Methodology Development for Topology Optimization of Power

Methodology Development for Topology Optimization of Power Transfer Unit Housing

Structures

Povendhan Palanisamy

Master of Science Thesis TRITA ITM EX 2020:518

KTH Industrial Engineering and Management

Machine Design

SE-100 44 STOCKHOLM

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Examensarbete TRITA ITM EX 2020:518

Metodutveckling för topologioptimering av växellådshusstrukturer i kraftöverföringsenheter


Godkänt

2020-09-16

Examinator

Ulf Sellgren

Handledare

Ulf Sellgren

Uppdragsgivare

GKN ePowertrain Köping AB

Kontaktperson

Simon Samskog

Sammanfattning

Simuleringsdriven design är en metod och process som har utvecklats i många år, och med dagens

avancerade programvaror ger möjlighet att få in simulering direkt i designprocessen. Fördelarna

med att använda simuleringsdriven design i produktutvecklingsprocessen är välkända och jämfört

med en mer traditionell designprocess kan den simuleringsdrivna designprocessen ge användaren

möjlighet att utforska, optimera och designa produkter med reducerade ledtider som följd.

En av de metoder som tillämpas i simuleringsdriven design är användning av topologioptimering

(strukturoptimering). Topologioptimering är något som GKN använder i designprocessen. På

grund av komplexiteten hos produkterna GKN designar och tillverkar kräver designprocessen

mycket ingenjörsarbete och tid. Produktionen har också problem med att tolka

topologioptimeringsresultaten.. Syftet med avhandlingen är att utforska olika simuleringsverktyg

som används för topologioptimering och förbättra metodiken och processen för att öka

designtolkningen av en statisk topologioptimering. Detta kräver en god förståelse för komponenten

och produktutvecklingsprocessen. För att förbättra osäkerheterna i resultaten från optimeringen,

är det nödvändigt att dessa resultat är lätta att tolka, och visualiseringen av resultaten ska vara

tydliga och visa hur lastvägarna går och därmed vart ribbor ska läggas.

Programvarorna som användes för att utföra topologioptimering i denna avhandling är Inspire,

SimLab, HyperMesh och OptiStruct (HyperWorks suite). Statisk topologioptimering är utförd och

tillverkningsbegränsningar för gjutningsprocesser har inkluderats.

Den metod som utvecklats är robust för liknande växellådshusstrukturer, och processen som

föreslås är mera effektiv. Den föreslagna metoden har verifierats genom att den tillämpats för ett

växellådshus.

Det resulterande topologikonceptet antas ha en bättre designtolkningsbarhet, vilket möjliggör en

förbättrad kommunikation och kunskapsöverföring i konstruktionsprocessen, jämfört med den

nuvarande processen. Produktens vikt minskas, och en mer optimal design nås med färre

iterationer.

Nyckelord: Topologioptimering, designtolkbarhet, husstruktur, designvolym, svarsfunktioner och

parametrar

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Master of Science Thesis TRITA ITM EX 2020:518

Methodology Development for Topology Optimization of Power Transfer Unit Housing Structures


Approved

2020-09-16

Examiner

Ulf Sellgren

Supervisor

Ulf Sellgren

Commissioner

GKN ePowertrain Köping AB

Contact person

Simon Samskog

Abstract Simulation driven design is a method and process that has been developed over many years, and

with today’s advanced software, the possibility to embed simulation into the design process has

become a reality. The advantages of using simulation driven design in the product development

process is well known and compared to a more traditional design process, the simulation driven

design process can give the user the possibility to explore, optimize and design products with

reduced lead time.

One of the methods that is applied in simulation driven design is the use of topology optimization

(structural optimization). Topology optimization is something that GKN uses in the design

process. Due to the complexity of the products GKN design and manufacture, the output from the

topology optimization lacks good design interpretability and the design process requires a lot of

time and effort.

The purpose of the thesis is to explore different simulation tools used for topology optimization

and improve the methodology and process with higher design interpretability for a static topology

optimization. This requires a good understanding of the component and the product development

process. It is imperative that the topology result must have high design interpretability, and the

visualization of the result must show the formation of clear rib structures.

The software’s used for performing topology optimization in this thesis are Inspire, SimLab,

HyperMesh, and OptiStruct (HyperWorks suite). Static topology optimization is conducted, and

manufacturing constraints for the casting process are considered. The methodology developed is

robust for similar gearbox housing structures, and the process is set up to be efficient. The proposed

method is verified by implementing it on a housing structure.

The resulting concept from the topology optimization is deemed to have higher design

interpretability which improves knowledge transfer in the design process when compared to the

current topology results. The weight of the product is reduced, and a more optimum design is

reached with a lesser number of iterations.

Keywords: Topology Optimization, Design interpretability, Housing structure, Design Volume,

Response Functions and Parameters.

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FOREWORD

Every engineering accomplishment counts, and the thought of contributing for a greater purpose

gives me immense fulfilment. When doing this thesis, I found myself in a position where I can use

my acquired engineering knowledge and skills in engineering simulations to contribute to reducing

carbon emissions through developing an optimal product with reduced weight.

I am thankful and proud that I got an opportunity to work on improving the product development

process at GKN, Köping. I would like to thank Simon Samskog, my supervisor at GKN, for his

support and guidance in the thesis work. I am thankful to Karthik Pingle, the Manager of the CAE

and calculation team, for welcoming me into the team and for providing the necessary facilities

and support during this unprecedented time of the Covid-19 pandemic. I would like to

acknowledge Zarad Abdallah, Rafal Czech, and my colleagues at GKN for the stimulating

discussions and the encouraging environment. I thank Lina Larsson for her help with the CAD

modelling and for the many meetings.

I also thank Fredrik Idberg from Altair for giving me a big helping hand with the software and for

his valuable insights in topology optimization.

I would like to extend my gratitude to Ulf Sellgren, my supervisor at KTH, for his guidance and

constructive feedback.

Last but definitely not least, I would like to thank my family for always being there for me, for

their wise counsel and their steadfast belief in me.


Köping, August 2020

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NOMENCLATURE

Notations

Symbol Description

E Young´s modulus (Pa)

r Radius (m)

t Thickness (m)

K Stiffness matrix

ρ Density vector

𝐾 Penalized stiffness matrix

𝜌𝑒 Elemental density

𝐶 Compliance

𝑢 Displacement vector

𝑓 Force

𝐾𝑒 Elemental stiffness

𝑉 Volume

𝑃 Penalization factor

𝑔 Response quantity

𝑥 Design variable

𝑈 Objective function

𝑒1, 𝑒2 Base vectors

Abbreviations

CAD Computer Aided Design

CAE Computer Aided Engineering

PLM Product Lifecycle Management

RDU Rear Drive Unit

PTU Power Transfer Unit

NDS Non-design space

T.O. Topology Optimization

DV Design Volume

D. O. E Design of Experiments

FEM Finite Element Methods

FEA Finite Element Analysis

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PDE Partial Differential Equations

AWD All-Wheel Drive

B. C. Boundary Conditions

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TABLE OF CONTENTS

FOREWORD 5

NOMENCLATURE 7

TABLE OF CONTENTS 9

LIST OF FIGURES 12

LIST OF TABLES 15

1 INTRODUCTION 16

1.1 Background 16

1.2 Purpose 17

1.3 Objectives 17

1.4 Delimitations and limitations 17

1.5 Method 18

2 FRAME OF REFERENCE 19

2.1.1 Real-world problems, Engineering models and FEM 19

2.1.2 Optimization 21

2.1.3 Topology Optimization 22

2.1.4 Optimization processes 31

2.2 Component 32

2.3 Current product development Process 33

2.4 Current optimization methodology at GKN 34

2.5 Problem identification 34

2.6 Areas of focus for this thesis 35

2.7 Planned studies 36

3 IMPLEMENTATION 37

3.1 Modelling 37

3.1.1 FE modelling 37

3.1.2 Contacts. Load Cases and Boundary Conditions 39

3.2 Studies 40

3.2.1 Design Volume Study 41

3.2.2 Investigative Study on Software and Process 44

3.2.3 Parameters and Response Function study 47

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3.3 Methodology Development 50

3.4 Methodology Implementation 51

3.4.1 Loop 1 51

3.4.2 Loop 2 53

4 RESULTS 57

4.1 Studies 57

4.1.1 Design Volume Study 57

4.1.2 Investigative study on Software 59

4.1.3 Response Function and Parameter study 61

4.2 Methodology 65

4.3 Result comparison of current and proposed process 66

4.4 Process/ Methodology Differences 67

5 DISCUSSION AND CONCLUSIONS 70

5.1 Discussion 70

5.1.1 Design Volume study 70

5.1.2 Discussion on Software 70

5.1.3 Response function and Parameter Study 70

5.1.4 Methodology 71

5.1.5 Methodology Implementation 72

5.1.6 Post Processing Result 73

5.1.7 Topology result Validation 73

5.2 Conclusions 74

7 REFERENCES 77

APPENDIX A: GANTT CHART 79

APPENDIX B: RISK ANALYSIS 79

APPENDIX C: CHECKLIST FOR DESIGN VOLUME HANDOVER 80

APPENDIX D: INSPIRE PRE-PROCESSING PROCEDURE 81

APPENDIX E: ISO VALUES, RESULT INTERPRETATION AND DESIGN HANDOVER LOOP 82

APPENDIX F: CADDOCTOR GUIDELINES 85

APPENDIX G: BOOLEAN OPERATION PROCEDURE 86

APPENDIX H: METHODOLOGY 88

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APPENDIX I: EXPLORING UNIQUE FEATURES OF INSPIRE 89

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LIST OF FIGURES

Figure 1 AWD components ........................................................................................................... 16

Figure 2 Relative stiffness Vs Density for different penalization factors. Figure courtesy [10] .. 24

Figure 3 Representation of gradient based optimization method. Figure courtesy [11] ............... 28

Figure 4 Mesh refinement increases from a) to c) and the topology solution differs is shown.

Figure courtesy [3] ........................................................................................................................ 30

Figure 5 Checkerboard effect in 3D structure used in the thesis................................................... 30

Figure 6 a) Design Volume b) Checkerboard effect c) Solution. Figure courtesy [3] .................. 31

Figure 7 PTU cross section ........................................................................................................... 32

Figure 8 Model 1 (Oval concept used in Experimentation) .......................................................... 33

Figure 9. Model 2 (Round concept used in Implementation phase) ............................................. 33

Figure 10 Flowchart: Product development process for housing structures at GKN .................... 33

Figure 11 Flowchart of current topology optimization methodology at GKN.............................. 34

Figure 12 Comparison of topology results for simple and complicated geometries ..................... 34

Figure 13 Split CAD model of partitioning the housing structure ................................................ 35

Figure 14 Illustration of thesis methodology ................................................................................ 36

Figure 15. FE meshed model ......................................................................................................... 37

Figure 16. Locations of application of bearing reaction forces and the RBEs used ..................... 38

Figure 17 Bolts modelling ............................................................................................................. 38

Figure 18 Mounting points ............................................................................................................ 38

Figure 19 Freeze contact between the inner layout and the housing............................................. 39

Figure 20 Baseline design ............................................................................................................. 41

Figure 21 Packaging environment ................................................................................................. 41

Figure 22 Updated Design volume ................................................................................................ 42

Figure 23 NDS Bearing seats and bolts ........................................................................................ 43

Figure 24 Internal layout ............................................................................................................... 44

Figure 25 NDS Internal layout ...................................................................................................... 44

Figure 26. Inspire model ............................................................................................................... 45

Figure 27. SimLab model .............................................................................................................. 45

Figure 28. HyperMesh model ........................................................................................................ 45

Figure 29 Illustration diagram of model used for the studies ....................................................... 48

Figure 30 Design volume .............................................................................................................. 51

Figure 31 NDS 1 ........................................................................................................................... 52

Figure 32 Design space 1 .............................................................................................................. 52

Figure 33 Loop 1 topology result .................................................................................................. 52

Figure 34 Loop 1 result handover ................................................................................................. 53

Figure 35 Realized design (top view)............................................................................................ 53

Figure 36 Realized design (cover)................................................................................................. 53

Figure 37 Realized design (side view) ......................................................................................... 53

Figure 38. Design space ................................................................................................................ 54

Figure 39. NDS ............................................................................................................................. 54 Figure 40. Design volume ............................................................................................................. 54

Figure 41. Loop 2 (top view) ........................................................................................................ 55

Figure 42. Loop 2 (bottom view) .................................................................................................. 55

Figure 43. Loop 2 (side view) ....................................................................................................... 55

Figure 44 Realized design (iso view) ............................................................................................ 55

Figure 45 Realized design (bottom view) ..................................................................................... 55

Figure 46 Realized design (side view) .......................................................................................... 56

Figure 47 Realized design (cover)................................................................................................. 56

Figure 48 T.O result Design volume 1 .......................................................................................... 58

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Figure 49 T.O result Design Volume 2 ......................................................................................... 58

Figure 50 T. O result Model 1 ....................................................................................................... 59

Figure 51 T. O result Model 2 ....................................................................................................... 59

Figure 52 Inspire T.O result .......................................................................................................... 59

Figure 53 SimLab T.O result ......................................................................................................... 59

Figure 54 HyperMesh T.O result .................................................................................................. 59

Figure 55 Inspire T.O result with highlighted connections ........................................................... 59

Figure 56 SimLab T.O result with highlighted connections ......................................................... 59

Figure 57 HyperMesh T.O result with highlighted connections ................................................... 59

Figure 58 Software characteristics comparison............................................................................. 61

Figure 59 Response 1 DV1 T.O result .......................................................................................... 61

Figure 60 Response 1 DV2 T.O result ......................................................................................... 61

Figure 61 Response 2 DV1 T.O result ......................................................................................... 62




Figure 65 Different T.O results by varying Parameter 1 ............................................................... 63

Figure 66 T.O result ...................................................................................................................... 63

Figure 67 T.O results with Parameter 2 ........................................................................................ 63

Figure 68 T. O result Parameter with holes................................................................................... 64

Figure 69 T. O result Parameter without holes ............................................................................. 64

Figure 70 T.O results without symmetric constraints ................................................................... 64

Figure 71 T.O results with symmetric constraints ........................................................................ 64

Figure 72 Methodology flow chart - Loop 1 and Loop 2 .............................................................. 65

Figure 73 Result from current process .......................................................................................... 66

Figure 74 Results Loop 1 .............................................................................................................. 66

Figure 75 Results Loop 2 ............................................................................................................. 66

Figure 76 Principal Stress ............................................................................................................. 68

Figure 77 Von Mises Stres (MPa) ................................................................................................. 68

Figure 78 Stress analysis result for concept developed without topology optimization ............... 69

Figure 79 Stress analysis result for the 1st design concept based on T.O results .......................... 69

Figure 80 Mesh size 5 mm ............................................................................................................ 72

Figure 81 T:O result Mesh size 5 mm ........................................................................................... 72

Figure 82 Mesh size 2 mm ............................................................................................................ 72

Figure 83 T.O result Mesh size 2 mm ........................................................................................... 72

Figure 84 Stress analysis result (top view) (left) and T.O result (right) ....................................... 73

Figure 85 Stress analysis result (side view) (left) and T.O result (right) ...................................... 74

Figure 86 Updated design volume according to checklist ............................................................ 81

Figure 87 Design volume .............................................................................................................. 82

Figure 88 Partitioned bearing seats ............................................................................................... 82

Figure 89 Partitioned bolts ............................................................................................................ 82

Figure 90 Design space (brown parts) and NDS (Gray parts) ...................................................... 82 Figure 91 T.O result ...................................................................................................................... 83

Figure 92 Topology concept in .stp format ................................................................................... 83

Figure 93 Partitioned internal layout ............................................................................................. 83

Figure 94 Internal layout superimposed on T.O results ................................................................ 84

Figure 95 Simplified fillets ........................................................................................................... 85

Figure 96 Simplified chamfers ...................................................................................................... 85

Figure 97 Design space partitioned through BOOLEAN operation ............................................. 86

Figure 98 Meshed models of Design and Non-design space ........................................................ 86

Figure 99 Surfaces of Design and Non-design space .................................................................... 87

Figure 100Volume mesh of design and non-design space ............................................................ 87

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Figure 101 Bearing forces modelled in Inspire ............................................................................. 89

Figure 102 3D bolts modelled in Inspire ....................................................................................... 89

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LIST OF TABLES

Table 1. Bearing reaction forces.................................................................................................... 40

Table 2 software accuracy model set up ....................................................................................... 45

Table 3 Software combinations ..................................................................................................... 47

Table 4 Response function study .................................................................................................. 49

Table 5 D. O. E .............................................................................................................................. 54

Table 6 Software characteristic comparison ................................................................................ 61

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1 INTRODUCTION

This chapter describes the background, purpose, objectives and the limitations of the project. The

method by which this thesis was performed is briefly discussed. This thesis is commissioned by GKN

to improve the existing topology optimization methodology of housing structures with the focus on

improving the design interpretability of the topology results.

1.1 Background

Generally, in the traditional design process, simulation is used for validation at the final design

stages, based on which only minor modifications can be made. A better way to generate a good

design concept is to use simulation for creating early design proposals. This is the idea of

simulation driven design. Optimization is used as a practical design tool to change the design

process to a process driven by computational analysis. In the last decade, topology optimization

has become a topic of growing interest in the industry. Finite element-based optimization

algorithms have advanced, and hence, the use of commercial software is increasing rapidly.

GKN Automotive is the industry leader in engineering drive system technologies. GKN Köping

AB specializes in design, engineering and development of All Wheel Drive (AWD) components.

Their innovative AWD intelligently transfers torque between wheels based on traction

requirements. The main components of the AWD driveline solution consists of a Power Transfer

Unit (PTU), Propeller shaft and a Rear Drive Unit (RDU) as show in Figure 1. The PTU transmits

power from the vehicle transmission gearbox to the RDU through the propeller shaft. These PTUs

and RDUs manufactured are hypoid gearboxes.

Figure 1 AWD components

The current method at GKN for designing the gearboxes is an iterative process. The initial design

is proposed by the design team and then computationally analyzed by the CAE team. With these

results, possible design improvements are made by the design team in order to satisfy the structural

and design requirements. This modified design is again analyzed, and the process is iterated. This

back and forth iteration process may increase lead time for product development. Even though it

is a partially iterative process with topology optimization being used in the product development

process, the current process could be improved with a better results.

Simulation driven design brings in the idea of using computational analysis right from the early

stages of product development to make calculated engineering judgments [12]. Here, the initial

concept is developed using mathematical optimization to result in an optimal design with minimal

material usage. Assisting design engineers with simulation during early design stages helps reduce

the number of back and forth iterations between the design and CAE engineers. The well-

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established numerical modeling methods and modern computational capabilities open new

possibilities for simulation backed product development to be used in the industry. If a clear design

is suggested through topology optimization the number of back and forth iterations could be

reduced.

1.2 Purpose

The thesis aims to improve the product development process through improving the existing

topology optimization method. The purpose of the thesis is to propose a methodology that can be

followed to produce an improved optimal initial design concept for housing structures.

The purpose of proposing a methodology can be further synthesized as follows:

To obtain better, precise and clear results from topology optimization

To interpret the results in a more useful way in the design process To reduce the number of iteration loops in the design process

1.3 Objectives

The main objectives of this work are to:

1. Propose an improved topology optimization methodology for the PTU housing structure

2. Improve design interpretability of the optimization results

To attain the objectives, the following research questions are to be answered:

How, and where, in the design process will Topology Optimization be most useful for the

designers?

How can the HyperWorks software suite be utilized to its maximum capabilities to perform

topology optimization efficiently and appropriately?

From what dimensionalities should the problem be approached to achieve desired results?

What should the developed methodology entail?

Detailed and specific questions for the different dimensionality aspects through which the

problem is approached are listed in their respective sections.

To what extent can the methodology be standardized for PTU housing structures?

1.4 Delimitations and limitations

The thesis was restricted to not investigate and significantly modify the product

development process. Only minor alterations in the overall process could be made to fit in

the developed methodology.

Only static load cases are used in both the experimentation and implementation phases.

Effect of vibrations and noises are not considered in this thesis.

Modal load cases are used to show that this methodology can be expanded for other load

cases as well. D. O. E’s for selecting optimization parameters for modal load cases are not

in the scope of this thesis.

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Due to the complex nature of the housing structure, the design realization of the proposed

topology concept is performed by the design team using CATIA.

Multiple manual iterations in the design realization are avoided due to the time frame of

the thesis.

Stress analysis is performed for the implementation phase and is done using HyperMesh

OptiStruct.

Non-linear structural analysis and optimization was deemed not necessary for the thesis

considering the computational time.

The volume of the design space is considered a constant and is not modified in this project

1.5 Method

The project starts with collecting theoretical knowledge and information on the topics of CAE

fundamentals, topology optimization, and on the product and Way of Work at GKN. The current

performance of Topology Optimization at GKN is comprehended, and the problems are identified.

Requirements and objectives for the project are set based on discussions with the design and CAE

teams. This is followed by a literature study on state-of-the-art optimization procedures and

relevant case studies. Knowledge pertaining to general optimization processes and configurations

was gathered through literature. D. O. Es were conducted to gain knowledge specific to topology

optimization of housing structures and to the know-how of improving design interpretability and

for the selection of design volume, objective functions, constraints, geometric and manufacturing

parameters.

A detailed appraisal of the functionalities of commercial topology optimization software

(HyperWorks suite) is done to understand how the various software can be utilized to the

maximum capacity to perform topology optimization. The commercial software used in this thesis

are Inspire, SimLab, HyperMesh and OptiStruct. In some optimization procedures, HyperView is

used for post-processing. CADDoctor is used for design simplification.

With the knowledge gathered from all these sources, a methodology for topology optimization of

PTU housing structures that can be integrated into the current design process is proposed. This

methodology is verified by implementing it on a design model. Detailed implementation steps for

executing this method is described in Appendix H.

Design realization of the proposed topology concept is done by the design team using CATIA.

Linear stress analysis is performed on the realized design for validating the proposed topology

concept.

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2 FRAME OF REFERENCE

This chapter starts by explaining the basics of solid mechanics and builds on it, the theory of Finite

Element Method and the underlying mathematics involved in it. Introduced then are the concepts

of Optimization and Solution methods, followed by numerical methods implemented in the

commercial software to form the optimization problem. Finally, the chapter discusses optimization

from various perspectives, the current status of optimization at GKN, and is concluded by the

problem formulation along with solutions.

2.1 Theory

When an engineering problem deals with solid structures, the theory of solid mechanics is used to

define the governing physics of the problem. The analytical application of solid mechanics has its

limitations when applied to real-world engineering problems. When the boundaries of solid

mechanics are reached, the concept of FEM can be introduced. FEM is regarded as, by far, the

most successful approach in the engineering world to solve complicated geometries [11].

2.1.1 Real-world problems, Engineering models and FEM

A system refers to the collection of entities that performs a function(s). In the real world, a system

produces a particular output for specific input. In engineering, a model is considered an

approximate and abstract description of the complex functions of this real-world system.

Generally, a system is modeled at various levels of complexities depending on the engineering

requirement.

Different kinds of models are developed to understand the functions of a system from different

perspectives. Analysis models are built based on engineering science to determine how the entities

of a system are related to each other to perform a function. Design Models are constructed from

the analysis models for prediction and decision-making tasks. In this thesis, an analysis model is

set up for Static and Modal analysis, and a design model is constructed for structural optimization.

A model can be expressed mathematically by a set of mathematical relations, consisting of y as output, x as input, and the function f(x) representing the system. This function f(x) represents the

laws of physics of the system.

Structural design can be modeled mathematically by defining its geometric configuration,

materials used, boundary conditions, and loads and by quantifying them. These assigned values

must satisfy the mathematical relationship describing the task performed by the structure.

When a physical system in the real world is a continuous solid structure, it translates into a

continuous variable problem. Functions used to express such Continuous variable problems are

derived from the principles of Solid Mechanics in the form of Partial Differential Equations

(P.D.E) with Boundary Conditions (B.C) that can be solved analytically. But it becomes expensive

and sometimes impossible to computationally solve these analytical equations for complicated

geometries and loads, which is the case for almost every engineering application.

To reduce the computational complexity, the analytical models developed to represent such

systems are generally oversimplified. To attain a realistic output (y) for these simplified analytical

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models, a safety factor according to established standards and based on experience, is introduced.

But for engineering applications, the accuracy requirements are very high.

Hence, solid structure problems cannot be satisfactorily solved by this simplified analytical

method. The concepts discussed in this section are referred from literature [11].

Finite Element analysis

To solve engineering problems with high accuracy FEA is used. FEA is a numerical method based

on the solid mechanics theory. It discretizes the continuous complex geometric structure into

smaller entities of regular geometric shapes, interconnected at common points. These smaller

entities are called elements and the connection points are the nodes [17]. A set of algebraic

equations are formed at the nodes. Initially, the unknown first nodal quantity, i.e., displacement,

is determined in the context of this thesis. All other secondary quantities that are to be calculated

(in our context - Stress and Strain) are expressed in terms of the primary nodal quantity. These

algebraic equations are now expressed in the form of matrices of a higher order, which is a

convenient and computationally effective way to solve when there are a number of unknowns. In

a structural component, there are usually millions of unknowns to be solved for.

FEM Terminologies and concepts used in this thesis

The concepts discussed in this sub-section are referred from [5] and [17].

Shape functions - The structural responses at non-nodal points are calculated using an

approximation function called shape functions between two nodal points. These shape functions

are polynomial functions of various orders – the complexity and accuracy of the solution increase

with increasing order of this polynomial functions.

Degree of Freedom - Represents the number of field variables needed to describe a nodal point

in a FE model.

Isotropic material and anisotropic material - Anisotropic materials are dependent on the

orientation of the axis. These materials are used when there is a need for mechanical property to

differ when measured from different axes.

Isotropic material is independent of direction and has identical material properties in all directions.

For this thesis, isotropic materials are used for all the FE models.

Linear, Non-Linear analysis - Linear analysis is used on a structural problem when the linear

relations represent the equation defining the structural properties. In static analysis, linear analysis

is applied when the stress is in an elastic range. Stiffness matrix remains constant during this linear

static analysis. All the FE models used in this thesis are applied using linear analysis.

In non-linear analysis, the equations governing the structural properties are non-linear; hence, in

the static analysis, the stiffness matrix is not constant. This non-linear relationship of the governing equations arises from the material or geometric nonlinearity of the structure when the structure

reaches the plastic stress.

Material and Geometric Nonlinearity - When the stress applied to a structure is above the plastic

limit, the material starts to deform, and this deformation is non-linear. This non-linear deformation

of the material and the geometry is captured by using non-linear materials and defining geometries

as nonlinearity.

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Mesh Refinement - The accuracy of the FE solution increases as the mesh is refined. When the

mesh is refined, more nodal quantities are available, and hence the approximation for non-nodal

points is more accurate. The mesh is refined either by increasing the number of elements (h-

refinement) or increasing the order of the polynomials (p-refinement) used as the interpolation

functions.

Stiffness matrix - For a structural problem, the linear or non-linear behavior of a structure is

captured by the stiffness matrices.

2.1.2 Optimization

Once the basics of FEM is understood, the concept of optimization and the mathematics involved

is introduced, followed by the application of commercial software using HyperWorks suite. Next,

the specific theory pertaining to this thesis is presented. If further reference is needed, the reader

is referred to [3].

Optimization Model

A design model becomes a decision-making model when evaluation criteria are used to choose the

best design for a specific purpose, in this case, optimization [11]. In an optimization model, the

evaluation criteria are the objective functions, and the best design selected is the optimal design

obtained from the optimization. The definition of an optimal or best design is subjective and must

be based on the project's requirement. Finding this best design is the primary purpose of

optimization.

Mathematical Model

The optimization problem can be mathematically expressed as either a minimization or

maximization function f(x) of a desired property subjected to constraints. In the engineering

domain, some common properties of interest are minimization of compliance, minimizing mass, and minimizing volume fraction. This f(x), called the objective function, is a function of design

variables x and provides the optimum solution for the problem. According to [10], this

optimization problem can be numerically expressed as

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓(𝑥) (1)

𝑊ℎ𝑒𝑟𝑒 𝑥 =

{

𝑥1𝑥2..𝑥𝑛}

Subjected to {𝑔𝑖(𝑥) ≤ 0, 𝑖 = 1,2… .𝑚)ℎ𝑖(𝑥) = 0, 𝑗 = 1,2… . 𝑛)

}

x is the vector of design variables, which are independent entities that define a model. Design

variables can define the geometries either directly (e.g. dimensions) or indirectly.

𝑔(𝑥) and ℎ(𝑥) are the state variables i.e. dependent variables that record or capture the response

of the model. These variables are usually used to define the constraints for the optimization

problem.

22

Response function – The behavior of a system, modeled using the design variables x is captured

by the state variables 𝑔(𝑥), ℎ(𝑥), and 𝑓(𝑥). These are the outputs or the responses of the systems

and hence are called response functions.

Objective function – For an optimization model built with a response function 𝑓(𝑥). as an

evaluation criterion to choose the optimum design, the function 𝑓(𝑥). becomes the objective of

the function. It is a scalar value formulated from a set of design responses of all nodal points. [18].

Constraint function – A constraint is a condition for an optimization problem that must be satisfied

by the solution. If constraints are violated, then the resulting design is not feasible. In reference to

equation (1), they can either be an equality function (ℎ(𝑥)) or an inequality function (𝑔(𝑥)). These

constraints are formulated from a single scalar value. [18].

Structural engineering optimization problems are non-linear in nature. A non-linear optimization

model has either objectives or constraints that are non-linear.

Structural Optimization

In this section, the concept of optimization is applied to engineering structures to find the optimum

layout for a linear elastic structure. Layout in the structural design context means the size, shape,

or topology of the structural features. Hence, size optimization, shape optimization and topology

optimization are the three aspects of structural optimization. The optimization model is built on

the analysis model developed using the Finite Element method mentioned in section to perform

structural optimization.

Taken from [3], size, shape and topology optimization are as follows:

Size optimization – Here, the design variables representing structural parameters of aspects of size

like thickness is varied.

Shape Optimization – Here, the design variables representing the boundaries of the design like

diameter of a hole are varied.

Topology Optimization – Here, the design variables x represents the location of material in space.

These design variables are varied to find the optimum distribution of material and voids.

2.1.3 Topology Optimization

23

T.O is a mathematical procedure that optimizes material layout within a given design space, for a

given set of loads, boundary conditions and constraints in order to maximize the performance of

the structure. Basically, topology optimization decides where to place the isotropic material and

where to remove material in the given design domain to reach the optimum design. [3]. For the

material placement to be decided, the design domain is discretized into Finite Elements. Then the

links between every element are decided based on the optimization methods (section). [4].

The optimal design as a result of topology optimization will contain minimum number of elements

as the subset of the whole design domain. The design variable is represented as elemental densities

ρe, and the relation between stiffness and density is linear, as shown below. [9]

𝐾(𝜌) = 𝜌 ∗ 𝐾 (2)

𝜌 = {0 = 𝑉𝑜𝑖𝑑 1 = 𝑀𝑎𝑡𝑒𝑟𝑖𝑎𝑙

}

The design variable is the elemental density vector, assigned a value of either 1 or 0. Numerically, value 1 represents that the material is present at a particular coordinate, and 0 represents that the

point in space is a void.

Since FEM discretizes the complex design domain into a large number of discrete elements (N),

the optimization becomes a large-scale discrete optimization problem [10]. The number of

combinations here is 2^N, and it becomes an NP-hard problem [7]. The time consumption and

computation cost exponentially increase with the geometric complexity. It becomes an impossible

problem to solve this large set of matrices using existing algorithms.

The discrete design variable function from Equation (2) is mathematically converted to a

continuous variable problem using established methods to solve this problem. This allows calculus

to be applied for solving the minimization or maximization optimization problem. A well-

established method, which is applied in this project, is the SIMP method. SIMP is the abbreviation

for "Solid Isotropic Material with Penalization."

Material Distribution Method – SIMP Interpolation

In this method, the material density, which is the design variable, is continuously varied from 0 to

1. The intermediate values between 0 and 1 are assigned assuming that the density varies linearly

from 0 to the density of the material corresponding to the stiffness value. Stiffness is dependent on

the density and is directly proportional and shown as

𝐾(𝜌) 𝛼 𝜌 (3)

𝐾(𝜌) = 𝜌 ∗ 𝐾 (4)

0 < 𝜌 < 1

Because of the nature of the SIMP interpolation, intermediate densities are formed in the material

distribution of the topology optimization results. In engineering applications, these topologies with

intermediate densities are difficult to interpret and impossible to manufacture. The analysis result

of the component with intermediate densities will be largely offset from the actual structure. The

presence of intermediate densities stiffens the structure and does not give a realistic interpretable

concept as the topology result. This intermediate density problem is eliminated by introducing the

idea of penalization. [10][11][3].

24

Penalization

The penalization factor is introduced to avoid intermediate density values and to convert the

topology results into a manufacturable design. The penalized stiffness of the material is

represented as K(ρ) and the penalization factor is represented as p. The value of the penalization

determines how accurately the intermediate density values are avoided. As illustrated in the graph,

the penalization value of 3 is recommended for elements with a Poison ration of 0.3. [10][11][3]

𝐾(𝜌) = 𝜌𝑃 ∗ 𝐾 (5)

Figure 2 Relative stiffness Vs Density for different penalization factors. Figure courtesy [10]

Drawbacks of The SIMP Method

1. Even though sophisticated penalization techniques are applied to remove elements with

intermediate densities, it is theoretically impossible to remove all such elements. The

presence of these intermediate densities affects the structural performance of the

component, and the significance of its influence is inversely proportional to how well they

are penalized.

2. Comparing topology structures formed using different penalization techniques might lead

to a misleading result interpretation. [3].

Solutions Concerning Penalization

Some solutions that can be implemented in OptiStruct, directed to solving the drawbacks

concerning penalization, are discussed here. The topology structures resulting in an indiscrete

solution can be rectified in the follows ways [1].

1. The volume fraction constrains assigned too low for the mesh refinement used for the FE

model should be increased.

2. The maximum number of iterations set in the solver is insufficient to get convergence in

the result for a discrete design. The default - maximum number of iterations should be

changed and increased.

3. Sometimes the penalty factor must be changed to a value other than the default value of 3.

25

4. Objective tolerance can be reduced.

Problem Formulation

The optimization can now be formulated using the SIMP interpolation as follows.

Min or Max 𝑓(𝜌) (6)

Subject to constraints 𝑔(𝜌) and ℎ(𝜌)

Where 𝜌 a vector of all elemental densities is 𝜌𝑒. The value of 𝜌 varies between 0 ≤ 𝜌 ≤ 1. 𝜌𝑒 belongs to the entire design space.

The constraints 𝑔(𝜌) and ℎ(𝜌) can be either equality or inequality constraints.

The important objective function and constraints used in this thesis are minimizing compliance

and minimizing the volume fraction.

Minimizing compliance

Compliance is defined as the equivalent strain energy of a structure. When the strain energy of the

structure is minimized, the structural stiffness increases. Therefore, the problem of maximizing

the stiffness of the structure is formulated by minimizing the compliance of the FE model. The

compliance for this project can be defined as according to, [1][7],

𝐶 =

1

2𝑢𝑇𝑓 (7)

Here, 𝑢 and 𝑓 are the displacement and force vectors of respectively containing all the elemental

displacements and force.

When the equation of motion is expressed as

𝑓 = 𝐾 ∗ 𝑢 (8)

where 𝐾 is a vector of all elemental stiffness, 𝐾𝑒

When the optimization problem is formulated according to ()

Min 𝐶(𝜌) =

1

2𝑢𝑇𝑓 (9)

Subject to 𝐾(𝜌) ∗ 𝑢 = 𝑓

Minimize Volume Fraction

The optimization problem can be formed for minimizing volume fraction. [9].

Min 𝑉(𝜌) = 𝜌𝑃 ∗ 𝑉0 (10)

26

Here, 𝜌 is the density vector of all elemental densities. 𝑉0 is the initial volume.

Note: When minimizing volume fraction is used as the objective function, stress constraint or

displacement constraints must be used to avoid all the volume being removed.

Computational Procedures

In the HyperWorks software, the optimization problem is solved using an iterative procedure

known as the local approximation method. Please refer to [1] for further reference. This follows

the following steps

1. Finite element Analysis of the structural problem.

2. Verification of whether convergence is achieved.

3. Screening of response function to retain the active responses.

4. Sensitivity Analysis on the retained responses from step 4.

5. Optimization using dual gradient based optimization method formulated using sensitivity

information.

Convergence verification

Convergence of an optimization problem is verified using two types of convergence tests. [1]

1. Regular convergence is achieved when the objective function change in consecutive iterations

is less than the objective tolerance and the constraint function violation is within a 1% range.

2. Soft convergence is achieved when the design variable change is minimal for two consecutive

iterations.

The optimization software displays the message “Design is feasible” when the convergence criteria

is achieved and displays “No feasible design is achieved” otherwise.

Sensitivity Analysis

As mentioned earlier, in the structural FEM analysis all the secondary nodal quantities i.e.

responses are calculated in terms of the primary nodal quantity i.e. displacement. The response

quantity, g, is expressed in terms of displacement as

𝑔 = 𝑞𝑇 ∗ 𝑢 (11)

Sensitivity of a response function can be calculated through two approaches. [1][3].

The sensitivity of a response function w.r.t design variable can be defined as the derivative of this

response w.r.t design variable x as

𝜕𝑔

𝜕𝑥=𝜕𝑞𝑇

𝜕𝑥∗ 𝑢 + 𝑞𝑇 ∗

𝜕𝑢

𝜕𝑥 (12)

When the equation of motion is expressed as

𝑓 = 𝐾 ∗ 𝑢 (13)

Its derivatives w.r.t design variable x and the sensitivity of displacement vector is

𝐾 ∗𝜕𝑢

𝜕𝑥=𝜕𝑓

𝜕𝑥−𝜕𝐾

𝜕𝑥∗ 𝑢 (14)

27

This is used in the equation (1). One forward-backward substitution is required for each design

variable. Typically, one to three design variables exist for every element and this makes the

computation expensive.

To reduce the computational cost, the second approach can be implemented by introducing a vector

a as shown below.

𝐾 ∗ 𝑎 = 𝑞 (15)

Using this, the sensitivity of the response constraint can be calculated as

𝜕𝑔

𝜕𝑥 =

𝜕𝑞𝑇

𝜕𝑥∗ 𝑢 + 𝑎𝑇 (

𝜕𝑓

𝜕𝑥 −

𝜕𝐾

𝜕𝑥 ∗ 𝑢) (16)

This second approach requires only one set of forward-backward substitution to calculate vector

a, for each of the retained constraints from the previous response screening step.

Gradient Based Dual Optimization Algorithm Method

The concept of gradient based optimality method is explained in a simplified way by using an

objective function of 2 independent variables x1 and x2. This method works by calculating the

steepest ascent or descent from a point and taking steps for optimization in this direction. The dual

optimizer algorithm is explained on a conceptual level in the later part of this section. First, the

partial derivatives of the objective function are formed as

∇𝑈 =

𝜕𝑈

𝜕𝑥1𝑒1 +

𝜕𝑈

𝜕𝑥2𝑒2 (17)

Here e1 and e2 are the base vectors. The initial direction for the steepest ascent or descent is

obtained by calculating the gradient vector at the values 𝑥10 and 𝑥2

0. The lengths to move in the

direction of the gradient vector is calculated by

𝜕𝑥1𝜕𝑥2

= 𝜕𝑈

𝜕𝑥1 𝜕𝑈

𝜕𝑥2⁄ (18)

At the new point (𝑥11, 𝑥2

1) = (𝑥10 + 𝜕𝑥1 , 𝑥2

0 + 𝜕𝑥2) the gradient vector ∇𝑈(𝑥11, 𝑥2

1), and the length

is obtained for the next steepest ascent or descent. This step is repeated until an optimum at least

corresponding to the local optimum is reached. In HyperWorks software which is used in this

thesis the method of steepest descent is used.

28

Figure 3 Representation of gradient based optimization method. Figure courtesy [11]

If a numerical approach is used instead of analytical method like this problem, the gradient is

calculated at 𝑥𝑖𝑗 and at 𝑥𝑖

𝑗 + 𝜕𝑥𝑖 and by following the equation

∇𝑈𝑖𝑗 = 𝑈𝑖

𝑗+1 − 𝑈𝑖

𝑗 (19)

The gradient vector is calculated in a straightforward way. [10].

𝜕𝑥𝑖𝑗

𝜕𝑥1𝑗 ≈

∇𝑈𝑖𝑗

∇𝑥𝑖𝑗⁄

∇𝑈1𝑗

∇𝑥1𝑗⁄

(20)

There exist many methods to calculate the step length in a numerical approach. [1]. In HyperWorks

the step lengths are calculated according to

𝑥𝑖+1 = 𝑥𝑖 − 𝛾∇𝑈(𝑥𝑖) (21)

{𝑖 | 1 ≤ 𝑖 ≤ 𝑛}

Here γ is a constant and is mentioned in the reference paper [6] and referred from [7].

As per the reference [18,] the dual algorithm is regarded as well-equipped for this kind of topology

optimization problems, where the number of design variables exceeds the number of constraints.

This algorithm [8] is implemented in the software used in this thesis.

Convex problem

An important characteristic which determines if a solution converges to an optima is the convexity

of an optimization problem. [10].

An optimization problem is of convex nature, when the feasible design space enclosed by the

constraints forms a convex or concave geometry. Linear objective and constraint functions by

29

default forms convex problems. Even when the constraints are convex, and the objective is either

convex or concave functions reliable solutions can be formed up to very large size of variables.

The term non-convex optimization is used to determine when the feasible region enclosed by the

constraints are neither convex nor concave geometry. The time taken for convergence of solution

to such a problem is exponentially high.

The initial problem is linearized and converted into separable explicit convex problems using the

linearizing scheme [6].

𝑐(𝑥) = 𝑐(𝑥0) + ∑ 𝑐𝑖

0(𝑥𝑖 − 𝑥𝑖0) −∑ (𝑥𝑖

0)2𝑐𝑖0 (1

𝑥𝑖−1

𝑥𝑖0)

−+ (22)

This equation is applied to an optimization problem described by objective and constraint

functions described as:

Objective: 𝐶0(𝑥) (23)

Constraints: 𝐶𝑗(𝑥) ≤ 0, (𝑗 = 1, , ,𝑚)

𝑥𝑖 ≤ 𝑥𝑖 ≤ 𝑥𝑖

and the design variables are normalized and transformed using the Lagrangian multiplier method

described in [17]. This makes the objective function separable and it can be solved using the

procedure explained in [17].

Drawbacks of Gradient optimization method

This section discusses the drawbacks [13] on a conceptual level, the nature and cause of these

drawbacks and their solution. The full extent of the solutions is explained in [3].

1. Mesh Refinement and Existence of solutions

For a topology optimization problem formulated on a design volume with the material being

discretized and refined using the FEM technique, it is found that multiple solutions exist for the

same problem description. The core reason for the existence of multiple solutions to the same

design volume defined with the same loading and boundary conditions is the inherent nature of

the concept of material discretization. The logic behind this is that the task assigned for the

algorithm is to find the most efficient design for the problem and thereby when the mesh is refined,

the algorithm can find a more efficient way of placing holes and solid elements as shown below.

30

Figure 4 Mesh refinement increases from a) to c) and the topology solution differs is shown. Figure courtesy [3]

If this is viewed from a design perspective, this quality of the optimization method is undesired. It

leads to ambiguity in how the best result must be quantified. From an output perspective when a

mesh is refined, clearer output of the same design is expected rather than a qualitatively different

result.

There are solutions to control the mesh dependency of the topology optimization problem.

1. Geometry Constraint methods - Perimeter control, Member size control

2. Gradient methods – Local and Global

3. Filter methods

The local gradient and filtering methods remove thin structures. The geometry and the global

gradient methods do not restrict the formation of thin structures.

2. Checkerboard effect

Checkerboard effect refers to the pattern formed when alternating solid and void elements are

placed over the domain of the design space during topology optimization. According to [3],

topology optimization gives solutions with checkerboard pattern when material distribution

method is directly applied on displacement based finite element method. The origin of this problem

is related to the features of finite element approximations in which the numerical modelling

overestimates the stiffness of the checkboards according to Bendsoe and Sigmund (mentioned in

section 1.3.2 in the literature [3]).

Figure 5 Checkerboard effect in 3D structure used in the thesis

31

Optimization configurations for Optimization configurations for

The effect can be suppressed by many ways like

Sensitive filters

Geometry constrains

Higher order elements

Figure 6 a) Design Volume b) Checkerboard effect c) Solution. Figure courtesy [3]

Design Volume, checkerboard effect and the solution are shown in Figure 6.

3. Non-Uniqueness, Local Minima and Dependence on Data

a. Global Optimum

The nature of the structural topology optimization problem and the solution methods makes the

problem non-convex and there exists multiple solutions. It is impossible to be sure that the design

attained is the global minima and one can find several different local minima for the same problem.

It should also be observed that one can obtain different solutions to the same discretized problem

when choosing different parameters of the algorithm.

However, this problem can be addressed by applying continuation method to move closer towards

convergence of reliably good design. The overall idea is to gradually change back the nature of

convex problem to non-convex problem in a step by step manner. In each step the convergence is

ensured using the gradient based optimization algorithm.

b. Dependence on data

The solution obtained through topology optimization heavily relies on the input data used for

applying the optimization procedure. This means that the optimum design obtained is extremely

sensitive to all the entities that builds the topology problem. It should be understood that the design

domain, loads, boundary conditions, the geometry control and other parameters must be very

specifically identified and defined for a particular problem.

2.1.4 Optimization processes

32

From literature, the optimization process can be carried out in few ways

1 Single step optimization

2 Sequential optimization – This method suggests increasing the complexity of the problem in

subsequent iterations.

3 DomainSub Domain optimization/Domain decomposition [12] – This method addresses the

problem of complex geometries by splitting the CAD into simpler geometries and applying

loads accordingly.

2.2 Component

AWD provides better traction control and driving dynamics for the vehicle by distributing power

and torque between the rear and front wheels. The Power Transfer Unit (PTU) is an integral

component of the All-wheel Driveline (AWD). The PTU transfers rotatory power from the Engine

to the rear wheels through the propeller shaft.

The housing structure provides the structural integrity for the PTU. It is a complicated geometry

with a thin sheet of average minimum thickness 4 mm and is casted in Aluminum. There are

various internal components to the PTU such as the ring gear welded to tubular shaft, pinion gear

and shaft, clutches, spacers, bearings, seals, washers, flanges, circulating oil and breather nipples.

Most of these components are rotatory and not connected directly to the Housing. Bearings are the

only machine element that hold all these rotatory components by supporting the shafts mounted

onto the housing. All the reaction forces are transferred to the housing through these bearings. The

housing facilitates oil circulation within PTU.

Figure 7 PTU cross section

The component considered in this thesis for building the optimization model are the housing

structure, cover, gearbox and the bracket. Even though optimization is performed on the housing

structure and the cover, the bracket and the gearbox are considered for a more realistic

representation of the boundary condition in order to avoid infinitely stiff boundary conditions at

the mounting points of the housing and cover.

The CAD models of the design volumes used in the optimization problem can be seen in Figure

6Figure 8 and Figure 9. The models are provided by the design team at GKN. Model 1 (Oval

concept) is used during the experimentation phase to develop the methodology and Model 2

(Round concept) is used for the implementation phase for verifying the methodology. Using

separate models checks the robustness of the methodology.

33

Figure 8 Model 1 (Oval concept used in

Experimentation)

Figure 9. Model 2 (Round concept used in

Implementation phase)

2.3 Current product development Process

A flowchart depicting the product development process at GKN is shown in Figure 10. The process

of product development is complicated, involving communication and knowledge transfer among

3 teams – design, systems calculation and CAE. The design process moves parallel to the

calculation process and hence suggesting an initial concept for design through topology

calculations as early as possible is preferred.

The T.O has to be placed in this process considering two things:

1. When the inputs for T.O will be available

2. When the results of T.O will be useful for the design team

Simulation driven design [12] is used to improve the process by producing computationally backed

up T.O results that can be used in the early stage of product design. Using these results early on in

the process provides a right trajectory for designing and reduces the lead time.

Hence, T.O is placed once the initial gear layout and bearing reaction forces are calculated as

shown.

Figure 10 Flowchart: Product development process for housing structures at GKN

34

2.4 Current optimization methodology at GKN

Figure 11 Flowchart of current topology optimization methodology at GKN

GKN performs topology optimization using the HyperWorks suite. The FE Model is setup using

the HyperMesh as pre-processor and OptiStruct is the solver used for carrying out the optimization

problem. The results are visualised using the post processor HyperMesh. These three tools are part

of Altair HyperWorks suite.

The steps involved for the current optimization methodology is straightforward: loads and

boundary conditions are applied on the design volume. Optimization configurations including the

objectives, constraints and other parameters are applied to the model. The design volume is created

by design team and the CAE team performs topology optimization on the produced design volume.

The current optimization methodology is illustrated in the flowchart as shown in Figure 11.

The goal is to remove as much material as possible from the design volume at locations where it

is not necessary and form clear rib structures. But as can be seen in Figure 11, sufficient material

is not removed to form clear rib structures that can be better interpreted. Hence, the visualization

of the results is not clear and knowledge transfer from the T.O to the finalized results is poor.

2.5 Problem identification

Three problems are identified for forming an undesirable design concept from topology

optimization. These problems are identified from different perspectives of T.O. discussed in

section 2.6.

1. A major reason is due to the complex nature of the housing geometry. Optimization tends

to give clear results for simple structures and the results become messy when the

complexity increases as shown in Figure 12.

Figure 12 Comparison of topology results for simple and complicated geometries

35

2. Another problem identified is over constraining the optimization model with load cases

and the following input parameters:

Objective functions and constraints

Manufacturing, Member thickness, stress and frequency parameters

3. Thirdly, a conventional linear process for topology optimization is not efficient when

applied on complicated structures such as housing.

2.6 Areas of focus for this thesis

The thesis will deal with three stand points as focal perspectives to solve the problems identified

above

2.7.1 Design perspective

The DomainSub-domain method mentioned in the section 2.1.4 Optimization processes is taken

as the inspiration for this perspective. Unfortunately, in our case, it is not possible to partition the

housing into smaller parts since forces at the split sections cannot be measured.

Hence, this approach cannot be used directly but the principle of this approach is carried forward

in simplifying the design volume. Translating this method into this thesis is done by choosing the

appropriate geometry of design volume to get the best T.O results.

Figure 13 Split CAD model of partitioning the housing structure

2.7.2 CAE perspective

From a calculation perspective, another solution to consider is the choice of appropriate response

functions for the optimization inputs. The objectives and the responses used for the optimization

models should be standardized.

2.7.3 Process perspective

Sequential optimization method discussed in section suggests increasing the complexity of the

optimization problem iteratively. From a process perspective for this thesis, it would be wise to

increase the geometric complexity and the optimization configuration complexity iteratively. The

process should be flexible to incorporate modularized design volume, loads, inputs and

parameters.

36

2.7 Planned studies

As the topology optimization result heavily depends on design volume, response functions and

parameters, it is crucial to conduct experiments and select them accordingly. Hence, three studies

are conducted from which conclusions are derived. Conclusions are documented for a design

volume study, software study, and a response function and parameter study as illustrated in the

flowchart shown in Figure 14. As the first step, the influence of CAD input is understood and the

appropriate geometry for design volume definition is chosen. Next, the software are selected for

the chosen design volume. Lastly, the optimization configurations are selected. These conclusions

are used in proposing a new methodology.

Figure 14 Illustration of thesis methodology

Studies were not only conducted for methodology development in this thesis but also for

knowledge creation from a broader perspective. These are added in the appendix. This can be

useful for other projects on housing structure optimization.

Studies are conducted since the knowledge is not readily available. Each configuration and each

parameter for a topology model has to be identified through the experimentation of many different

combinations e.g., shape and size of design volume.

37

3 IMPLEMENTATION

In this chapter, the first section gives a general modelling specification. Then the studies conducted

are explained. At the end, the methodology and its implementation are discussed.

3.1 Modelling

Following are the steps of Topology optimization model setup.

1. FE Modelling of components

2. Contacts, Loads and Boundary conditions

3. Analysis step

4. Optimization model

3.1.1 FE modelling

Meshing

A linear finite element grid is modeled with solid 3D tetra elements. A reasonable mesh quality is

essential for a good result. Hence, an average element size of 5 mm is used for the design volume,

unless otherwise mentioned. This element size provides the so-called coarse mesh for the FE

model. A more refined mesh with an average size of 3mm was used for fillets; meshes with size

of 2 mm were used for the bolt holes. This meshing size was used to bring down the computational

time for optimization analysis.

To mesh complicated solid geometries, SimLab is preferred based on its advantageous

functionalities available. A two-dimensional tetrahedral surface mesh was generated and used to

create the solid mesh. Once a component is meshed, the element quality is checked.

Figure 15. FE meshed model

Bearing Force Modelling

The bearing forces obtained from the system calculation department at GKN are applied to the PTU housing through 1 dimensional rigid body elements (RBE). All the nodes on the internal

surface of the bearing seat are connected to a center node using RB elements. A RBE is a node to

node connection with infinite stiffness which transfers all the forces and moments or the DOF’s

from one node to another. The RBE2 allows connecting one node to multiple nodes and distributes

the forces evenly among all the connected nodes. These RBE2 elements are used to apply the

bearing reaction forces to housing. The reaction loads are applied on the center node and they are

evenly distributed to the nodes on the bearing surfaces, as mentioned. In reality, even though the

forces are not evenly distributed on the bearing surface, this simplification is the standard followed

at GKN driveline to model the Bearing loads.

38

Figure 16. Locations of application of bearing reaction forces and the RBEs used

Bolts

The 4 components constituting the housing assembly model used for optimization are connected

using bolt connections. During FE modelling, the RBE2 elements are used as modelling

approximation of the bolts (as shown in Figure 17) which hold the parts together. The nodes on

the entire surface of bolt holes are connected to a center node using an infinitely stiff connector.

All the nodes behave as a single entity since they have the same DOF.

Figure 17 Bolts modelling

The Mounting points

Similar to the bolts, the mounting points at the gearbox and also at the brackets are modeled using

RBE2 elements.

Figure 18 Mounting points

39

Modelling of the Shafts, Simplified Gears and Bearings

1) Simplified Gears and Shafts

The FE models of the shafts and the gears without gear teeth and spline are simplified. This

simplification is done as an acceptable compromise between result accuracy and computation time.

This model simplification results in the intersection of elements between the gear models at the

gear contact. The gear contact is defined by RBEs. This simplified gear modelling doesn't have a

significant effect on the result for modal analysis.

The gears and shafts are meshed in HyperMesh using the automesh feature. First a 2D mesh is

created for the upper cross section. This 2D mesh is spun 360 degree with 80 elements on the

circumference. This is a relatively coarse mesh. The common set of nodes at the start and end

location of the spin are paired using the equivalence option.

2) Bearing

The complete bearing model is not used for the FE modelling. Instead, a simplified model with

only the outer ring and the inner ring is modeled. These rings are meshed similar to the gears as

discussed above. First order Hexagonal mesh with a relatively coarse mesh is used. The inner

surface of these 2 separate rings are connected to their respective center nodes using RBE2

elements. These 2 center nodes are connected using a CBUSH element. CBUSH is a generalized

spring-damper structural element. The CBUSH allows to define rotational stiffness in one desired

direction and assigns infinite stiffness in all other D.O.F.

3.1.2 Contacts. Load Cases and Boundary Conditions

a) Contacts

When the CAD model has to be partitioned and meshed as separate components, the elements in

the respective components will not have any interactions with each other. This means that forces

are not transferred, and the components are left either without support or without boundary conditions. Then, the interactions between such elements are modeled using contacts. These

interactions are modeled as being glued together using the FREEZE or TIE contacts, since both

the static and modal analysis performed in the project are linear. The 3 contacts modeled are the

bearing seats and the housing, the bolts and the housing, and the inner layout and the housing as

shown in Figure 19.

Figure 19 Freeze contact between the inner layout and the housing

40

b) Static Loads

The reaction forces and the moments from the bearings are calculated from the road load data

using Romex, a systems simulation tool and are shown in Table 1. The PTU is loaded in 2 different

ways, namely, Drive and the Coast. Drive is the forward loaded condition of the PTU at and Coast

is the reverse loaded condition.

Table 1. Bearing reaction forces

Hard points are a so-called term used to collectively refer to the points of application of the reaction

forces and boundary constraints. The drive and the coast loads are applied in bearing locations as

shown in Figure 16. These are applied on the bearing center points as separate loads. In

HyperMesh, these 2 loads are assigned separate time steps, and both are synchronously considered

in the stress analysis.

c) Boundary conditions

The corresponding center nodes of the RBEs shown in location of mounting points as shown in

the Figure 18 are fixed in all 6 directions at the mounting points. This means that these points are

grounded with respect to space.

3.2 Studies

The HyperWorks suite has various software to perform topology optimization, namely, Inspire,

SimLab and HyperMesh. The topology optimization analysis using these HyperWorks software

has many different functionalities and parameters. To get a deeper understanding of the software

and their various functionalities, studies are planned and conducted. This benefits one to choose

them specific to the housing structure and effectively apply them to build the optimization

problem.

For satisfying the requirements set for the thesis (mentioned before) of obtaining clear rib

structures, the question arises whether it can be achieved by just controlling the parameters and

functionalities in the optimization tools. By understanding the optimization procedures and the

algorithms running the commercial tool, it is observed that the design input is a significant part in

determining the topology output (refer part).

41

Hence, focus should be given on understanding the housing structure and the design input, in

addition to studying the software and optimization parameters.

Studies are therefore conducted in three perspectives, namely,

1. Design Input

2. Software and Process

3. Optimization inputs / Configurations

3.2.1 Design Volume Study

The CAD input i.e. the Design Volume, is partitioned into two spaces - Design space and Non-

design space. The optimization tool explores the design space to find the optimum design layout,

whereas the Non-design space is untouched by the algorithm.

This study was conducted to define the geometric input / the CAD model and to make the process

of preparing and partitioning the cad inputs in the most time efficient way. Hence, answers to the

following questions are sought in this study:

1. What is the influence of the initial size of the Design Volume?

2. What is the effect of the geometry of Design space and Non-design space?

3. How to select the design space for the D.O.E study and methodology development?

4. What is the most time efficient Way of Work for creating the design Volume i.e work

delegation among the design and CAE teams?

Process Perspective to identify the most time efficient process

The motivation behind this study is to see if, when the CAE Engineers create the design volume,

the process is more time efficient and the freedom and capability to modify the design volume is

worth the effort taken.

Design volume can be created through the following two approaches

● From Baseline design.

● From the internal gear and bearing layout and packaging space.

Figure 20 Baseline design Figure 21 Packaging environment

Design volume is created using CAE tools for both approaches.

42

For the baseline design approach, Inspire has options like Simplify – Rounds, Holes, Plug etc. for

geometric simplification. The drawback is that Inspire is not capable of simplifying complex

geometries with concave and convex fillets intersection [Information collected from Altair

support]. In the quest for searching for a tool which could effectively simplify complex geometries

in the HyperWorks suite, CAD doctor (Appendix F) was identified to be capable of removing all

the fillets automatically. After removing the fillets, the next step would be to fill the sockets with

material and to create a solid geometry on top of it, which is again done in Inspire.

The second approach was worked on in another internal project with Altair.

The conclusions for the best Way of Work are discussed in the results section.

Design volume creation

The stage by stage designing of the PTU housing was closely studied in order to get a feeling of

the structure and understand how it’s sequentially built to know where and how optimization could

be of support to the process.

The initial design volume provided by the design team needed many backs and forth iterations and

proved to have many complications; hence, it was not possible to work with during

implementation. The author of this thesis worked closely with the design team to create a design

volume as shown in Figure 22. In order to avoid such a situation, a checklist to establish the

standard for the design handover was proposed.

Figure 22 Updated Design volume

This new model was used during the methodology implementation phase. The reason for the new

model not being used in the other studies is that this design volume was created when working

parallel on the other studies. An advantage of using separate models during the experimentation

and implementation phase is that it checks the robustness of the developed methodology against

geometry.

Geometry of NON-DESIGN Space

Now that the Design Volume has been produced, it has to be partitioned into the design and non-

design space for the optimization tools to work on the optimization solution. The solution to the

optimization problem depends heavily on the geometry of the non-design space. Non design space

was defined and studied in many different combinations. Two such geometries which proved to

be most useful, and hence used in this project, are explained in detail under this section and will

hereafter be referred to as Design Volume 1 and Design Volume 2.

1) Design Volume 1

43

To build the optimization model with this approach, a non-design space as shown in Figure 23 is

defined as the four bearing seats and the bolt holes. The Inspire software is used for splitting the

geometry and is preferred over the other HyperWorks tools since this is a quick analysis. The

Partition option under the Geometry Modify menu is used to perform this operation. The

remainder of design volume is assigned to be the design space by enabling the Design Space check

box under property editor. [2]

An optimization model was set up with the following optimization configurations.

Analysis Type: Static

Objective: Minimize Mass

Constraint: Volume fraction 30%

Thickness – min 5.9948 mm

Load cases – Drive and coast

Loads and Boundary conditions are to be applied only on the non-design space. FREEZE contact

is applied between the bearing rings and bolts.

Figure 23 NDS Bearing seats and bolts

2) Design Volume 2

For the design volume 2, the non-design space is defined as the entire internal layout of the PTU

housing, with design space being the remaining design volume.

Modelling the internal layout in FE Tools

Design volume 2 when modeled using inspire proved to be inefficient due to the geometric

complexity of the non-design space. HyperMesh is the best suited for this design complexity.

It was observed that for the Design Volume 1, Inspire software can handle it well because of the

geometric simplicity.

HyperMesh

First, any element in the internal layout is selected. Next, in order to select all elements of the

internal layout, tools Elements, attached by face is used. Using the organize toolbar, the selected

elements are arranged in a separate component. This becomes the non-design space.

Inspire

44

As mentioned earlier, the design volume was too complicated to partition the internal layout using

Inspire. While creating the new design volume mentioned in that section, this issue was taken into

consideration and it was made possible to split the internal layout from the design volume CAD,

as shown in Figure 24 below. All the faces in the internal layout are selected and geometry → Partition is applied with a thickness of 3 mm. This splits the internal layout into a separate

component and makes it possible to assign it as the non-design space.

Figure 24 Internal layout

The following Optimization configurations are used for the model setup


Objective: Minimize Mass

Constraint: Volume fraction 30%

Thickness – min 5.9948 mm

Load cases – Drive and coast

This is the same configuration as for Design Volume 1. This checks for the influence of NDS on

the T.O result.

Figure 25 NDS Internal layout

3.2.2 Investigative Study on Software and Process

The purpose of this study is to explore and maximize the utilization of various available tools in

the HyperWorks Suite, to understand the functionalities available in different tools and to develop

competence in working with the commercial software. The following questions are to be answered

by this study

45

1. Is the modelling accuracy the same with the different software? Can they be used

interchangeably?

2. What software should be used for what kind of design Volume?

3. What are the special functionalities that can either improve the model accuracy or simplify

the model setup time?

4. Can software be used in a combination to utilize the advantages of their different features?

Will the file transfer be reliable and is it compatible?

5. How to categorize the software based on the chosen set of parameters to get a comparative

overview?

Comparing optimization solution to study the influence of software on the topology

result

To understand the influence of the software on the resulting topology design, optimization models

were set up with similar configurations in HyperMesh, SimLab and Inspire. The framework,

Graphic User Interface (GUI), capabilities, level of automation and the functionalities constituting

the optimization setup are different for these different software. Hence, for this study, objectives

and constraints were carefully chosen such that they were common and available in all three

software. To define certain functionalities the exact same features were not available in all the

software; in such cases, the closest resembling features are used.

The three steps for optimization modelling are pre-processing, solving and post-processing. The

FE model is set up in the pre-processing step. Solving process involves analysis and optimizing

the material distribution. Post-processing is the result interpretation.

The software can have different functionalities for these different steps.

As mentioned in the section (design volume creation), the Oval concept of GKN PTU is used.

Table 2 software accuracy model set up

Inspire SimLab HyperMesh

Figure 26. Inspire model

Figure 27. SimLab model

Figure 28. HyperMesh model

Optimization configuration


Objective: Max stiffness

Constrain: Stress and 30% vol

Loads: Static Drive/Coast

Non-design space: Design volume 1



Objective: Min compliance

Constrain: Stress and 30% vol

Loads: Static Drive and Coast




Objective: Min Compliance

Constrain: Stress and 30% Mass



46

Pre-processing- Inspire

● CAD model is partitioned into design and non-design space as discussed in section.

● The meshing process is automated and does not have to be done manually.

● Contacts don't need to be defined manually.

● RBE2 elements were created as shown in section and bearing loads and boundary conditions are applied as shown in the section.

Pre-processing- Inspire &

SimLab

● CAD model is partitioned into design and non-design space using Inspire as discussed in section.

● Meshing is done in SimLab. Solid tetrahedron mesh as discussed in section is used.

● FREEZE or TIE contacts are defined at the partitioned design and non-design space.

● For modelling loads and boundary conditions, refer section and section.

Pre-processing- HyperMesh

● Design and non-design space is created by organizing the elements in the FE model as discussed in section.

● Meshing is done in SimLab. Solid tetrahedron mesh as discussed in section is used.

● Since the non-design space was created by reorganizing the meshed model, there is no need to define contacts.

● For modelling loads and boundary conditions, refer section and section.

Solving - Inspire Solving - SimLab Solving - OptiStruct

● .fem file is created

from HyperMesh and

exported to

OptiStruct for

solving.

Post-processing - Inspire Post-processing - SimLab Post-processing -

HyperView

● .h3d file is generated

from OptiStruct and

imported in

HyperView for post-

processing.

Software selection

When different software is used for the three stages of optimization, namely pre-processing,

solving, post-processing, choosing the best combination of software helps improve the process

efficiency and reduces model set up time. Hence, different combinations are tried out as shown in

the Table 3 below and the best combinations are identified for the two design volumes. Since it is

a very time-consuming task to implement all 5 combinations with the two design volumes, some

of the combinations have subjectively been assumed to not have high process efficiency and are

not experimented with.

47

Table 3 Software combinations

3.2.3 Parameters and Response Function study

The purpose of this study is to aid the selection of Optimization objectives, constraints and

parameters in order to obtain the respective desired outcome (mentioned in the results section) for

the 2 design Volumes from the optimization analysis. This study is divided into two steps.

Step 1: Selecting Response Functions

Step 2: Selecting Parameters

Model used for the study

The Oval concept of GKN PTU as shown below, explained in the section, is used for the purpose

of this study. Design Volume 1 and 2 are created from this model as shown in Figure 29. Response

functions and their respective parameters for the 2 design volumes are to be selected from the

study.

48

Figure 29 Illustration diagram of model used for the studies

Plan and formulation of D.O.E

Response functions record the behavior of a particular quality like stress and strain, during an

analysis. Theoretically in the software, any of the available response functions can be used as either

an objective or a constraint. But it may not be a meaningful combination to be used; also a large

number of combinations is possible as shown below, but it is not feasible to experiment with every

such combination.

Hence, the following objective functions and constraints are selected for the study through

literature studies and quick experimentation with software:

Response Function Selection

Objective

Min Compliance/Weighted Compliance or Max Stiffness

Min volume fraction

Min mass

Max Frequency

Constraint

Stress constraints

Mass targets

Volume Fractions

Frequencies

These Objective and Constraint functions are used to formulate meaning combinations of

optimization configuration as written below [16]:

49

1. Max Stiffness/ Min compliance with volume fraction constraint

2. Max Stiffness/Min compliance with volume fraction & stress constraints

3. Min mass with stress constraint

4. Minimize Volume Fraction with stress constraints

5. Max Frequency

These optimization configurations were applied on both Design Volume 1 and 2. The particular

configuration which achieves the desired result for the respective design volumes is selected for

the implementation phase in the section. For this D.O.E study, Inspire, SimLab and HyperMesh

are used interchangeably as concluded from the accuracy study.

Table 4 Response function study

Response

Functions

Problem Formulation Optimization

Configuration DV1

Optimization

Configuration DV2

Min Weighted

compliance

objective with

volume fraction

constraint

Min 𝐶(𝜌) = 1

2𝑢𝑇𝑓


Constraint

V ≤ Volume fraction

Analysis Type:

Static

Objective: Max

Stiffness

Constraint: Volume

fraction 30%

Thickness – min

5.9948 mm

Load cases – Drive

and coast


Objective: Min

Compliance

Constraint: volume

fraction 30%

Optimization

Parameter: Loads: Static Drive

and Coast

Max

Stiffness/Min

compliance with

volume fraction

& stress

constraints

Min 𝐶(𝜌) = 1

2𝑢𝑇𝑓


Constraint

V ≤ Volume fraction

𝜎 ≤ Stress

Analysis Type:

Static

Objective: Min

Compliance

Constrain: 30%

Mass and Stress

Loads: Static Drive

and Coast


Objective: Min

Compliance

Constraint: volume

fraction 30% and

stress

Optimization

Parameter: Loads: Static Drive

and Coast

Min mass with

stress constraint Min 𝑀 = ∑ 𝑚𝑖

𝑛𝑖

Constraint

𝜎 ≤ Stress

Objective: Min Mass

Stress constraint:

F.O.S 1.2

Thickness

parameter – min

5.9948 mm

Load case – Drive

-

Minimize

Volume

Fraction with

stress

constraints

Min 𝑉(𝜌) = 𝜌𝑃 ∗ 𝑉0

Constraint

Analysis Type:

Static

Objective: Min

Volume fraction

Constraint: Stress

constraint

-

50

𝜎 ≤ Stress Optimization

Parameter:

Min and max

thickness – 1 mm &

2 mm

Load cases: Static

Drive and Coast

Note:

1. In response function 1, Volume fraction was varied between 30% and 60% but the later

one proved to be ineffective.

2. In response function 2, the optimization configuration is almost the same as the Response

function 1 with an addition of stress constraint to the volume fraction constraint.

In general, stress constraint on optimization problems is recommended to be avoided as

mentioned in this reference paper [19]. This paper talks about the difficulties of

implementing the stress constraints on optimization algorithms of a FEM problem and of

the solution methods for solving the difficulties.

3. In response function 4, according to [19][20], stress constraints with relaxed limits were

used for the initial stage optimization and more discrete topology results were observed.

Parameter Selection

The purpose of this study is to tweak the optimization solution in order to direct the design into a

more realistic solution which is as close to a realizable design as possible. The two types of

parameters used in this optimization problem are geometric parameters and manufacturing

parameters. Useful optimization parameters are identified for the chosen response functions and

corresponding design volumes.

Model and the Optimization configuration used for this study are described as:

Design Volume 1

Objective: Minimize Volume Fraction

Constraint: Stress constraint

Design Volume 2

Objective: Minimize compliance

Constraint: Volume fraction and Stress constraint

The Parameters chosen for this study are as follows:

1. Member Thickness

2. Casting draw Direction

3. Cast with and without holes

4. Symmetry plain constraint

3.3 Methodology Development

51

The objective of this thesis is to develop a Topology Optimization Methodology for PTU housing.

The new methodology is proposed based on the knowledge gathered from the studies conducted,

from literature and through discussion with the engineers and designers at GKN. This methodology

should be implemented as a part of the overall product development process and should support

the design process without eliciting major changes to it.

The proposed method is a hybrid of Sequential and DomainSub-domain optimization mentioned

in the Literature section. In order for the methodology to be robust, a different CAD model of PTU

housing to the one used in the studies is made use of1.

The methodology is presented as a flowchart in the Results section and the steps and

recommendations on how to perform the T.O process are provided.

3.4 Methodology Implementation

The methodology is verified by implementing it on the PTU housing throughout the component

development process. Resultantly, the housing structure is developed as shown in the results

section.

In this methodology, the T. O process is divided into two loops to implement the concept of

DomainSub-domain optimization and sequential optimization. Hence, this methodology is

coined to be a hybrid methodology.

3.4.1 Loop 1

The purpose of the first loop is to form the primary rib structures along the major load paths.

The implementation steps of Loop 1 are explained as follows.

1) Design Input

The design volume presented in Figure 30 is developed using approach 2 mentioned in section and

is acquired from the design team. The machining tolerance is considered in the design, sharp fillets

are avoided, and the volume covers the maximum available space. This gives the optimization

algorithm maximum room to explore.

Figure 30 Design volume

1 A different housing model with an updated design volume is used at this stage as decided by GKN because by this

stage of the thesis a design update was made in the project and a decision was made to shift from internally called

oval concept to the round concept.

52

2) Pre processing

As shown in Figure 31 and Figure 32, the design volume was split into non design and design

space in Inspire. The non-design space is the bearing seats and the bolts, and the remaining design

volume is the design space.

Figure 31 NDS 1 Figure 32 Design space 1

3) Model setup configuration

The design volume is transferred to SimLab for FE modelling and Optimization model set-up.

Element size of 4mm is used in order to ensure a reasonable quality mesh. All other modelling

procedures are followed as specified in the section (FE modelling) for static analysis.

The Optimization model is set up with the following configuration


Objective: Min Volume fraction

Constraint: Relaxed stress constraint

Parameter: Min and max thickness – 1 mm & 2 mm

Load cases: Static Drive and Coast

This configuration produced primary load paths but were not upto the expected standard. Many of

the bolts were not connected to the bearing surface. In order to improve the result, the analysis was

iterated with additional symmetric constraint parameters. Cyclic symmetry constraint for the cover

and the symmetry constraint over the Y axis of the housing were identified to produce the expected

result as shown in Figure 33.

Figure 33 Loop 1 topology result

4) Result handover

iso plots are a visual representation of the material density of the T. O results. They are varied and

set to the value that displays the best visualization and the result file is exported for that particular

value. The value of iso plot set for Figure 33 is 0.8. The results from SimLab are exported in .sh

and .FEM files to the OSSMOOTH tool in HyperMesh, which creates .STP files that are handed

over to the design team.

53

Figure 34 Loop 1 result handover

The internal layout of the Design Volume is split in Inspire and extracted as a .STP file. This

internal layout is superimposed on the Topology result as shown in Figure 34. This superimposed

model will be helpful for the designers to identify the location of the material concentration on the

load paths.

5) Design realization

The optimization result is interpreted by the designer and is realized into a CAD model

considering further design and manufacturing aspects.

Figure 35 Realized design (top view) Figure 36 Realized design (cover)

Figure 37 Realized design (side view)

3.4.2 Loop 2

54

The purpose of Loop 2 is to add additional rib structures as reinforcement to the realized design of

Loop 1. The implementation steps of Loop 2 are as follows:

1) Design Volume Preparation

The realized design from loop 1 is used as the non-design space. The design space is formed by

cutting out this realized design from the Design volume of loop 1 (Figure 38,Figure 39 and Figure

40). This is done using the BOOLEAN operation in HyperMesh. The different ways of performing

the procedure for this operation is explained in the Appendix G. Approach 1 mentioned in this

Appendix is implemented here. Once the Boolean operation is performed, we get the Loop 2 design

space as shown in Figure 38. These 2 components are then meshed separately with an average

mesh size of 5 mm. The interaction between them is defined using FREEZE contact type.

Figure 38. Design space

Figure 39. NDS

Figure 40. Design volume

2) Optimization Model Setup

The following optimization configuration was used for the first iteration:


Objective: Min Compliance

Constraint: Vol fraction 30% Stress 300Mpa

Optimization Parameter: Thickness min 7 & max 14

Manufacturing: Split DV 2 - Y axis, DV 1 – Y axis


3) Solution Iterations

Once the first result is obtained, the iterations for the parameter modification are carried out. The

configurations as mentioned in the Table 5 are tested. 14 runs were conducted in total by modifying

the parameters sequentially.

Table 5 D. O. E

55

4) Result Handover

After the results were obtained, an appropriate iso value is chosen in which the clear visibility of

the reinforced rib structures is observed.

Figure 41. Loop 2 (top view)

Figure 42. Loop 2 (bottom view)

Figure 43. Loop 2 (side

view)

5) Design realization

The optimization result was interpreted and realized by the design team, taking into consideration

the design perspective and manufacturing feasibility. It was also tried to make the design as close

to topology result as possible.

Figure 44 Realized design (iso view) Figure 45 Realized design (bottom view)

56

Figure 46 Realized design (side view) Figure 47 Realized design (cover)

57

4 RESULTS

In this chapter, the results that are obtained with the methods and requirements of Chapter 3 are

compiled and analyzed. They are presented as per the three study perspectives and the questions

are answered.

4.1 Studies

4.1.1 Design Volume Study

Process Perspective

Creation of design volumes using CAE tools in method 1 i.e from baseline design, was difficult to

be implemented. Even after simplifying the design volume using CAD Doctor, performing

geometric modification using Inspire was ineffective.

The second method of creating the design volume from the internal gear and bearing layout and

packaging space both used in the Project with Altair as mentioned, seems to provide some useful

results. But the amount of work involved is high and the result accuracy is to be validated since

the design volume created is an approximation of the original internal layout.

Thus, it turns out that both methods mentioned in the section are not proceeded for this

project. However, an important finding from this study is that the design volume when created by

the design team proved to be the most time efficient way forward. And the geometric requirements

for these design volumes are to be provided as per the checklist listed in section.

Design volume creation

The following decisions was taken regarding the creation process of design volume

It was agreed that partition of non-design space as the internal layout for the Design Volume was

time consuming to be made by the design team and is to be created by CAE team. The same

decision applies for partitioning the bearing seats and the bolting hard points in loop 1.

Geometry of Non design space

The nature of topology optimization is to form organic results. Since it is desired to form a more structured result, the way it’s seen possible in this thesis is by forming the primary rib structures

as a first step And as a next step forming the secondary rib structure in addition to the primary rib

structure.

Design Volume 1

This design volume is best suited for visualizing primary load paths taken by the bearing reaction

forces.

58

Figure 48 T.O result Design volume 1

In this case, first, the bearing reaction forces applied on the inner surface of the bearing seats are

transferred all over the NDS, which is the bearing rings. From the outer surface of the bearing

seats, the loads are transferred to the bolts through the stiffest paths. To satisfy the objectives and

the constraints, the FE materials are redistributed along this path. This results in the material

distribution being clearly concentrated on the major load paths connecting the bearing seats and

the bolts as shown in Figure 48.

Design Volume 2

Figure 49 T.O result Design Volume 2

DV2 gives an idea for the reinforcements needed on the internal layout which could be interpreted

as the location for secondary rib structures. Hence, DV1 and 2 are chosen as is. They are chosen

to be used sequentially in the methodology development.

In Design Volume 2, the forces applied on the inner surface of the bearing rings are transferred to

the entire internal layout. From the outer surface of the non-design space, the loads are transferred

through the bolts via the stiffest paths. Since the non-design space is much larger compared to that

of Design volume 1, it acts as a much bigger obstacle for the optimization algorithm to redistribute

the materials through the most effective path. Less clear connections to the bolts are formed and

more material concentration is observed over the internal layout to satisfy the objective and the

constraints.

Size of design volume

It is generally considered that large design volumes are advantageous for performing Topology

Optimization since there is a lot of freedom for the algorithm. The reality is that this is not always

the case. When the DV is large, the stiffness along the load path could be minimum/very short and

there might be a low number of connections between the bolts and the bearing seats. Hence, it is

good practice to experiment with the volume of the design input. Same Topology configurations

were applied on model 1 and 2 (shown in Figure 8 and Figure 9), which have different sizes of

design volumes. The resulting Topology design is shown in Figure 50 and Figure 51. It shows that

59

even though model 2 had a larger design volume, many major load path connections were not

formed.

Figure 50 T. O result Model 1 Figure 51 T. O result Model 2

4.1.2 Investigative study on Software

The influence of Software on topology solution for HyperMesh, SimLab and Inspire are

compared as shown in the Figure 52 - Figure 57.

Figure 52 Inspire T.O result Figure 53 SimLab T.O result Figure 54 HyperMesh T.O result

Figure 55 Inspire T.O result with

highlighted connections

Figure 56 SimLab T.O result with

highlighted connections

Figure 57 HyperMesh T.O result

with highlighted connections

Even though all the software uses Optistruct as the background solver, the influence of different

software has to be verified because of the difference in functionalities, automation, GUI and

compatibility of the 3 software. The above figures show that the topology results are similar

irrespective of the software used for performing the optimization. This conclusion coincides with

60

the results mentioned in the reference paper [14]. However, it can be observed that the visualization

of the results is heavily dependent on the software used. Hence, it can be reasoned that the software

must be selected for the different design volumes used in this thesis.

Software Selection

Software is done based on the following criteria: geometric complexity of design and non-design

space, functionalities of interest and visualization of topology results. The results of the study for

selecting the most effective combination of software for the 2 design volumes as discussed in the

section 3.2.1 are summarized below.

Design Volume 1

Inspire (Pre Processing) SimLab (FE Model and Optimization Setup) SimLab (post-

processing)

Since the NDS is bearing seats and bolts and is a simple geometry, Inspire is used to partition the

design and non-design space. SimLab is used for optimization since the model set up is time

efficient for this N.D.S and the required optimization configuration. The in-built Post processing

in SimLab can study and present the results in a good way.

Design Volume 2

HyperMesh (Geometry Boolean) HyperMesh (FE Model and Optimization Setup)

HyperView (Post –Processing)

In D.V.2, the geometric complexity is higher than N.D.S used in D.V.1 and hence Inspire and

SimLab are not good choices for pre-processing and solving. Optimization configurations also

become complex since at this stage manufacturing constraints have to be considered. So,

HyperMesh is used for that operation. HyperView is an excellent tool for post processing and

topology result interpretation.

Uniqueness of the different software Comparison of Advantages of different Software

⮚ Inspire

▪ Automated realistic bearing forces can be input.

▪ 3D bolts plug ins can be used for a more realistic model setup.

▪ Useful for working directly with the CAD

▪ Meshing need not be done manually, it is automatic.

⮚ SIMLAB

▪ Optimization process can be automated in few steps.

⮚ HyperMesh

▪ It offers the possibility to choose from a variety of objective functions and optimization

constraints and makes it possible to set up the model in any desired way.

▪ HyperMesh is a software best suited for complex parameters, geometries and

constraints.

Through extensive usage of the software in the HyperWorks suite, a good understanding of the

software from a broad range of perspectives was gained. Based on this knowledge, the software

was compared for different characteristics. These entities have been compared for the three

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software as illustrated in Figure 58 below and the software have been ranked as shown in the Table

6.

Table 6 Software characteristic comparison

Note: Entities which are marked in blue are of subjective nature.

Subjective quantities may change depending on the user and his/her familiarity

with tool. Here listing has be done assuming a novice user.

Figure 58 Software characteristics comparison

4.1.3 Response Function and Parameter study

Response Function Study

The results of topology optimization for the Response function 1 to 4 and for Design volumes 1

and 2 as mentioned in section 3.2.3 is shown in Figure 59 to Figure 64. Figure 60

Figure 59 Response 1 DV1 T.O

result Figure 60 Response 1 DV2 T.O result

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Figure 61 Response 2 DV1 T.O result Figure 62 Response 2 DV2 T.O result

Figure 63 Response 3 DV1 T.O result Figure 64 Response 4 DV1 T.O result

The conclusions from the study are summarized as below.

For DV 1, it is observed through visual inspection that Figure 64 (which corresponds to Response

function 4) produces the most distinct and clear load paths.


Constraint: Stress constraints

For DV 2, the result interpretability is difficult because of the complicated internal layout and

because parameters are not refined. Even then, the model is set up this way for the study because

then the effect of a particular response function can be viewed unbiased. For DV2, Response

function 2 is observed to provide the best reinforcements.

Objective: Max Stiffness/Min compliance

Constraint: Volume fraction 30% and stress constraints

Parameter Study

The following parameters are applied on the two design volumes and the some of the important

results are summarized as showing in the figures below (Figure 65 to Figure 71).

1. Member thickness

2. Draw Direction Constraints

3. Manufacturing Constraints

4. Symmetric Plane Constraints

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Member thickness

Figure 65 Different T.O results by varying Parameter 1

Draw Direction Constraints

Figure 66 T.O result Figure 67 T.O results with Parameter 2

64

Manufacturing Constraints

Figure 68 T. O result Parameter with holes Figure 69 T. O result Parameter without holes

Symmetric Plane Constraints

Figure 70 T.O results without symmetric constraints

Figure 71 T.O results with symmetric constraints

The parameters chosen for the respective DVs are summarized below.

DV1 - min and max member thickness constrain proved to be helpful to refine the load paths. To

improve the number of bolt connections, symmetric plane constraints were introduced.


Constraint: Stress constraints

⎻ Member Thickness

⎻ Casting draw direction

⎻ Symmetry & cyclic planes

DV2 - min and max member thickness constraint and cast draw direction constraints with holes

was observed to aid the Topology results.

65

Objective: Max Stiffness/Min compliance

Constraint: Volume fraction and stress constraints

⎻ Member Thickness

⎻ Casting draw direction

⎻ Symmetry planes

⎻ Holes/without holes in cast

4.2 Methodology

In the previous sections of the Results chapter, answers to the questions devised in the studies

Section are noted. Using those results, a theoretical methodology is built. This methodology is

then verified by implementing it on a PTU Housing Structure as discussed in the section. The main

steps of this proposed methodology is presented in the flow chart as shown below. The step by

step procedure of the methodology can be followed in the section.

Figure 72 Methodology flow chart - Loop 1 and Loop 2

This methodology is developed as a combination of two of the optimization methodologies

mentioned in the theory section i.e domain→ sub-domain and sequential optimization

methodology.

Domain→ sub-domain optimization methodology

To achieve the goal of obtaining clear rib structures, a few rib structures are formed in Loop 1 and

the additional reinforcements are formed in Loop 2 as can be seen from the results of the respective

Loops in the flowchart. This concept is inspired from the principle of domain→ sub-domain

optimization method. Any optimization model should not be initially constrained with many

response functions and parameters; rather, the complexity should be added incrementally. In this

methodology, the geometric complexity of non-design space is sequentially increased, and the

complexity of the optimization parameters is also increased iteratively. Hence, initially, primary

rib structures are formed with less model complexity and the reinforced rib structures are formed

with higher model complexity, leading to results with clear visualization.

Loop 1 is done on Design Volume 1 and Loop on Design Volume 2. So, the above discussed

concept is implemented.

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Sequential optimization methodology

Loads and constraints are given in increments from Loop 1 to Loop 2. This is based on the principle

of sequential optimization methodology. Structures that are optimized for particular loads are

extremely susceptible to failure when the structure is subjected to loads that are not considered in

the analysis. In order to design a structure that is robust for various loading conditions like static,

dynamic, fatigue loads, mount stiffness, etc. all the required loads must be included into the

optimization process. In this thesis, all these loads can be applied sequentially in increments of the

loop and hence it can be ensured that the topology result is robust towards the required load.

Since both these concepts are combined in this proposed methodology, the term hybrid

methodology is used in this thesis.

From literature, it is understood that in order to attain the Global Minimum, the initial point in the

design space from which the optimization problem is started is important. Therefore, when the

initial design volume is large, large sets of design points are utilized in the optimization. Hence,

solutions will not be missed, and the solution will be directed close to the global minimum.

4.3 Result comparison of current and proposed process

Figure 73 Result from current process

Figure 74 Results Loop 1 Figure 75 Results Loop 2

The main objective for the thesis is to increase the design interpretability of the results of the T.

O. Figure 73 shows the T. O result from the current optimization methodology followed at GKN.

As can be clearly seen, distinct rib structures are not visible. Hence, the knowledge transferred,

and the usability of the T. O result is lesser during the design realization.

This is tackled with, in this thesis successfully. The results from the proposed methodology give

useful guidelines for the placements of the rib structures and can be used in the design process to

extract useful information from the T. O results. High interpretability is achieved.

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4.4 Process/ Methodology Differences

Current Process

Proposed Process

The main difference is that the optimization process is divided into two loops. 2 different

model set ups and DVs. Number of software is more. Software utilization is increased, and

their unique features can be used to rescue time.

Optimization methodology is more integrated into the component development process.

The concept of simulation driven design is implemented here.

Time set up will be more. but the results improved. So in the overall design process,

iteration time will be reduced.

This process is flexible to add lot of forces and other thing can be added modularly and the

model can be applied to different PTU models.

Topology Result Validation

The topology result is validated by observing the stress distribution pattern on the realized design

when static linear analysis for the coast and drive load cases are performed.

Stress patterns should appear only on the location of the predicted rib structures. If the stress

pattern form at any unpredicted location on the housing structure, it either means that the topology

optimization did not suggest the deposition of material at that location or that it has been

disregarded during design realization due to manufacturing and design constraints.

The Principal and Von Mises stress are computed and shown in the Figure 76 and Figure 77. A

discussion on the stress patterns is presented in section (5).

68

Figure 76 Principal Stress

Figure 77 Von Mises Stres (MPa)

69

Reducing number of iterations using optimization

As an example to show how optomization can be beneficial in reducing the number of iterations

in the design process, the first concept of the PTU without optmization is compared to the result

of the first concept developed using topology optimization. Figure 78 shows the stress patters of

the concept developed without optimization. A large stress pattern is observed here. After this step,

to reduce the stress pattern, rib structures would have to be added iteratively. Figure 79 has

comparitively much less stress patterns after the first loop of optimization. This shows the potential

to reduce the number of iterations in the design process.

Figure 78 Stress analysis result for concept developed

without topology optimization

Figure 79 Stress analysis result for the 1st design

concept based on T.O results

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5 DISCUSSION AND CONCLUSIONS

5.1 Discussion

5.1.1 Design Volume study

1. Discrete rib structures are the expected outcome of the T.O result. In a similar housing

structure problem [15], the T.O results were obtained by experimenting only with the

optimization parameters and not by experimenting with the design or non-design space.

They were able to obtain only organic results. This supported the idea of the splitting the

design volume into two loops in the proposed methodology.

2. Defining NDS as bearing outer race and bolts is suitable for visualizing Force flow.

Defining NDS as Inner layout is suitable for identifying rib structures.

Hence the choice for NDS in Loop 1 and 2.

3. The CAD model creation from baseline design approach using CAD Doctor was deemed

inefficient and dropped because it was too much CAD work for the CAE engineers.

4. When the CAD model creation is done in the second approach, internal layout is created

by giving constant clearance from the internal gear and shaft geometries.

5. As stated in this reference paper [9], the size of the design volume must be iteratively

reduced to obtain a desirable result. This conclusion is also discussed in the results section

4.1.1. But in this thesis, the size of the DV was not iterated due to thesis time limitations.

5.1.2 Discussion on Software

1. Software can be used interchangeably if model setup is similar since they give similar

results in test run.

2. Inspire is time efficient for the study of simple optimization models and Hypermesh

efficiently handles more complicated optimization models.

3. When Software is used interchangeably, transition of files between software works

flawless.

4. The GUI of HyperMesh is difficult to intuitively navigate and the learning curve is slow.

Whereas SimLab and Inspire is more intuitive and easier to work with.

5. Topology optimization and the methodology developed is not bound by the software

limitations of using only one particular software. These limitations are overcome by using

the best features of multiple software in the HyperWorks suite.

5.1.3 Response function and Parameter Study

1. The objective, Weighted compliance is chosen because it forms reinforcements on the NDS

and Volume fraction is chosen because it gives clear discrete load paths.

2. It is instructed not to use strict stress constraints because even if the stress values are

exceeded in any one particular hotspot, that design is considered not feasible since it does

not satisfy the constraint. But these hotspots can be rectified with slight design alteration

manually and that solution does not need to be terminated. Hence the stress constraint

feature is misleading.

3. Response functions must be chosen appropriately according to the - geometry of the structure

- Results of interest

4. Parameters must be added sequentially based on the optimization requirement and

understanding of how the optimization problem has to be guided. e.g. if bolts are not

connected, then symmetry planes must be added . Design thinking and understanding the

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geometry is needed to direct the topology optimization software to desired meaningful

results. 5. Optimization problem must start with minimum number of optimization parameters to

avoid the algorithm from restricting possible design solutions.

6. Alternatively, manufacturing and geometric constrains can be used at later later iterations

depending on the model requirement

7. For a model if we observe the effect of one response function or parameter without

activating any other parameter, the topology looks fuzzy but the effect of a particular

response type can be viewed unbiased.

8. For a model to look clear, the appropriate combination of the response functions and

parameter must be used. This combination differs depending on the geometry of the design

volume.

9. Refining Optimization to get clear results is an iterative process. It is not possible to define

hardcoded steps to be followed for all the different geometries.

10. The manufacturing and the geometric parameters which produces the best design for design

models is mentioned in the section 4.1.3. These can be used as an initial guess of parameters

for any similar housing structure.

5.1.4 Methodology

1. This methodology was created for PTU housing structures. Robustness is checked so that

this methodology is applicable for different PTU models.

2. An important principle that applies to the T.O process is that a few iterations should be

performed for every optimization. By changing only the parameters in an iterative way,

this methodology can be used for other housing structures. When this method is applied

for other models during the first iteration, mesh size of 5 mm, same response parameters

have to be applied. If the design is not feasible, then first depending upon the topology

output, the parameters have to be iteratively changed. After many iterations, if the results

are not discrete or distinct, it can be reiterated with a smaller mesh size.

3. When this methodology has to be applied for other products like RDU housing comparatively many iterations might be needed. However, many of the concepts and

conclusions drawn from the methodology of T.O. of PTU housing structures can also be

applied for other products like RDU structures. It is recommended to start the optimization

process with fewer constraints and gradually increase the constrains to avoid missing the

optimum design structure.

4. The first loop of topology optimization should be performed once the bearing loads are

calculated and the initial concept of internal layout is defined.

5. The second loop should be performed when the loop 1 design is realized and primary rib

structures are formed.

6. Many D.O.E and iterations were conducted to make any conclusion. Since it is not feasible

to give recommendations after just one optimization.

7. T. O results are not robust to loads which are not included as input for the analysis. Hence,

it is ensured that this methodology is scalable to add additional loads eg. dynamic loads.

but in the implementation phase of this thesis, static loads are used.

8. As discussed in the section 2.1.2 under 3, topology optimization could result in entirely

different designs when the element size is varied. There is a general notion that, better the

mesh quality, better the topology result. In order to verify this and to select the optimum

mesh size for loop 2, a mesh size of 2 mm for the design space is used and the optimization

model is set up with the same configuration and the results are as shown in Figure 85 and

Figure 83. Mesh size of 5 mm is chosen to be most suited for this T.O. and this mesh size

should be used in the first iteration when the method is used for another model.

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Figure 80 Mesh size 5 mm Figure 81 T:O result Mesh size 5 mm

Figure 82 Mesh size 2 mm Figure 83 T.O result Mesh size 2 mm

9. While performing loop 2, a separate D.O.E study was conducted as shown in Table 5 D.

O. E to verify if the results obtained from the studies are applicable even when applicable

since the geometry of housing structures is modified. The D.O.E studies show that the

similar set of response functions and parameters as chosen from the previous studies are

applicable for the changed housing geometry. It should be noted that additional parameters

are used on top of the parameters selected from studies mentioned in section 4.1.3.

10. The realized design after Loop2 is considered to be conceptually feasible. The stress

hotspots must be found after stress analysis and must be manually corrected. This manual

correction step can’t be eliminated.

11. The realized design manufactured through optimization process will be closer towards the

global optimal design.

12. One of the aspects to achieve Global Optimal design is how the loads are applied. When

some load is applied in the first stage to get a design realization and then some additional

load to the second stage, global min is already restricted at the point after the first stage

when compared to all loads being applied at the first stage. If in the proposed method, all

the loads could be applied in Loop 1, the soln. would not be restricted and would try to

attain G. M. But that is not possible in the proposed method because then the results are

not clear, the design requirements are not attained and the interpretability of the soln. is

not good. All this is done/Avoided intentionally because since certain requirements were

set, the 2 loop concept cannot be avoided.

13. To perform T.O using this method, software knowledge is needed in HyperMesh, inspire

and SimLab.

5.1.5 Methodology Implementation

1. From the diagrams Figure 35 Figure 36 Figure 37 the loop 1 design realization, one can

observe that extensive design change has to be made in order to satisfy the design

requirements and the manufacturing requirements demanded by conventional

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manufacturing processes. These requirements add a lot of mass to the structure than is

suggested by the topology.

2. The observations and the results made from the studies are not extremely sensitive to the

non-design space. They are robust to a certain extent to the nature of the geometry.

3. Performing Loop 2 has two additional advantages. The stress requirement set for Loop 1

optimization may not be satisfied by the realized design because of the vast difference

between the T. O. result and the realized design. This requirement will be satisfied during

Loop 2, which has little difference between the T. O. result and the realized design. The

second advantage is the possibility of introducing other load cases into the simulation

driven design process.

5.1.6 Post Processing Result

1. One of the most difficult parts turned out to be interpreting and realizing the resulting topologies. The result consists of lots of elements with intermediate densities and some

estimate has to be done on which parts and features that are important. It is also difficult to

estimate properties such as strength and mass for the finished product from the design

concept. 2. The post processor HyperView makes it possible to study and present results in a good and

usable way 3. The results are extracted as .STL format from Inspire, SimLab and HyperView. The files

are also exported as .STP format using HyperMesh.

4. Iso plots are a visual representation of the material density of the topology. Appropriate iso

values are to be chosen before the T.O result is exported.

5. The implementation phase of the thesis uses the realistic and exact loading, boundaries

conditions and geometric complexities. Hence, this methodology can be followed as

instructed in the product development process to obtain optimum results.

5.1.7 Topology result Validation

Figure 84 Stress analysis result (top view) (left) and T.O result (right)

The major stress patterns can be observed on the realized design as shown in Figure 84 Stress

analysis result (top view) (left) and T.O result (right) and Figure 85 Stress analysis result (side

view) (left) and T.O result (right). These stress patterns are compared to the topology results and

it is observed that material deposition was predicted by the topology optimization at the location

of these stress patterns. This is shown by the highlighted parts in Figure 84 Stress analysis result

(top view) (left) and T.O result (right) and Figure 85 Stress analysis result (side view) (left) and

T.O result (right).

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Figure 85 Stress analysis result (side view) (left) and T.O result (right)

In this thesis since the rib structures are not generated in a single stage optimization the

possibility for eliminating the globally optimum design has increased because of the manual

interference in the optimization.

The applicability of this approach of performing two loops of optimization can be extended to

other problems with different geometries. The limitation of this approach as mentioned in the

previous point has to be remembered while it is applied.

Since this thesis project is performed at GKN ePowertrain Köping AB using Altair Hyperworks

software package, the proposed methodology is heavily dependent of this software. Even though

this approach can be implemented in other available commercial software, significant amount of

work has to be invested to investigate the software specific functionalities.

5.2 Conclusions

Visualization of major load paths is achieved when approach 1 is used in the optimization

procedure. These major load paths are interpreted as primary rib structures.

Reinforcements for these primary rib structures are formed when DV2 is used in the

optimization process. These reinforcements are realized as secondary rib structures.

For visualizing primary rib structures,

o volume fraction as objective function with relaxed stress constraints is found to

give the best results. This is done in SimLab.

o Minimum and maximum member thickness along with manufacturing constraints

for split planes is chosen.

o Cyclic-symmetric and axial-symmetric constraints are used for improving the bolts

connections.

For visualizing secondary rib structures,

o Minimum compliance as objective function and constraints as relaxed stress and

30% volume fraction is used.

o Minimum and maximum member thickness along with manufacturing constraints

for split planes is chosen.

Similar topology optimization results are produced, irrespective of whether Inspire,

SimLab or HyperMesh is used for the model set up.

The methodology developed is robust when used for similar housing structures.

The methodology is flexible to include further load cases like modal, mount stiffness, point

mobility.

This methodology can be followed as instructed in the product development process,

without needing major modifications in the existing process, to obtain optimum results.

75

The iso plot can be varied to understand and estimate the importance of the structural

features formed.

The topology results are converted into a .stp format. The result handover in this format is

useful for interpretation and design realization. This is done using OOSMOOTH operation

in HyperMesh.

The topology result is closer to a design that can be manufactured using conventional

manufacturing methods.

Better visualization of results is achieved when compared to the current topology result.

By using the methodology that is proposed, knowledge transfer and interpretability of results

is improved to a large extent and the research questions formulated as herby answered as

follows.

The housing structure is topologically optimized when the design for internal layout

and the bearing reactions force calculation are available.

Software available in the Hyperworks suite is efficiently combination as mentioned in

the Section Software Selection.

The problem of increasing the design interpretability in topology results is addressed

from 3 perspectives respectively Geometry of Design and Non Design Volume,

Software used and the choice of Optimization parameters used.

Addition to following the proposed methodology the user may be required identify the

appropriate geometric and manufacturing parameters if the geometry of the product

differs significantly.

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6 FUTURE WORK

Design Realization studies should be conducted, for the purpose of thesis must

concentration was not given to how the topology result is interpreted and how can the result

be maximum utilized in the manufacturing. More intuitive approach is taken. It could be

further investigated how the topology result can be further investigated and more

knowledge can be transferred into the realized design.

The modal and static combined optimization should be experimented more to direct the

result towards the expected results by varying the optimization configuration and the

parameters.

The MMO model could be perfected and the optimization configuration could be found to

tweak and direct the topology towards the expected result.

As the next step towards Simulation driven design, Shape Optimization could be performed

on the realized design after topology optimization.

A sensitive study could be conducted to identify which loads can be used at which stage of

optimization. Is worthwhile to use combine multiple load cases in topology optimization

or will it save time if it is used in shape optimization since it will produce same results.

A comparative study can be made to check the possibility to produce the housing additive

manufacturing methods which allows much freedom to manufacture the topology result

without many changes to it. This could reduce the weight significantly. However, the cost

to mass produce and the cost involved in transition will be huge and may not be possible

immediately.

Further study how to interpret/realize the resulting design concept from the topology optimization and how to estimate the properties (stress, mass, etc.) of the finished product

from the design concept

77

7 REFERENCES

[1] Altair, “Alair OptiStruct user manual”, 2019.

[2] Altair, “Practical aspects of Structural Optimization”, 2018.

[3] Bendsoe M P., and Sigmund O., Topology Optimization: Theory, Methods and Applications.

Springer Verlag, 2003.

[4] Borvall T., Topology optimization of elastic continua using restriction. Archives of

Computational methods in Engineering 8, pp 34, 2001.

[5] Femto Engineering, “In FEA, what is linear and nonlinear analysis”, [Online] Available at:

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[6] Fleury C., “Conlin: an efficient dual optimizer based on convex approximation concepts”,

Structural Optimization 1, pp 81-89, 1989.

[7] Idaberg F., “Optimization Study of Frame Components for a Truck”, Master’s thesis,

Department of Solid Mechanics, Stockholm: KTH, 2018.

[8] Jog C.S., "A robust dual algorithm for topology design of structures in discrete variables,"

International journal for numerical methods in engineering, pp 1607-1618, 2001.

[9] Larsson R., “Methodology of Topology and Shape Optimization: Application to a Rear Lower

Control Arm”, Master’s thesis, Department of Applied Mechanics, Göteborg: Chalmers

University of Technology, 2016.

[10] Olason A and Tidman D., “Methodology for topology and shape optimization in the design

process”, Master’s thesis, Department of Applied Mechanics, Göteborg: Chalmers University

of Technology, 2010. [Available online:

https://odr.chalmers.se/bitstream/20.500.12380/130136/1/130136.pdf].

[11] Olsson M., Tellner M., Sadek S. and Sandberg D., An Introduction to design optimization

[Lecture handout], Department of Solid Mechanics, Stockholm: KTH, accessed 2019.

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thesis, Department of Machine Design, Stockholm: KTH, 1999.

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on Procedures Dealing with Checkerboards, Mesh-dependencies and Local Minima, Structural Optimization, 16, 68-75, 1998.

[14] Wook-han Choi, Cheng-guo Huang, Jong-moon Kim, Gyung-Jin Park, “Comparison of

some commercial software systems for structural optimization”, Proceedings of the 11th World

Congress of Structural and Multidisciplinary Optimisation, Vol 1, Sydney, Australia, pp 71-

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[15] Zhuang S., “Gearbox housing topology optimization with respect to gear misalignment”,

Master’s thesis, Department of Management and Engineering, Linköping: Linköping

University, 2012.

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content/uploads/2011/03/optistruct_optimization_10-0.pdf, accessed 2020-08-04, 2009.

[17] [Online] Available at: https://www.engr.uvic.ca/~mech410/lectures/FEA_Theory.pdf,

accessed 2020-08-04, n.d.

[18] [Online] Available at: https://abaqus-

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optobjectives.htmhttps://www.engr.uvic.ca/~mech410/lectures/FEA_Theory.pdf, accessed

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[19] Verbart A., “Topology optimization with stress constraints”, Master’s thesis, Department

of Mechanical Engineering, The Netherlands: Technical University Delft, 2015.

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stress limit as a criteria in formulation," FELIP International journal of Engineering Analysis,

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APPENDIX A: GANTT CHART

APPENDIX B: RISK ANALYSIS

80

APPENDIX C: CHECKLIST FOR DESIGN VOLUME HANDOVER

● Avoid fillets in the internal envelop

● Avoid internal ribs from start – a” clean” design space

● Split the internal design volume domain for ribs from that of internal envelope

81

● Avoid guiding diameter across sealing interfaces (no small steps)

● Avoid smooth shoulder for bearing seats (preferred 90 degree)

● Avoid oil plug

● Avoid small, shallow lubrication channels

● Outer envelope considering machining clearances and tolerance

● Include partition line for casting and draft angles for casting

Figure 86 Updated design volume according to checklist

APPENDIX D: INSPIRE PRE-PROCESSING PROCEDURE

STEP 1: Import .STP in Inspire File Import .STP file

82

Figure 87 Design volume

STEP 2. Non design space – Bearing Seats Partition Select Bearing seats 8mm thickness Partition all

Figure 88 Partitioned bearing seats

STEP 3. Non design space - Bolts Partition Select all Bolts 3mm thickness Partition all

Figure 89 Partitioned bolts

STEP 4. Design space – Housing and cover

Select Housing & Cover Property editor check Design space

Figure 90 Design space (brown parts) and NDS (Gray parts)

APPENDIX E: ISO VALUES, RESULT INTERPRETATION AND DESIGN HANDOVER LOOP

STEP 1: Optimization Results from SimLab Store .sh , .fem files from run folder.

83

Figure 91 T.O result

STEP 2. OSSMOOTH covert results to .STP (HyperMesh) Use .sh files and .fem files to create .stp files of the optimization results.

(Note: iso surface in the format of .stl can is normally extracted using Tool 🡪 Export 🡪 Iso_Surface

)

Figure 92 Topology concept in .stp format

STEP 3. Partition Internal layout (Inspire) Partition Select all internal surface 3mm thickness Partition all

Figure 93 Partitioned internal layout

STEP 4. Superimpose Optimization results and internal layout (Inspire) Import .STP files of the Optimization results

and Internal layout in same model for better visualization during design realization.

84

Figure 94 Internal layout superimposed on T.O results

85

APPENDIX F: CADDOCTOR GUIDELINES

Model simplification Procedure

Figure 95 Simplified fillets

Figure 96 Simplified chamfers

86

APPENDIX G: BOOLEAN OPERATION PROCEDURE

Approach 1

● Geometry Solid edit (Boolean operation)

● Operation type : advanced

● Solids ⎻ A : Design Volume ⎻ B : N.D.S

● Operation ⎻ A-B (del B Parts)

● Combine through ⎻ None

Figure 97 Design space partitioned through BOOLEAN operation

Mesh the parts separately.

Combine them.

Add contacts between them.

Figure 98 Meshed models of Design and Non-design space

87

Approach 2

Surface Creation Outer, Mid Surface and inner Layout

● Geometry Solid edit (Boolean operation)

● Operation type : advanced

● Solids ⎻ A : Design Volume ⎻ B : N.D.S

● Operation ⎻ A+B (Keep common Parts)

● Combine through ⎻ AB faces in B (B cuts A)

Figure 99 Surfaces of Design and Non-design space

● Move 2D Mesh form 2 geometries to one file.

● Create Volume Mesh.

Figure 100Volume mesh of design and non-design space

88

APPENDIX H: METHODOLOGY

89

APPENDIX I: EXPLORING UNIQUE FEATURES OF INSPIRE

Inspire is a special purpose tool developed for optimization. It allows the geometric modification

possible and makes the optimization model set up easy for the user.

Special Features in Inspire

Loads

Using the Loads menu and by enabling the bearing force in the property editor window, the bearing

forces are more realistically modeled instead of being evenly distributed on the entire bearing inner

wall.

Figure 101 Bearing forces modelled in Inspire

Connectors

Inspire has an option to use 3 dimensional bolts directly from its library. It simplifies the modelling

and use of realistic bolts in the optimization analysis.

Figure 102 3D bolts modelled in Inspire

Documents

Methodology Development for Topology Optimization of Power