt-holes of the solid supports where the RIS is mounted to the gearbox housing.

4.3.2. Service load cases

The service load cases are designed to replicate the forces that the gearbox housing experiences

during typical racing conditions: acceleration, braking, bump and cornering. These load cases are

established to evaluate the structure’s stiffness and were projected in collaboration with Optimal Structural

Solutions.

When designing the service load cases, several assumptions were made:

The LMP1 car in question has a 40/60 weight distribution, i.e., 40% of the weight sits at the front

of the car and 60% at the rear. These weight percentages are equally split between the two tires

of each axle (20% for each front tyre and 30% for each rear tyre);

The car’s center of gravity (CG) sits in the longitudinal symmetry plane of the car, 0.338m above

the ground and 1.784m behind the front axle;

The LMP1 car in question has a mass of 970kg, is carrying 70kg of fuel and an 80kg driver. Total

mass: 1050kg;

The gearbox housing’s center of gravity and the center of the rear wheels are at the same

distance from the ground;

The LMP1 car in question would have to endure the following accelerations: 2g for acceleration,

5g for cornering (right and left), -5g for braking and 3g for bump.

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The service load cases are characterized as follows:

Acceleration: since the LMP1 car in question is rear-wheel driven, a horizontal force acts on

each rear tyre’s contact point with the ground. The force applied on each tyre is equal to half the

car’s inertia, which is equal to the car’s total mass times its acceleration;

Braking: the car’s force of inertia (𝐹𝑖𝑛𝑒𝑟𝑡𝑖𝑎) in the braking load case is split 60/40 between the two

axels (60% braking force applied at the front and 40% at the rear) and split equally between the

two tires of each axle;

Cornering: in order to produce 5g of acceleration when cornering, the LMP1 car exerts a

centripetal force that has a 60% contribution from the rear tires and a 40% contribution from the

front tires;

Bump: this load case simulates the LMP1 car going over a curb or over a hole in the track’s

pavement. In order to mimic this effect in the bump load case, an equal upwards force is applied

on each rear tyre.

In order to better demonstrate the method used to determine the resulting forces applied on the

gearbox casing, figure 4.8 illustrates the acceleration load case as an example:

Figure 4.8 - Acceleration load case

Figure 4.8 shows the position of both the car’s and the gearbox’s center of gravity. Also

represented are the inertia forces and the weight transfer forces (𝑊𝑇). The latter account for the transfer of

weight that occurs when this rear-wheel driven car accelerates, shifting its weight to rear and thus exerting

downward forces at the rear tires, which then result in a reaction force (represented in figure 4.8) applied

to each tyre. Furthermore, the transfer of weight when accelerating tends to lift the nose of the car,

resulting in a downward reaction force acting on each front tyre.

Car

CG

Gearbox

CG

𝐹𝑖𝑛𝑒𝑟𝑡𝑖𝑎

𝐹𝑖𝑛𝑒𝑟𝑡𝑖𝑎2

𝐹𝑖𝑛𝑒𝑟𝑡𝑖𝑎2

𝑊𝑇

𝑊𝑇 𝑊𝑇