Structural Optimization of Mechanical Systems

Topology optimization with parametrization of hardpoint positions

Master’s thesis in Applied Mechanics

Aliki Berndtsson
Andreas Mattiasson

Department of Industrial and Materials Science
CHALMERS UNIVERSITY OF TECHNOLOGY
Gothenburg, Sweden 2019





Master’s thesis 2019

Structural Optimization of Mechanical Systems

Topology optimization with parametrization of hardpoint positions

ALIKI BERNDTSSON
ANDREAS MATTIASSON

Department of Industrial and Materials Science
Division of Material and Computational Mechanics

Chalmers University of Technology
Gothenburg, Sweden 2019



Structural Optimization of Mechanical Systems
Topology optimization with parametrization of hardpoint positions
ALIKI BERNDTSSON
ANDREAS MATTIASSON

© ALIKI BERNDTSSON, 2019.
© ANDREAS MATTIASSON, 2019.

Supervisor: Robin Larsson, Volvo Cars
Examiner: Martin Fagerström, Department of Industrial and Materials Science

Master’s Thesis 2019
Department of Industrial and Materials Science
Division of Material and Computational Mechanics
Chalmers University of Technology
SE-412 96 Gothenburg
Telephone +46 31 772 1000

Cover: Wheel suspension system of the the second generation Volvo XC90.

Typeset in LATEX
Printed by Reproservice
Gothenburg, Sweden 2019

iv



Structural Optimization of Mechanical Systems
Topology optimization with parametrization of hardpoint positions
ALIKI BERNDTSSON
ANDREAS MATTIASSON
Department of Industrial and Materials Science
Chalmers University of Technology

Abstract
This is the report for a master’s thesis project on structural optimization of me-
chanical systems, at Volvo Cars. The shift towards electric propulsion and high cost
of battery capacity puts further need for lightweighting since it can bring signifi-
cant cost advantages in terms of price per range. Topology Optimization (TO) is
a method that finds its true potential in context with additive manufacturing due
to high level of design freedom, but it can also be beneficial in context with other
manufacturing techniques such as casting. Previous thesis projects at Volvo Cars
have emphasized TO in the context of casted components and it is widely used to-
day. However, TO is often used to refine the geometry relatively late and applied to
single components with predetermined boundary conditions such as joint loads and
joint positions, i.e. constraints that narrows the solution space. The main idea for
this project is therefore to broaden the solution space by introducing TO on system
level together with an outer parametric loop for joint positions. This method can
be applied to mechanical systems in general, but the main focus is to provide a
prestudy for applications within automotive wheel suspension systems.

A Finite Element (FE) model of a rear wheel suspension system is developed and
validated with respect to force signals from an existing, dynamic, full vehicle model.
The linkages are then optimized with respect to stiffness and weight based on cur-
rent joint positions. The proposed scheme for parametrization of joint positions is
limited to a two-component system.

Over all, the FE-model correlates well with respect to force signals, but there are still
room for improvements, especially with respect to modeling of dampers. In order
to introduce proper stiffness constraints during optimization, there is also a need
for correlation of the FE-model with respect to displacements. The future potential
for TO on system level seems promising. In addition to a broader solution space
through parametrization of joint positions the relative mass distribution between
different components are treated within one single system optimization.

Keywords: Structural optimization of a multibody systems, Topology optimization,
Parametrization of joint positions.

v





Acknowledgements
The report consists of documentation from a Matser’s thesis project at Volvo Cars.
The work was carried out during the spring of 2019, by two student from the master
program in Applied Mechanics at Chalmers University of Technology.

We would like to thank our supervisor Robin Larsson for great support during the
entire project, Joakim Zakrisson for support regarding Road Load Data and Åsa
Sällström as well as the rest of the group CAE Body, Chassis and RLD for good
input during the project. Furthermore, a great thanks to Anton Jurinic from Das-
sault Systémes for a nice introduction course in Tosca Structure.

Last but not least, we would also like to thank our optimization guru Harald Has-
selblad, our examiner Martin Fagertröm and our manager Per Sjövall for believing
in this project and for recognizing the potential of system level optimizations.

Aliki Berndtsson & Andreas Mattiason, Gothenburg, June 2019

vii





Contents

List of Figures xi

List of Tables xv

1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Earlier studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Specification of issue under investigation . . . . . . . . . . . . . . . . 4

2 Theory 7
2.1 Technological platforms . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Strong format . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Weak format . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3 FE-format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Topology Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 TO example: Minimum compliance using SIMP . . . . . . . . 12
2.3.2 TO in Tosca . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 FE-modeling in Abaqus . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Connector elements . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.2 Coupling constraints . . . . . . . . . . . . . . . . . . . . . . . 14

3 Methodology 15
3.1 Prestudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Strength events . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 FE-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1.1 Bushing attachments and material models for com-

ponents . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.1.2 Steel sheet components . . . . . . . . . . . . . . . . . 17
3.2.1.3 Aluminum cast components . . . . . . . . . . . . . . 18
3.2.1.4 Boundary conditions . . . . . . . . . . . . . . . . . . 18
3.2.1.5 Assembly . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1.6 Linear elastic bushing characteristics . . . . . . . . . 19
3.2.1.7 Damper loads . . . . . . . . . . . . . . . . . . . . . . 19

ix



Contents

3.2.1.8 Drawing state loads and pretension forces . . . . . . 20
3.2.2 Verification with respect to force signals . . . . . . . . . . . . 21
3.2.3 Sensitivity Analysis: Bushing Stiffness . . . . . . . . . . . . . 21
3.2.4 Sensitivity Analysis: Component Stiffness . . . . . . . . . . . 22

3.3 Structural Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.1 Two-component problem . . . . . . . . . . . . . . . . . . . . . 22
3.3.2 TO of the wheel suspension system . . . . . . . . . . . . . . . 24

3.3.2.1 Design volumes of the linkages . . . . . . . . . . . . 24
3.3.2.2 Concept load cases . . . . . . . . . . . . . . . . . . . 25
3.3.2.3 Topology optimization configuration . . . . . . . . . 25

4 Results 27
4.1 Verification of FE-model . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Force signals with poor correlation . . . . . . . . . . . . . . . 30
4.1.2 Static analysis and dynamic events . . . . . . . . . . . . . . . 31
4.1.3 Sensitivity analysis: Bushing stiffness . . . . . . . . . . . . . . 32
4.1.4 Sensitivity analysis: Component stiffness . . . . . . . . . . . . 35

4.2 Structural Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.1 Optimization of the two-component model . . . . . . . . . . . 36
4.2.2 TO of the wheel suspension system . . . . . . . . . . . . . . . 38

4.2.2.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.2.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Discussion 43
5.1 FE-model of the wheel suspension system . . . . . . . . . . . . . . . . 43

5.1.1 Rigid bushing attachments . . . . . . . . . . . . . . . . . . . . 43
5.1.2 Quasi-static analysis with linear assumptions . . . . . . . . . . 44
5.1.3 Boundary conditions and bushing stiffness . . . . . . . . . . . 44

5.2 TO on system level . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2.1 Load distribution during optimization . . . . . . . . . . . . . . 46
5.2.2 Optimization strategies . . . . . . . . . . . . . . . . . . . . . . 46

6 Conclusion 47
6.1 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Bibliography 49

A Linearization of bushing characteristics I

B Results from verification of FE-model V

C Results from sensitivity analysis: bushing stiffness XXIII

D Additional Results from TO of the system model XXV

E Proposed workflow for development XXVII

x



List of Figures

1.1 Proposed scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Current workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Five links constraining one DoF each . . . . . . . . . . . . . . . . . . 7
2.2 The studied rear wheel suspension system; an integral link rear axle,

viewed from behind . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 System components and hardpoints for linkages . . . . . . . . . . . . 9
2.4 Bushing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Loaded body and kinematically admissible displacements . . . . . . . 10

3.1 Overview of project and methodology . . . . . . . . . . . . . . . . . . 15
3.2 Snapshots from studied strength events . . . . . . . . . . . . . . . . . 16
3.3 Toelink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 UCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Subframe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.6 LCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.7 Knuckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.8 Boundary conditions similar to a chassis rig test fixture . . . . . . . . 18
3.9 Assembly and global reference frame . . . . . . . . . . . . . . . . . . 19
3.10 Damper force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.11 Pretension forces from the leaf spring at drawing state . . . . . . . . 20
3.12 Two-component model: Problem definition and Abaqus FE-model . . 23
3.13 Linkages as design volumes . . . . . . . . . . . . . . . . . . . . . . . . 24
3.14 LCA design volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.15 UCA design volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.16 Toelink design volume . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1 Force signals in HP6 during BIP, FE-model ( ) vs RLD ( ) . 27
4.2 High RMSnorm: BIP and DOC, FE-model ( ) vs RLD ( ) . . . 30
4.3 High RMSnorm: SAC, FE-model ( ) vs RLD ( ) . . . . . . . . 30
4.4 Non-physical deflections caused by peak damper loads during DOC . 31
4.5 FRH

z for DOC: HP12 and 14, FE-model ( ) vs RLD ( ) . . . . 31
4.6 System response of bushing stiffness in HP 3 local y-direction . . . . 33
4.7 System response of bushing stiffness in HP 31-32 local y-direction . . 34
4.8 Change in peak force in HP4 for DOC when changing the stiffness of

the LCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.9 Performance plot: Minimize mass (rigid bushings) . . . . . . . . . . . 36

xi



List of Figures

4.10 Topology evolution: Minimize mass (rigid bushings) . . . . . . . . . . 36
4.11 Performance plot: Minimize mass (flexible bushings) . . . . . . . . . 37
4.12 Topology evolution: Minimize mass (flexible bushings) . . . . . . . . 37
4.13 Strain energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.14 Tip Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.15 Topology evolution: Minimize strain energy (flexible bushings) . . . . 38
4.16 Topology of the optimized components for case 1 with all load cases,

BIP, DOC and SAC . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.17 Topology of the optimized components for case 2 with all load cases,

BIP, DOC and SAC . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.1 Straight UCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Boundary for hardpoint position during parametric optimization . . . 45

A.1 Bushing characteristics: HP1 & 7, FE-model ( ) vs RLD ( ) . I
A.2 Bushing characteristics: HP3, FE-model ( ) vs RLD ( ) . . . II
A.3 Bushing characteristics: HP4, FE-model ( ) vs RLD ( ) . . . II
A.4 Bushing characteristics: HP12 & 14, FE-model ( ) vs RLD ( ) III
A.5 Bushing characteristics: HP18, FE-model ( ) vs RLD ( ) . . . III
A.6 Bushing characteristics: HP31 & 32, FE-model ( ) vs RLD ( ) IV
A.7 Bushing characteristics: HP33 & 34, FE-model ( ) vs RLD ( ) IV

B.1 F norm in HP1 during BIP, FE-model ( ) vs RLD ( ) . . . . . VI
B.2 F norm in HP3 during BIP, FE-model ( ) vs RLD ( ) . . . . . VI
B.3 F norm in HP4 during BIP, FE-model ( ) vs RLD ( ) . . . . . VII
B.4 F norm in HP6 during BIP, FE-model ( ) vs RLD ( ) . . . . . VII
B.5 F norm in HP7 during BIP, FE-model ( ) vs RLD ( ) . . . . . VIII
B.6 F norm in HP12 during BIP, FE-model ( ) vs RLD ( ) . . . . . VIII
B.7 F norm in HP14 during BIP, FE-model ( ) vs RLD ( ) . . . . . IX
B.8 F norm in HP18 during BIP, FE-model ( ) vs RLD ( ) . . . . . IX
B.9 F norm in HP31-32 during BIP, FE-model ( ) vs RLD ( ) . . . X
B.10 F norm in HP33-34 during BIP, FE-model ( ) vs RLD ( ) . . . X
B.11 F norm in HP1 during DOC, FE-model ( ) vs RLD ( ) . . . . . XII
B.12 F norm in HP3 during DOC, FE-model ( ) vs RLD ( ) . . . . . XII
B.13 F norm in HP4 during DOC, FE-model ( ) vs RLD ( ) . . . . . XIII
B.14 F norm in HP6 during DOC, FE-model ( ) vs RLD ( ) . . . . . XIII
B.15 F norm in HP7 during DOC, FE-model ( ) vs RLD ( ) . . . . . XIV
B.16 F norm in HP12 during DOC, FE-model ( ) vs RLD ( ) . . . . XIV
B.17 F norm in HP14 during DOC, FE-model ( ) vs RLD ( ) . . . . XV
B.18 F norm in HP18 during DOC, FE-model ( ) vs RLD ( ) . . . . XV
B.19 F norm in HP31-32 during DOC, FE-model ( ) vs RLD ( ) . . XVI
B.20 F norm in HP33-34 during DOC, FE-model ( ) vs RLD ( ) . . XVI
B.21 F norm in HP1 during SAC, FE-model ( ) vs RLD ( ) . . . . . XVIII
B.22 F norm in HP3 during SAC, FE-model ( ) vs RLD ( ) . . . . . XVIII
B.23 F norm in HP4 during SAC, FE-model ( ) vs RLD ( ) . . . . . XIX
B.24 F norm in HP6 during SAC, FE-model ( ) vs RLD ( ) . . . . . XIX
B.25 F norm in HP7 during SAC, FE-model ( ) vs RLD ( ) . . . . . XX

xii



List of Figures

B.26 F norm in HP12 during SAC, FE-model ( ) vs RLD ( ) . . . . XX
B.27 F norm in HP14 during SAC, FE-model ( ) vs RLD ( ) . . . . XXI
B.28 F norm in HP18 during SAC, FE-model ( ) vs RLD ( ) . . . . XXI
B.29 F norm in HP31-32 during SAC, FE-model ( ) vs RLD ( ) . . . XXII
B.30 F norm in HP33-34 during SAC, FE-model ( ) vs RLD ( ) . . . XXII

C.1 Sensitivity analysis of bushing stiffness in HP3 and HP4 showing
RMSEvent vs bushing stiffness for all events . . . . . . . . . . . . . . . XXIII

C.2 Sensitivity analysis of bushing stiffness in HP31-32 and HP33-34 show-
ing RMSEvent vs bushing stiffness for all events . . . . . . . . . . . . . XXIV

D.1 Topology of the optimized components for case 1 with only BIP . . . XXV
D.2 Topology of the optimized components for case 1 with only DOC . . XXV
D.3 Topology of the optimized components for case 1 with only SAC . . . XXVI
D.4 Topology of the optimized LCA with M ≤ [0.6− 0.9]M̄Ref . . . . . . XXVI
D.5 Topology of the optimized UCA with M ≤ [0.6− 0.9]M̄Ref . . . . . . XXVI
D.6 Topology of the optimized Toelink with M ≤ [0.6− 0.9]M̄Ref . . . . . XXVI

E.1 Proposed workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXVII

xiii



List of Figures

xiv



List of Tables

3.1 Configuration of TO of case 1 . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Configuration of TO of case 2 . . . . . . . . . . . . . . . . . . . . . . 26

4.1 RMSnorm for all force signals during the BIP event, with subtraction
of drawing state loads . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 RMSnorm for all force signals during the DOC event, with subtraction
of drawing state loads . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 RMSnorm for all force signals during the SAC event, with subtraction
of drawing state loads . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Summary of the system response with respect to bushing stiffness . . 32
4.5 Results from optimization case 1 with all load cases, BIP, DOC and

SAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.6 Results from optimization case 2, M ≤ M̄Ref . . . . . . . . . . . . . . 41
4.7 Results from optimization case 2, with M ≤ [0.5− 0.9]M̄Ref . . . . . 42

B.1 RMSnorm for all force signals during the BIP event, with subtraction
of drawing state loads . . . . . . . . . . . . . . . . . . . . . . . . . . V

B.2 Comparison of mean RMSnorm during BIP . . . . . . . . . . . . . . . V
B.3 RMSnorm for all force signals during the DOC event, with subtraction

of drawing state loads . . . . . . . . . . . . . . . . . . . . . . . . . . XI
B.4 Comparison of mean RMSnorm during DOC . . . . . . . . . . . . . . . XI
B.5 RMSnorm for all force signals during the SAC event, with subtraction

of drawing state loads . . . . . . . . . . . . . . . . . . . . . . . . . . XVII
B.6 Comparison of mean RMSnorm during SAC . . . . . . . . . . . . . . . XVII

xv



List of Tables

xvi



1
Introduction

This is the report for a master’s thesis project on structural optimization of me-
chanical systems, at Volvo Cars. This chapter defines and motivates the study. It
also covers the main ideas behind the project.

1.1 Background

The shift towards electric propulsion of automotive vehicles brings higher demand for
lightweight structures since the high prices of battery capacity put further need on
energy efficiency. Lightweighting is one among several ways, e.g. reduce drivetrain
losses, aerodynamic drag and rolling resistance, to improve energy efficiency and it
can bring significant cost advantages in terms of price per range [1–4].

Product development using structural optimization tools plays an essential role
in the process of decreasing weight while maintaining mechanical performance.
Lightweighting often involves substitution towards high-strength, lightweight mate-
rials [4], but it can also refer to optimization of material distribution, i.e. Topology
Optimization (TO), a method that finds its true potential in context with additive
manufacturing due to high level of design freedom [5–7]. However, TO can also
be beneficial in context with other manufacturing techniques such as casting [6].
Previous thesis projects at Volvo Cars have emphasized TO in context of casted
components (see [8, 9]) and it is widely used today.

Computer Aided Engineering (CAE) is a key component during the product develop-
ment process and there is a demand for improved structural optimization methods.
Looking at wheel suspension system design, Finite Element (FE) based optimization
techniques (e.g. TO) are often used to refine the geometry relatively late in the de-
velopment process. It is applied to single components with predetermined boundary
conditions such as joint loads and joint positions, i.e. constraints that narrows the
solution space. In fact, an optimization is generally performed in subject to some
constraints and the end result will never be better than what is allowed by those
constraints. Hence, it is relevant to explore methods for structural optimization in
context with early phase system design.

1



1. Introduction

The main idea for this project is to broaden the solution space by introducing FE-
based optimization earlier in the concept phase and by adopting a perspective on
system level rather than on component level. The sooner these techniques can be
applied, the less parameters (e.g. joint positions) have to be considered as fixed
during optimization.

Figure 1.1: Proposed scheme

In Figure 1.1 a scheme is proposed on how to
combine TO with an outer loop of parametriza-
tion, which may become a powerful tool for
targeting structural performance during concept
generation. The proposed scheme updates the
input parameters in the outer loop according to a
predefined Design Of Experiments (DOE), which
is a systematic method to determine the rela-
tions between input and output variables. The
topology optimization result is then saved for
each set of parameter values and can be studied
as a function of the parameters, used as input
for conceptual decisions. This method can be
applied to mechanical systems in general, but
the main focus of this project is to provide a
prestudy for application within chassis.

Two main loops can be identified within the durability and vehicle dynamics parts of
the product development process. A simplified overview of the current workflow is
presented in Figure 1.2. Today, TO is merely used during the component phase but
the idea is to provide a tool that can introduce TO earlier on, i.e. during the system
phase. Hence, with the proposed methodology, it may be possible to target system
weight at an earlier stage, i.e. when hardpoint positions (joint positions) are not
considered as fixed. The hardpoint positions can then be used as input parameters
in the outer loop in Figure 1.1.

Figure 1.2: Current workflow

2



1. Introduction

1.1.1 Earlier studies

There are plenty of studies on TO of multibody systems (e.g. [10–13]) and a pre-
vious thesis work covered parametrization of design volumes with components that
competes for packaging volume [14]. However, non of these involves parametriza-
tion of joint positions. One of the most relevant studies found involves TO of a ski
binding with parametrization of bolt hole positions [15], but application was limited
to a one-component TO.

Only a few papers have been found that cover TO of multiple components combined
with parametrization of joint positions (see e.g. [16–20]). In 1997, Chickermane and
Gea [16] proposed a method to address the issue of multicomponent topology opti-
mization in context with layout of joint positions. However, the proposed method
involved overlapping design domains that defines admissible joint positions as well
as applications to very simple structures. Similar approaches have been adopted
by Qing and Steven [19] in 2001 and by Qian and Ananthasuresh [18] in 2004, but
applications were still limited to rather simple and overlapping structures. Later
attempts have been made by Yildiz and Saitou [20] in 2011 and Liu [17] in 2016,
but applications were still merely academic, involving very simple FE-models for
both structures and joints.

Other papers cover optimal layout of support structures with constraints on joint
loads [21, 22]. Qian and Ananthasuresh [18] covered optimal embedding of com-
ponents where they adopted a similar approach as for the embedded joints. This
was later developed by Gao et. al [23] in 2015, with applications to more advanced
FE-models. However, the TO was limited to the supporting structure, i.e. a one-
component TO. Thus, there are seemingly not many publications covering a suitable
methodology for this project.

Earlier studies have either been limited to a one-component TO or with applications
that are merely academic, i.e. only covering very simple structures. Hence, meth-
ods from earlier studies may be hard to extend into industrial level applications
for chassis components, i.e. 3D-structures where potential intersections between
components have to be avoided. The proposed scheme in Figure 1.1 may be compu-
tationally demanding, but it fits existing commercial softwares, i.e. a combination
between tools for parametrization [24–26] and tools for TO [27–29]. Therefore this
project is not only right in time by considering lightweighting within the automo-
tive industry. It is also highly relevant with respect to the rapid development within
generative design (see e.g. [30–32]).

1.2 Aim

The purpose is to study structural optimization of multibody mechanical systems,
especially an automotive wheel suspension system. The main goal is to provide a
basis for concept generation of multibody systems using FE-based structural opti-
mization. In particular, this work aims to:

3



1. Introduction

• Establish a simplified system model of an automotive wheel suspension system
in Abaqus [33] and link it to the optimization tools in Tosca [28].

• Verify the simplified system model against load data from an existing, dy-
namic, complete vehicle model in Adams Car [34].

• Perform sensitivity analysis with respect to system characteristics such as stiff-
nesses of components and bushings.

• Propose concept load cases for the wheel suspension system and optimize the
material distribution for linkages with respect to weight and stiffness.

• Implement structural optimization for a simple two-body system and propose
methods for TO with parametrization of hardpoint positions.

1.3 Limitations

The project is limited to involve:

• TO with simplified FE-modeling and linear assumptions.

• Verification with respect to force signals, i.e. not with respect to other quan-
tities such as moments, displacements, stresses or strains.

• External loads that are assumed to be constant during optimization, i.e. not
updated with respect to hardpoint positions or system weight.

Hardpoint positions and system weight may affect external loads, but with consid-
eration to computational effort this iteration in Adams Car is left outside the scope
of this study. Also note that non-linearities, i.e. geometrical non-linearities and
non-linear material models, are left out since it is generally recommended to start
with linear FE-models for new TO-problems. However, these limitations may put a
need for further development and follow-up projects in the future.

1.4 Specification of issue under investigation

The following questions are addressed in this work:

• What characteristics of components and joints needs to be included in the
system FE-model and what can either be simplified or left out in order to
benefit the phase of concept generation?

• What load cases are essential in order to capture key characteristics of the rear
wheel suspension system during the concept phase?

• How sensitive is the load response of the FE-model with respect to modeled
characteristics of bushings?

4



1. Introduction

• How does component stiffness affect internal loads in the system FE-model?

• What measures and requirements are essential for the wheel suspension system
at an early stage of development?

• What optimization setup, e.g. objective function and constraints, are suitable
for the proposed methodology in Figure 1.1?

• What magnitude of hardpoint repositioning is relevant to study with respect
to requirements from other disciplines?

• What is the future potential for topology optimization applied on system level?

By addressing these questions the aim is to provide a good foundation for future use
and development of structural optimization on mechanical systems.

5



1. Introduction

6



2
Theory

This chapter presents the theoretical framework for this project, starting with car
chassis and a brief description of the studied rear wheel suspension system. There-
after some fundamental frameworks within Finite Element Method (FEM) and
Topology Optimization (TO) are presented. Later parts involve some specifics about
TO in Tosca and FE-modeling in Abaqus .

2.1 Technological platforms

Traditionally car chassis consisted of a horizontal ladder framework on which other
chassis components where mounted, such as suspensions, wheels, steering system,
brake systems and transmission. Hence, a body on frame structure was used, similar
to the configuration for most trucks [35]. Today most cars use a unibody structure
where part of the chassis structure cannot be distinguished from the body floor.
This allows for a stiffer and lighter design, but at the expense of some modularity.
In order to keep up product diversification at reasonable costs, car manufacturers
standardize through technological platforms which include parts of the body as well
as the remaining chassis components [35].

Figure 2.1: Five links con-
straining one DoF each

The suspension system links the wheels to the body
and its main task is to enhance ground contact as
well as ride comfort while distributing loads [35]. The
kinematics of a functional wheel suspension system
should constrain five (out of six) Degree of Freedom
(DoF) of each hub. Hence, only allowing for vertical
travel. From a rigid body dynamics perspective, this
can be achieved through linkages, e.g. five bars with
spherical hinges (see Figure 2.1) that constrains one
DoF each. Some of the bars are often combined to
more advanced links that constrains more than one
DoF. See for example the concepts of McPherson or
Short Long Arm (SLA)/ Double wishbone suspension
systems [35].

7



2. Theory

The studied system is called an integral link axle [36] (see Figure 2.2). This configu-
ration of a rear wheel suspension system is beneficial in terms of packaging volume
which is good for boot space. However, the system is rather difficult to understand
and thereby tricky to tune and develop.

Each side of the system includes two simple links (upper control arm and toe link)
and one H-arm (lower control arm). An H-arm with both outer joints directly
connected to the knuckle, i.e the part that connects the links to the wheel, would
constrain four dofs on its own. To allow for one more dof, these type of suspension
systems are often equipped with an integral link between the knuckle and the outer
front hinge of the H-arm. However, for the studied system, that relative movement
is instead permitted by one of the bushings.

Figure 2.2: The studied rear wheel suspension system; an integral link rear axle,
viewed from behind

It should be noted that, in addition to the integral link, some other differences
may be encountered between different concepts within this category of suspension
systems; e.g. some have coil springs and two upper control arms on each side.
However, the studied system consists of the following components:

1. Two (one for each side) knuckles containing the hubs and attachments to the
suspension links.

2. Two (one for each side) upper control arms that acts as two point (upper) links
between the subframe and the corresponding knuckle.

3. Two (one for each side) lower control arms that acts as an H-arm between the
subframe and the corresponding knuckle. Recall the bushing that acts as an
intermediate integral link between the lower control arm and the knuckle.

4. Two (one for each side) toelinks that acts as two point links (at the lower front)
between the subframe and the corresponding knuckle.

5. A subframe that attaches the inner side of the links (opposite from the knuck-
les) to the body structure.

6. A leaf spring that acts as a beam between the lower control arms and the
subframe. It mainly stiffens the system with respect to relative vertical motion
between the knuckles and the subframe.

8



2. Theory

7. Two (one for each side) dampers that acts between the car body and and the
lower control arms to decelerates the wheels during vertical shocks.

8. An anti-rollbar that acts as a torsion bar and mainly stiffens the system with
respect to rolling motion.

Figure 2.3: System components and hardpoints for linkages

Components 1-5 are marked with circles on the left hand side of Figure 2.3. The
main joints between the components consists of ball joints and elastic bushings.
The theoretical centers of these joints, i.e. the pivot points between the moving
parts, are often called hardpoints (see the right hand side of Figure 2.3) and their
relative positions determines much of the suspension system’s kinematic properties.
It should be noted that the figure only shows hardpoints for the linkages and that
each of these hardpoints has a counterpart on the left hand side with the same
hardpoint number. Section 3.2.1 shows all hardpoints that are essential for this
project.

A ball joint is simply a spherical joint that permits free rotations while preventing
translations and these are used in HP6, i.e. the rear joint between the lower control
arm and the knuckle.

Figure 2.4: Bushing

The rest of the joints involve bushings. These consists of
an inner body that connects to one component and an
outer body that is fixed to the other. In-between the fixed
bodies is a layer of vulcanized rubber that determines the
stiffness characteristics of the bushing.

Bushings were initially introduced to avoid lubrication
of the joints and the elastic characteristics was mainly
seen as a drawback. Today they are used to enhance the
elasto-kinematic behavior of the system and to mitigate
noise in the coupe [35]. Figure 2.4 shows a principle sketch of a typical bushing.
The rotational stiffnesses are usually relatively small in order to allow sufficient link
motion. For this example, the translational stiffness in the x-direction is governed
by compression of the rubber beam whilst the other directions are governed by its
bending characteristics. Moreover, the bending motion in the y-direction is limited;
contact between the inner and outer rubber acts as a spring and increases the stiffness
[35].

9



2. Theory

2.2 Finite Element Method

Finite Element Method (FEM) is a method for solving differential equations nu-
merically with a discretized approximation and is often used to analyze complex
mechanical problems. The domain is subdivided into elements defined by a mesh
while the field of interest (e.g. displacement) is approximated by piece-wise polyno-
mials that make up a continuous function over the (discretized) domain [37, 38].

2.2.1 Strong format

Consider the body in Figure 2.5 with domain Ω, boundary Γ = ΓD∪ΓN and outwards
pointing normal vector n. It is loaded with the body force b and traction tp.

Figure 2.5: Loaded body and kinematically admissible displacements

The strong form for quasi-static linear elasticity reads
−σ ·∇ = b in Ω

u = uP on ΓD
σ · n = tP on ΓN

(2.1)

where u is the displacement vector and σ is the second order Cauchy stress tensor
[38]. The constitutive relation reads

σ = E : ε (2.2)

where ε = ε[u] = [u ⊗∇]sym is the second order strain tensor and E is the fourth
order stiffness tensor for isotropic linear elasticity [38].

10



2. Theory

2.2.2 Weak format

Consider an infinitesimal, kinematically admissible, virtual displacement δu (later
called test function) of the body in Figure 2.5. The weak format can then be derived
from the stationary condition in the principle of minimum total potential energy (see
for example [38, 39]) and is stated as; find u ∈ U such that:

a(u, δu) = l(δu) ∀δu ∈ U0 (2.3)

with the variational forms

a(u, δu) =
∫

Ω
ε[u] : E : ε[δu] dΩ

l(δu) =
∫

Ω
b · δu dΩ +

∫
ΓN
tP · δu dΓ

(2.4)

as well as the trial and test spaces

U =
{
v |

∫
Ω

[
|v|2 + |v ⊗∇|2

]
dΩ <∞, v = uP on ΓD

}
U0 =

{
v |

∫
Ω

[
|v|2 + |v ⊗∇|2

]
dΩ <∞, v = 0 on ΓD

} (2.5)

2.2.3 FE-format

Consider the FE-approximation uh as

u ≈ uh =
n∑

k=1
N k(a)k = N a ∈ Uh ⊂ U (2.6)

where the shape functions N and the nodal displacements a reads

N =

N1 0 0 N2 0 0 . . . Nn 0 0
0 N1 0 0 N2 0 . . . 0 Nn 0
0 0 N1 0 0 N2 . . . 0 0 Nn


a =

[
ux,1 uy,1 uz,1 ux,2 uy,2 uz,2 . . . ux,n uy,n uz,n

]T (2.7)

The FE-format is obtained by introducing the approximation in Equation (2.6) into
the weak format; Equations (2.3)-(2.4). On abstract form it reads

a(uh, δuh) = l(δuh) ∀δuh ∈ U0
h ⊂ U0 (2.8)

which is equivalent to the well known matrix equation

Ka = f (2.9)

where K is the global stiffness matrix and f is the global (external) force vector
[37, 38].

11



2. Theory

2.3 Topology Optimization

As described in [40], the general structural optimization problem takes the form
minimize f(ξ,η) with respect to ξ and η

subject to


design constraints on ξ
state constraints on η
equilibrium constraint

(2.10)

where f is the objective function, ξ are design variables and η are state variables
(both vectors in a general case). Structural Optimization is a wide term including,
Size, Shape and Topology Optimization. Here the focus is on the latter, which is
a tool that finds the optimal material distribution in a design space, given an op-
timization goal and a set of constraints, e.g. minimizing weight whilst maintaining
stiffness requirements. TO can either be gradient-based (i.e. optimization with gra-
dient techniques), or non gradient-based. The former is widely used and is arguably
more efficient when it comes to solving fine-resolution problems, although one dis-
advantage is that the result may be a local extrema rather than the global optimum
[41]. Non-gradient based methods uses global search and refers to heuristic algo-
rithms, e.g. evolutionary algorithms such as the genetic algorithm [42] and swarm
intelligence such as ant colony [43] or particle swarm [44].

Two common algorithms for gradient-based TO are the Method of Moving Asymp-
totes (MMA) and the Optimality Criteria (OC). MMA was originally introduced
by Krister Svanberg in 1987 [45] and has since then been developed further. The
method is iterative and in each step a strictly convex approximating subproblem is
generated (with respect to the so called ’moving asymptotes’) and solved using a
gradient method. The OC method uses Lagrange multipliers to define the topology
optimization where compliance is minimized under a volume constraint. In order
to handle inequality constraints Lagrange multipliers method is supplemented with
Karush- Kuhn-Tucker conditions [46].

2.3.1 TO example: Minimum compliance using SIMP

Minimum compliance (i.e. maximum global stiffness) is one of the simplest types
of design problem formulations within TO, where the squared energy norm ||u||2a =
a(u,u) is to be minimized [46]. Together with Equation (2.3), the minimum com-
pliance problem then takes the form

min
u∈U,E

l(u)

s.t. a(u, δu) = l(δu), ∀ δu ∈ U0,E ∈ Ead
(2.11)

where Ead denotes the set of admissible stiffness tensors. The discrete form reads
min
u,E

fᵀu

s.t. K(Ee)u = f , Ee ∈ Ead
(2.12)

12



2. Theory

where the stiffness matrix K depends on the element stiffness Ee and is written as

K =
N∑

e=1
Ke(Ee) (2.13)

There are various possible definitions of Ead. One possible definition is to formulate
a distributed, discrete valued, design problem as follows

E = 1ΩoptE0, 1Ωopt =

1 if x ∈ Ωopt

0 if x ∈ Ω \ Ωopt∫
Ω

1ΩoptdΩ = Vol(Ωopt) ≤ V.

(2.14)

where x is the coordinate vector that corresponds to the chosen reference frame
and E0 is the stiffness tensor for the given isotropic material. Furthermore, Ω is
the predefined design space and Ωopt is the sought optimal subset of Ω. The last
inequality expresses the limit on the amount of material that is at disposal, where
V is the upper limit for the volume.

One way to solve the discrete valued design problem in Equation (2.14) is to replace
the integer values with continuous variables and then introduce a penalty parameter
that steers the solution into discrete 0/1 values. The most common method is the
Solid Isotropic Material with Penelization (SIMP) method [46], which reformulates
Equation (2.14) as follows

E(x) = ρ(x)pE0, p > 1,∫
Ω
ρ(x)dΩ ≤ V ; ρmin ≤ ρ(x) ≤ 1, x ∈ Ω

(2.15)

where the design variable ρ(x) can be interpreted as a material density fraction and
p is the penalization factor that steers the intermediate densities towards 0 or 1. In
order to get true ’0/1’ design, p ≥ 3 is usually sufficient for materials with Poisson’s
ratio ν = 0.33 [47]. Furthermore, in order to prevent any possible singularity of the
equilibrium problem a lower bound ρmin is introduced, typically set to ρmin = 10−3.

2.3.2 TO in Tosca

Tosca Structure uses two different principle approaches for solving topology opti-
mization problems; the controller-based approach and the general sensitivity-based
approach. The former is based on the OC method and is restricted to minimum
compliance optimizations with a volume equality constraint. The sensitivity-based
approach uses MMA and enables other objective functions as well as constraints
(e.g. displacement, stress, and eigenfrequencies). Both approaches use SIMP as the
interpolation method with p = 3 as default for static load cases [48].

Tosca structure also allows for other types of manufacturing constraints such as
demold directions, symmetry constraints and member size constraints.

13



2. Theory

2.4 FE-modeling in Abaqus

This section is intended to give a brief overview of software specific terminology and
modeling in Abaqus, involving connector elements and coupling constraints.

2.4.1 Connector elements

Connector elements are used to model connections; e.g. rigid connections, ball
joints or bushings between different parts. Kinematic constraints are enforced with
Lagrange multiplier method. Hence, the system of equations grows in size for each
imposed kinematic constraint. A connector element is defined between two nodes
and governed by the defined connector type together with a corresponding connector
behavior, e.g. allowed degrees of freedom and what element stiffness those relative
motions are associated with [48]. Below is a description of connector types used:

• BEAM imposes kinematic constraints such that no components of relative mo-
tions (translations as well as rotations) are allowed [48].

• JOIN imposes kinematic constraints such that no components of relative trans-
lations are allowed [48]. Hence, it describes a typical frictionless ball joint.

• CARTESIAN does not impose any kinematic constraint, but allows translational
components of relative motion to be associated with a connector behavior, e.g.
stiffness components. [48].

• ROTATION does not impose any kinematic constraint, but allows rotational
components of relative motion to be associated with a connector behavior,
e.g. stiffness components. [48].

2.4.2 Coupling constraints

Coupling constraints defines coupling conditions between a reference node and a
group of coupling nodes with respect to selected degrees of freedom [48]. Below is a
description of the two available types of coupling constraints:

• KINEMATIC constrains the motion of the coupling nodes to the rigid body
motions of the reference node. This is imposed by eliminating degrees of
freedoms at the coupling nodes [48]. Hence, if all six degrees of freedoms are
selected, the nodes will be rigidly connected.

• DISTRIBUTING constrains the motion of the coupling nodes by distributing
loads, acting on the reference node, to the coupling nodes. (For an arbitrary
number of coupling nodes the equation system is underdetermined and there
is no unique solution for equilibrium. Weight factors in combination with node
positions are then used to determine the load distribution [48].)

14



3
Methodology

This chapter describes the methodology for the project. The work consists of three
main parts; Project Prestudy, Finite Element Analysis (FEA) and Structural Opti-
mization. Figure 3.1 shows an overview of the workflow.

Figure 3.1: Overview of project and methodology

The prestudy includes literature study, data collection and gathering of input from
pertinent departments. This is in order to establish sufficient knowledge within TO,
vehicle dynamics and FE-modeling. Data collection is mainly about load data from
previous simulations of the dynamic, full vehicle, model in Adams Car.

The second part involves development of an FE-model of the wheel suspension sys-
tem, including identification of characteristics of components, joints and boundary
conditions. It also involves verification with respect to force signals as well as sen-
sitivity analysis with respect to stiffness of components and bushings.

The structural optimization part covers concept load cases as well as optimization
strategies. TO is performed on the FE-model of the wheel suspension system, with
the linkages as design volumes. Since previous studies within TO combined with
parametrization of joint positions are limited, it is suitable to first learn from a
simple problem. Therefore this study is limited to provide a prestudy for future
development and the parametrization study is only performed on a simpler two-
component model.

15



3. Methodology

3.1 Prestudy

Except for planning, literature study and input from other diciplines the prestudy
mainly involves data collection of load signals.

3.1.1 Strength events

During the complete vehicle simulations in Adams Car, load signals are recorded for
each hardpoint and saved as Road Load Data (RLD). Three strength events are
chosen in consultation with employees at Volvo Cars, i.e. Brake In Pothole (BIP),
Drive Over Curb (DOC) and Skid Against Curb (SAC). Figure 3.2 shows snapshots
from the selected events.

Figure 3.2: Snapshots from studied strength events

BIP captures longitudinal loading while DOC involves vertical loading and SAC
covers lateral loading. Events governed by rolling loads are avoided in order to
enable the simplification of excluding the anti-rollbar within the FE-model.

3.2 Finite Element Analysis

There exists a wide range of commercial softwares for FEA and in this project Ansa
is used as a pre-processor, Abaqus as a solver and Meta as a post-processor. The
simplified FE-model is validated with respect to force signals corresponding to RLD
of the complete vehicle model. Outer load signals from the RLD are applied to the
system FE-model and force signals in wheel suspension joints are then compared
with the corresponding signals from the RLD.

3.2.1 FE-model

The development process is initiated by modeling of the individual components
which are then assembled into a system model. Modeling and parallelly validating
is an iterative process but this section is mainly intended to describe the final model.

16



3. Methodology

3.2.1.1 Bushing attachments and material models for components

Figure 3.3: Toelink

All components are modeled with isotropic linear
elastic materials, except for the glass fiber leaf spring,
which is modeled with an ortotropic linear elastic
material model. KINEMATIC constraints are used to
model rigid behavior between the theoretical centers
of the bushing’s outer bodies and the corresponding
interfaces on the components (see for example HP14
in Figure 3.3). Similarly, connector BEAM is used to
model rigid inner bushing bodies (see for example
HP12 in Figure 3.3). Attachments for ball joints are
modeled in the same manner (see HP6 in Figures 3.7
and 3.6).

3.2.1.2 Steel sheet components

The Upper Control Arm (UCA) and the Toelink (see Figures 3.3 and 3.4) consist
of steel sheets with thicknesses in the range of 2.0-2.8 mm. The components are
modeled with 3 mm mixed three node triangular (S3R in Abaqus) and four node
quadrilateral (S4 in Abaqus) shell elements. Furthermore, the UCA contains mul-
tiple seam and spot welds which are rigidly modeled, i.e. with connector BEAM and
KINEMATIC coupling respectively.

Figure 3.4: UCA Figure 3.5: Subframe

The subframe (see Figure 3.5) is mainly made out of steel sheets with thicknesses
in the range of 1.8-4.0 mm and is modelled with a similar methodology, but where
the element size is within the range of 3-5 mm.

17



3. Methodology

3.2.1.3 Aluminum cast components

The Lower Control Arm (LCA) as well as the main body of the knuckle consists
of cast aluminum. The components (see Figures 3.6 and 3.7) are modeled with
3 mm four node tetrahedral (C3D4 in Abaqus) solid elements. The LCA’s interface
with the damper (see HP56 in Figure 3.6) is modeled with connector BEAM and
the connection with the leaf spring (see HP68 in Figure 3.6) is modeled with a
DISTRIBUTING coupling constraint, such that the load is distributed over a surface,
corresponding to a type of bushing.

Figure 3.6: LCA Figure 3.7: Knuckle

3.2.1.4 Boundary conditions

On a vehicle, the subframe is attached to the car body through bushings in HP31-34
(see Figure 3.5). However, in a chassis rig test the car body bushings are attached
to a steel fixture (in general stiffer than the body). In this system FE-model the
car body/fixture is assumed to be rigid. Inner bushing bodies are fixed through a
KINEMATIC coupling with a fixed reference node, while the subframe is connected
to the outer bushing bodies. Hence, relative movement between the subframe and
the fixture depends on bushing stiffness. A simple illustration of the boundary
conditions are shown in Figure 3.8 where the spring elements (representing the
bushing’s stiffnesses) are fixed in one end and attached to the outer bushing body
in the other.

Figure 3.8: Boundary conditions similar to a chassis rig test fixture

18



3. Methodology

3.2.1.5 Assembly

Figure 3.9 shows the assembled model together with the global reference frame for
the car. The components are attached to each other through connector elements,
one in each hardpoint. Each connector element acts between two coinciding nodes,
i.e. between the center nodes of the corresponding inner and outer bushing bodies.

Figure 3.9: Assembly and global reference frame

The connector type JOIN is used to model the ball joints (HP6). The rest of the joints
consists of bushings which are modelled with a combination of the two connector
types; CARTESIAN and ROTATION. These are also associated with a corresponding
connector behavior that refers to the bushing’s stiffness characteristics. Note that
ROTATION is chosen based on the assumption that rotations can be considered small.

3.2.1.6 Linear elastic bushing characteristics

Non-linear bushing characteristics are linearized, but it is not obvious what stiff-
ness to choose in order to replicate load distributions over several strength events.
Translational characteristics are described by uni-axial force-displacement curves
and rotational characteristics are described by moment-rotation curves. The linear
bushing characteristics are shown in Appendix A where they are presented together
with the corresponding non-linear data (all without scales to protect corporate data).
Rotational stiffnesses uses the initial stiffnesses around the origin, while the trans-
lational stiffnesses are chosen to some value between the lowest and the highest
tangent of the corresponding bushing characteristics. The corresponding directions
refers to local reference frames for each bushings, i.e. not the global reference sys-
tem for the car. The values of the transalational stiffnesses are essentially chosen
by changing the stiffensses from low to high and then analyzing the response of the
system and choosing the value that corresponds well to the RLD.

3.2.1.7 Damper loads

Since there are no velocity during quasi-static analysis, damping characteristics
would have no effect. However, the damper behavior needs to be included somehow
in order to get reasonable load signals. This is solved by applying external forces

19



3. Methodology

from the dampers, acting on the lower control arms (see Figure 3.10). These forces
varies in time in accordance with the RLD.

Figure 3.10: Damper force

It should be noted that this method is associated with some drawbacks. Peak loads
from the dampers may result in some non-physical deflections of the system during
quasi-static analyses. This is because the entire damper force is used to statically
deform the system, while in a dynamic case the force mainly acts to decelerate the
unsprung mass.

3.2.1.8 Drawing state loads and pretension forces

The FE-model is unloaded in its initial state (the drawing state), at least if preten-
sion forces are not taken into consideration, while in reality (and in the dynamic
simulations) it is not. It is essentially loaded by the vehicle weight. This has to be
accounted for in order to get comparable load signals between the system FE-model
and the RLD. Two methods are used to handle this. The first involves determina-
tion of the drawing state loads, i.e. the loads that correspond to the vehicle standing
on the ground, and then to subtract these from the RLD before comparison. The
second method involves pretension forces with vertical force components that act in
the leaf spring. These forces have to act as internal forces to the system in the sense
that no forces are added to the system as a whole. The pretension forces are shown
in Figure 3.11 where the red components, acting on the lower control arms and the
subframe, are applied in the model.

Figure 3.11: Pretension forces from the leaf spring at drawing state

20



3. Methodology

The first method provides slightly better accuracy and involves less uncertainties,
why it is used during sensitivity analysis as well as for verification of the FE-model.
However, the second provides almost as good accuracy and since no loads are sub-
tracted it is preferable for use during optimizations.

3.2.2 Verification with respect to force signals

The FE-model is verified against the RLD with respect to force-time signals. In
order to get a broader view of how well it correlates, the verification is based on
time sequences from the strength events rather than just single snapshots in time.
For each event, a time interval is chosen such that the peak loads are captured.
The corresponding sequence for external loads, i.e. at the knuckles in HP9 and
the dampers in HP56, are then applied to the FE-model. Forces in intermediate
hardpoints are then validated against the corresponding load signals from the RLD.
Quasi-static analyses are used with load signals applied as 100 static load steps.
Hence, these are not dynamic analyzes. The force signals, from both the FE-model
and the RLD, are normalized according to Equation (3.1) before comparison.

F norm
ish =

Fish

max
i,s

(|F (RLD)
ish |)

(3.1)

Here h denotes hardpoint number, s denotes side, i.e. left (LH) or right (RH),
and i denotes force component in the global coordinate system for the car. The
normalization emphasizes that correlation of large magnitude components is more
important than correlation of small ones. In order to quantify the deviations, Root
Mean Square (RMS) values of the errors are calculated for the normalized force
components as

RMSnorm(i, s, h) =

√√√√ 1
N

N∑
n=1

(
F

norm (FE)
ishn − F norm (RLD)

ishn

)2
(3.2)

where n denotes the current time step. Hence, together with the normalization in
Equation (3.1), RMSnorm can be interpreted as the RMS value of the error, normal-
ized by the largest RLD-force component in the corresponding hardpoint.

3.2.3 Sensitivity Analysis: Bushing Stiffness

As mentioned previously, it is not obvious what linear characteristics to choose for
the bushings. Therefore, a sensitivity analysis of bushing stiffness is performed
where the bushing stiffnesses for different hardpoints are varied one at a time, while
everything else is held constant. They are varied between the lowest and highest
tangent stiffness for the corresponding bushing (see Appendix A).

21



3. Methodology

The sensitivity analysis is evaluated with respect to Root Mean Square (RMS) values
of the error, i.e. the difference between the FE-model and the RLD. The RMS values
are evaluated in two ways; RMSHP refers to the deviation in each hardpoint number
and RMSEvent to a total deviation for an event, see Equations (3.3) and (3.4).

RMSHP(h) =

√√√√ 1
I + S +N

I∑
i=1

S∑
s=1

N∑
n=1

(
F

(FE)
ishn − F

(RLD)
ishn

)2
(3.3)

RMSEvent =

√√√√ 1
H + I + S +N

H∑
h=1

I∑
i=1

S∑
s=1

N∑
n=1

(
F

(FE)
ishn − F

(RLD)
ishn

)2
(3.4)

Here h, i, s and n are defined in accordance with Subsection 3.2.2. The sensitivity
analysis of bushing stiffness aims to capture how sensitive it is to make an ansatz
about these values at an early stage, i.e. before the bushing characteristics are
determined, but also to measure the sensitivity of the results from the verification.

3.2.4 Sensitivity Analysis: Component Stiffness

In addition to bushing stiffness, a similar study is performed with respect to com-
ponent stiffness, where Young’s modulus for the linkages are varied. The aim is
to study how sensitive the load distribution is with respect to component stiffness.
Therefore, this analysis focuses on changes in peak forces rather than RMS val-
ues and serves as a prestudy for the optimization part, where component stiffness
depends on the topology.

3.3 Structural Optimization

There exists a wide range of commercial software for TO and in this project Tosca
GUI is used as a preprocessor, Tosca Structure as an optimizer, Abaqus as a FE-
solver and Tosca viewer as a postprocessor. The parametrization of hardpoint
positions are limited to involve a two-component problem.

3.3.1 Two-component problem

As mentioned in Chapter 1, there are seemingly not much literature on how to
combine TO with parametrization of joint positions, at least not within industrial
applications. Hence, general method development is needed within this area before
application to complex models such as the studied wheel suspension system. There-
fore a simple, made up, two-component problem is used to investigate different
optimization strategies in context with parametrization of hardpoint positions.

22



3. Methodology

The two-component model consists of two links that are connected to each other
(see Figure 3.12). It is modeled in accordance with the methodology for the wheel
suspension system. There are four hardpoints in total, where two of them are
connected to ground through bushings, represented by spring elements in the figure.
The components are also connected through a bushing and the upper three-point-
link is loaded in the hardpoint to the right.

Figure 3.12: Two-component model: Problem definition and Abaqus FE-model

The bushings are either set to be rigid or flexible in order to study how flexible
bushings affect optimization. The bushings have rotationally symmetric stiffnesses
and the parametrization is with respect to the horizontal position of the common
hardpoint between the links. It is varied between 100 mm (as currently seen in
Figure 3.12) and 300 mm with a step size of 25 mm. Both components are defined
as design volumes (the available design space Ω) but the topology for the zones
around the bushings are frozen through design constraints.

The two-component model is used to study different optimization strategies, where
the following three are included within this report:

• Minimize mass, with constraint on the vertical tip-displacement (∆ ≤ 1 mm)
for a model with rigid bushings.

• Minimize mass, with constraint on the vertical tip-displacement (∆ ≤ 10 mm)
for a model with flexible bushings.

• Minimize strain energy, with a constraint on system mass of 1.35 kg, for a
model with flexible bushings.

23



3. Methodology

3.3.2 TO of the wheel suspension system

This section explains the setup for topology optimization of the wheel suspension
system, where linkages are optimized with respect to weight and stiffness. The val-
idated FE-model is used as a reference model and provides benchmarks for stiffness
requirements and performance measurements.

3.3.2.1 Design volumes of the linkages

Linkages are exchanged for design volumes (see Figure 3.13) that defines the avail-
able design space Ω. More detailed views of the design volumes are shown in Figures
3.14-3.16. These uses four node tetrahedral elements with an element size of 6 mm
instead of 3 mm. This is simply in order to reduce the number of elements, making
the computational effort manageable and should be fine enough to represent stiff-
ness behavior. All design volumes are made of cast aluminium and are modelled
in accordance with the methodology described in Section 3.2.1. The zones around
coupling constraints and connector elements are frozen through design constraints.

Figure 3.13: Linkages as design volumes Figure 3.14: LCA design volume

Figure 3.15: UCA design volume Figure 3.16: Toelink design volume

24



3. Methodology

3.3.2.2 Concept load cases

The concept load cases consists of external loads applied at the knuckles in HP9,
damper loads in HP56 and pretension forces in the leaf spring. The pretension forces
are based on the car weight that is supported by the rear wheels as follows

F pret
z = PF ·RM (3.5)

where F pret
z is the magnitude of the applied vertical force components (illustrated

in Figure 3.11), RM is the total mass acting on the rear wheels and PF is the
Pretension load factor. Note that RM differs slightly between the strength events
due to different car loading, but PF should be fixed for a certain car, where PF = 7.1
in this case.

External loads at the knuckles in HP9 and at the dampers in HP56 are based on a
snapshot in time that corresponds to the governing peak load at the knuckle for each
event, i.e. the largest longitudinal force component for BIP, the largest vertical force
component for DOC and the largest lateral force component for SAC. Note that the
concept load cases involves asymmetry in accordance with the strength events.

3.3.2.3 Topology optimization configuration

Since the load cases are asymmetric a symmetry condition is applied in the xz-
plane to ensure geometrical symmetry with respect to the left and right hand side
of the car. The optimization strategy is to minimize strain energy subject to a mass
constraint. Two main cases are presented, the first without additional constraints
and the second with additional constraints on the displacements at the wheel center.

The configuration for the first optimization case is defined in Table 3.1. The objec-
tive is to minimize strain energy with a mass constraint, where M̄Ref refers to the
weight of the linkages in the reference model. The optimization is first performed
with respect to one load case at a time in order to study how different strength
events affect the topology. However, the last is with respect to an equally weighted
combination of all three load cases in order to gain robustness within the structure.

Table 3.1: Configuration of TO of case 1

Objective function Constraint Load case

Minimize strain energy M ≤ M̄Ref

BIP
DOC
SAC

BIP + DOC + SAC

The configuration for the second optimization case is presented in Table 3.2. As for
the mass, the allowed displacement is based on the reference model. However, the

25



3. Methodology

displacement constraints can be violated during the optimization and thus need to
be checked afterwards. The displacement constraints are such that they constrain
the governing displacement direction in the wheel center (HP9) for each event. Thus
for BIP, which captures longitudinal loading, the constraint is on the x-component.
Similarly the constraint is on the z-component for DOC and on the y-component
for SAC. The topology optimization is then iterated with lower and lower mass
constraints until the displacement constraints are just fulfilled. However, note that
system stiffness as well as the system weight will normally increase during the design
realization step. The corresponding optimization setups are presented in Table 3.2.

Table 3.2: Configuration of TO of case 2

Objective function Constraint Load case

Minimize strain energy

M ≤ M̄Ref

BIP + DOC + SAC| uBIP
x | ≤ | ūBIP

x |
| uDOC

z | ≤ | ūDOC
z |

| uSAC
y | ≤ | ūSAC

y |

Minimize strain energy

M ≤ 0.9M̄Ref

BIP + DOC + SAC| uBIP
x | ≤ | ūBIP

x |
| uDOC

z | ≤ | ūDOC
z |

| uSAC
y | ≤ | ūSAC

y |

Minimize strain energy

M ≤ 0.8M̄Ref

BIP + DOC + SAC| uBIP
x | ≤ | ūBIP

x |
| uDOC

z | ≤ | ūDOC
z |

| uSAC
y | ≤ | ūSAC

y |

Minimize strain energy

M ≤ 0.7M̄Ref

BIP + DOC + SAC| uBIP
x | ≤ | ūBIP

x |
| uDOC

z | ≤ | ūDOC
z |

| uSAC
y | ≤ | ūSAC

y |

Minimize strain energy

M ≤ 0.6M̄Ref

BIP + DOC + SAC| uBIP
x | ≤ | ūBIP

x |
| uDOC

z | ≤ | ūDOC
z |

| uSAC
y | ≤ | ūSAC

y |

Minimize strain energy

M ≤ 0.5M̄Ref

BIP + DOC + SAC| uBIP
x | ≤ | ūBIP

x |
| uDOC

z | ≤ | ūDOC
z |

| uSAC
y | ≤ | ūSAC

y |

26



4
Results

This chapter contains results from verification and sensitivity analysis of the system
FE-model as well as from the optimizations.

4.1 Verification of FE-model

This section is merely intended to highlight some of the most interesting findings.
However, complete results from the verification part is presented in Appendix B.
As can be seen in Figures B.1 - B.30, the FE-model tends to have high correlation
with the RLD and it captures most tendencies, such as force peaks and valleys. In
particular the force signals in the ball joints (HP6) are nearly identical to the RLD
for all strength events. The corresponding force signals for BIP are shown in Figure
4.1 and force signals for the other two events are presented in Figures B.14 and B.24.

Figure 4.1: Force signals in HP6 during BIP, FE-model ( ) vs RLD ( )

27



4. Results

The force signals from the FE-model are compared with force signals from the RLD.
The largest component from the RLD reaches ±1 due to normalization in accordance
with Equation (3.1).

Normalized RMS values, see Equation (3.2), are presented in Tables 4.1 - 4.3. Note
that RMSnorm is by no means a perfect measurement of deviation, but together
with Figures B.1 - B.30 it can offer some guidance. The correlation is usually very
high for RMSnorm ≤ 10 %, acceptable for RMSnorm ≤ 15 % and poor if RMSnorm ≥
20 %. Tables 4.1-4.3 also present means of RMSnorm, taken over all hardpoints and
calculated for each force component. A comparison, with respect to these mean
values, between the method that uses drawing state loads and the one that uses
pretension forces are presented in Tables B.2, B.4 and B.6. It can be noted that
both methods provide similar accuracy.

RMSnorm for all force signals during the BIP event is presented in Table 4.1. As can
be seen, the correlation is acceptable for all force signals except FLH

y in HP4. It is
highlighted in red and discussed further in Section 4.1.1.

Table 4.1: RMSnorm for all force signals during the BIP event, with subtraction of
drawing state loads

HP
Force

F LH
x F LH

y F LH
z FRH

x FRH
y FRH

z

HP1 3.2 % 12.5 % 1.0 % 2.2 % 12.3 % 0.8 %
HP3 4.0 % 5.0 % 1.3 % 3.6 % 3.8 % 1.2 %
HP4 2.5 % 26.9 % 3.2 % 1.9 % 7.0 % 0.9 %
HP6 2.7 % 4.0 % 2.7 % 2.1 % 1.3 % 1.9 %
HP7 2.1 % 12.5 % 1.2 % 1.2 % 12.3 % 0.8 %
HP12 1.3 % 8.0 % 1.0 % 1.0 % 5.5 % 0.6 %
HP14 1.7 % 8.0 % 0.8 % 1.2 % 5.6 % 0.5 %
HP18 1.6 % 5.6 % 2.5 % 1.5 % 5.5 % 2.1 %
HP31-32 8.5 % 4.1 % 4.6 % 7.2 % 3.4 % 3.8 %
HP33-34 8.3 % 8.3 % 11.8 % 7.6 % 10.1 % 9.4 %
Mean 3.6 % 9.5 % 3.0 % 3.0 % 6.7 % 2.2 %

RMSnorm for all force signals during the DOC event are presented in Table 4.2.
Here most force signals shows good correlation as well, but F LH

y and FRH
y in HP4

corresponds to high RMSnorm and this is discussed further in Section 4.1.1. F LH
z

and FRH
z in HP33-34 are also quite high, but may be considered as acceptable with

respect to the simplifications behind the system FE-model.

28



4. Results

Table 4.2: RMSnorm for all force signals during the DOC event, with subtraction
of drawing state loads

HP
Force

F LH
x F LH

y F LH
z FRH

x FRH
y FRH

z

HP1 1.3 % 2.5 % 0.5 % 1.3 % 2.5 % 0.5 %
HP3 2.5 % 4.0 % 3.3 % 2.5 % 3.9 % 3.3 %
HP4 2.5 % 39.3 % 6.1 % 2.4 % 39.0 % 6.1 %
HP6 2.1 % 2.9 % 4.9 % 2.0 % 2.9 % 4.9 %
HP7 1.1 % 2.4 % 0.6 % 1.1 % 2.4 % 0.6 %
HP12 2.3 % 10.5 % 1.7 % 2.4 % 10.4 % 1.7 %
HP14 2.3 % 10.6 % 1.0 % 2.3 % 10.6 % 1.0 %
HP18 0.9 % 3.9 % 6.0 % 0.9 % 3.9 % 6.0 %
HP31-32 6.7 % 4.5 % 5.8 % 7.1 % 4.9 % 5.7 %
HP33-34 9.7 % 6.3 % 16.6 % 9.2 % 6.0 % 16.8 %
Mean 3.1 % 8.7 % 4.6 % 3.1 % 8.7 % 4.7 %

Table 4.3 presents RMSnorm for all force signals during the SAC event. Here most
force signals show good correlation as well, but F LH

x and F LH
y in HP3 show poor

correlation and this is discussed further in Section 4.1.1. Also note that RMSnorm is
generally larger for the left hand side than for the right. This is because SAC is an
asymmetric strength event where the loads are much larger on the left hand side,
that hits the curb.

Table 4.3: RMSnorm for all force signals during the SAC event, with subtraction of
drawing state loads

HP
Force

F LH
x F LH

y F LH
z FRH

x FRH
y FRH

z

HP1 0.6 % 0.7 % 0.3 % 0.1 % 0.4 % 0.0 %
HP3 49.5 % 52.3 % 7.1 % 4.8 % 4.9 % 1.4 %
HP4 1.6 % 8.3 % 2.5 % 0.3 % 7.5 % 0.6 %
HP6 6.7 % 5.4 % 5.3 % 0.1 % 0.2 % 0.8 %
HP7 0.6 % 0.7 % 0.3 % 0.1 % 0.4 % 0.0 %
HP12 0.5 % 3.2 % 0.2 % 0.0 % 0.2 % 0.0 %
HP14 0.5 % 3.2 % 0.2 % 0.0 % 0.2 % 0.0 %
HP18 2.8 % 9.4 % 8.3 % 0.2 % 0.7 % 0.5 %
HP31-32 8.6 % 8.4 % 0.9 % 5.8 % 1.5 % 1.4 %
HP33-34 3.5 % 9.9 % 13.0 % 2.0 % 11.0 % 3.6 %
Mean 7.5 % 10.2 % 3.8 % 1.3 % 2.7 % 0.8 %

29



4. Results

4.1.1 Force signals with poor correlation

Figure 4.2: High RMSnorm: BIP and
DOC, FE-model ( ) vs RLD ( )

Despite that the correlation looks accept-
able overall, there are some exceptions
(five out of 180).

The first exception is found in strength
event BIP (see Figure 4.2) where F LH

y in
HP4 shows an RMSnorm = 26.9 %.

The second and third main exceptions are
found during strength event DOC (see
Figure 4.2) where both F LH

y and FRH
y in

HP4 show poor correlation, with RMSnorm

of 39.3 % and 39.0 % respectively.

The poor correlation for lateral forces in
HP4 can be referred to cases where the
dampers are more active. DOC is associ-
ated with large damper loads and in the
BIP event these forces are larger for the
left hand side than for the right. F LH

y

corresponds to the wheel that goes into
the pothole while FRH

y corresponds to the
wheel that stays on the road surface and
therefore shows better correlation with re-
spect to the RLD (see Figure B.3). Also
note that HP4 looks better for the SAC
event (see Figure B.23) where the damper
forces are smaller.

The fourth and fifth main exceptions occurs during SAC, where both F LH
x and F LH

y

in HP3 show poor correlation (see Figure 4.2). The corresponding values of RMSnorm

are 49.5 % and 52.3 % respectively.

Figure 4.3: High RMSnorm: SAC, FE-model ( ) vs RLD ( )

30



4. Results

4.1.2 Static analysis and dynamic events

The developed FE-model is only used during quasi-static analyses, but the applied
loads are from dynamic events. Hence, peak loads from the dampers result in un-
physical deflections. This effect is most significant during the DOC event, where
the large vertical forces statically pushes the LCA down, see Figure 4.4, where the
undeformed system (grey) is plotted on top of the deformed (red). Note that, in a
dynamic case, the peak damper force mainly acts to decelerate the unsprung mass.

Figure 4.4: Non-physical deflections caused by peak damper loads during DOC

Another noteworthy difference between the quasi-static FE-model and the RLD
refers to the difference between load signals for inner and outer hardpoints of the
toelink and UCA. Recall that HP12 and HP14 correspond to the outer and inner
bushings for the toelink. The corresponding signals for FRH

z during DOC are pre-
sented in Figure 4.5. As expected, equilibrium is fulfilled during static analysis of
the FE-model. In contrast, the RLD differs between these two hardpoints and there
are essentially two probable explanations for this difference; (1) the corresponding
local coordinate systems, within the dynamic model, are not perfectly aligned with
the global reference frame of the car during the whole event (they rotate with the
corresponding bushings) and (2) equilibrium is not present as the component accel-
erates. The FE-model has fixed coordinate systems and component accelerations
are not included in quasi-static analysis.

Figure 4.5: FRH
z for DOC: HP12 and 14, FE-model ( ) vs RLD ( )

31



4. Results

4.1.3 Sensitivity analysis: Bushing stiffness

This section is intended to focus on the most interesting results from the sensitivity
analysis with respect to bushing stiffness. However, additional results are found in
Appendix C and a summary is presented in Table 4.4. Comments on system reponse
is provided for each bushing stiffness that is varied. The analyzed hardpoints are
HP3 and HP4 since these show the highest discrepancy in the verification (see Table
4.1 - 4.3). HP31-34 are also analyzed to investigate the influence of car body bushing
stiffness. Over all, the system shows low sensitivity to bushing stiffness, but there
are a few exceptions, mainly with respect to bushing stiffness in the local y-directions
of HP3 and HP31-32.

Table 4.4: Summary of the system response with respect to bushing stiffness

HP Stiffness Event Comments on system response

3

Kx

BIP Negligible sensitivity
DOC Negligible sensitivity
SAC Some sensitivity in HP3, 12 and 14 for low stiff-

ness. Otherwise negligible

Ky

BIP Over all high sensitivity for low stiffness. Negligi-
ble sensitivity in HP4 and HP31-34

DOC Over all high sensitivity for low stiffness. Negligi-
ble sensitivity in HP4 and HP31-34

SAC Some sensitivity in HP3, 12, 14 and 18 for low
stiffness. Otherwise negligible

4

Kx

BIP Negligible sensitivity
DOC Negligible sensitivity
SAC Negligible sensitivity

Ky

BIP Negligible sensitivity
DOC Negligible sensitivity
SAC Some sensitivity in HP4. Otherwise negligible

31-32

Kx

BIP Negligible sensitivity
DOC Negligible sensitivity
SAC Negligible sensitivity

Ky

BIP Some sensitivity in HP31-34. Otherwise negligible
DOC Negligible sensitivity
SAC High sensitivity in HP31-34 for low and high stiff-

ness. Otherwise negligible

33-34

Kx

BIP Some sensitivity in HP31-34. Otherwise negligible
DOC Negligible sensitivity
SAC Some sensitivity in HP31-34. Otherwise negligible

Ky

BIP Negligible sensitivity
DOC Negligible sensitivity
SAC Some sensitivity in HP31-34. Otherwise negligible

32



4. Results

As can be seen in Figure 4.6, the bushing stiffness in local y-directions for HP3,
has a rather big influence on the system response, particularly for BIP and DOC.
For these two events it is clear that a higher stiffness results in a better correlation.
This is not the case for SAC. For some hardpoints the correlation is instead slightly
better for low stiffnesses. Thus a trade-off has to be made between the different
strength events.

The vertical dashed black line seen in the figures represents the value that is used in
the FE-model of the system. As explained in Section 3.2.3 the stiffnesses corresponds
to bushing specific material data, thus the lowest and highest tangent stiffnesses may
differ between hardpoints.

(a) RMS for each event (b) RMS for each hardpoint, BIP

(c) RMS for each hardpoint, DOC (d) RMS for each hardpoint, SAC

Figure 4.6: System response of bushing stiffness in HP 3 local y-direction

The sensitivity analysis of bushing stiffnesses in HP31-34 examines how sensitive the
system is with respect to stiffness of car body bushings, i.e. the boundary conditions.

As mentioned in Table 4.4, changes in car body bushing stiffnesses do not affect
force signals in hardpoints for linkages. This can be seen in Figure 4.7, where
the stiffness corresponding to the local y-direction in HP31-32, is analyzed. Thus,

33



4. Results

reasonable changes in stiffness does not affect linkages joints. However, by changing
the bushing modeling in hardpoints 31-34 to rigid elements, the correlation gets
much poorer in all hardpoints, not just for hardpoints 31-34. An indication to this
can be seen in Figure 4.7d where the plot shows that initially the result improves
with higher stiffness but eventually starts to worsen. Note that in these figures the
’High’ stiffness is very far from rigid.

(a) RMS for each event (b) RMS for each hardpoint, BIP

(c) RMS for each hardpoint, DOC (d) RMS for each hardpoint, SAC

Figure 4.7: System response of bushing stiffness in HP 31-32 local y-direction

The sensitivity analysis shows that the response in the intermediate joint exhibit
little sensitivity with respect to the bushing stiffness variation in car body bushings,
i.e. HP31-34. Similarly stiffness of bushings for linkages does not seem to affect the
reaction forces in HP31-34. It also shows that the system is mainly sensitive for
lower stiffnesses and that a trade-off has to be made with respect to RMS values for
different strength events.

34



4. Results

4.1.4 Sensitivity analysis: Component stiffness

The peak loads shows low sensitivity to component stiffness. This section is there-
fore limited to only involve results for the combination of hardpoint, component
and load case that shows the largest sensitivity, i.e. HP4, LCA and DOC (see Fig-
ure 4.8). The results are presented in terms of fractional changes of the largest
force magnitude within the hardpoint, all compared with the developed FE-model.
The component stiffness is varied with Young’s modulus and the force magnitude
corresponds to the largest norm that occurs during the event. Hence, the analysis
provides a measurement of how sensitive the load distribution is with respect to
component stiffness.

Figure 4.8: Change in peak force in HP4 for DOC when changing the stiffness of
the LCA

As can be seen in Figure 4.8, the loads are not very sensitive with respect to compo-
nent stiffness, at least not for reasonable values of Young’s modulus. As a reference
the LCA is made of cast aluminum with a Young’s modulus of 72 GPa, marked
with a vertical dashed line. During TO, the design volumes usually uses much more
material than the optimized topology. Hence, the corresponding components are
initially very stiff. The stiffness is then reduced as material is removed and the end
result can be expected to have a similar stiffness as the reference model. The results
indicates that changes in force magnitudes are within a few percent.

35



4. Results

4.2 Structural Optimization

This Section presents both the result from the optimization of the simplified model
and the result from the FE-model of the optimization of the wheel suspension system.

4.2.1 Optimization of the two-component model

The first optimization involves minimization of system weight, subject to a tip dis-
placement ∆ ≤ 1 mm for a model with rigid bushings. This setup gives an optimized
system mass that changes smoothly with respect to hardpoint position (see 4.9). The
corresponding topology evolution is shown in Figure 4.10. As expected, the system
needs less weight to fulfill the displacement constraint as the parametrized hardpoint
is re-positioned towards the tip. The slope of the curve in Figure 4.9 indicates that
systems with structurally inefficient joint positions will gain a larger weight reduc-
tion per unit length of repositioning compared to systems with structurally efficient
placement of hardpoints.

Figure 4.9: Performance plot: Minimize mass (rigid bushings)

Figure 4.10: Topology evolution: Minimize mass (rigid bushings)

36



4. Results

The corresponding optimizations with elastic bushings are shown in Figures 4.11
and 4.12. The optimization setup is minimization of system weight, subject to a tip
displacement ∆ ≤ 10 mm. The constraint is not fulfilled when the hardpoint is too
far away from the tip (due to flexibility in the bushings) and the corresponding data
points in Figure 4.11 are left out. The change in optimized system weight is rather
dramatic for hardpoint positions between 150 mm and 175 mm, then the slope levels
out.

Compared to the corresponding result for rigid bushings (see Figures 4.9 and 4.10) it
can be seen that the bushing stiffnesses affect the optimization results significantly.
The difference in slope (weight reduction per unit length of repositioning) is even
greater between structurally inefficient and efficient systems. Another noteworthy
difference refers to the geometrical outcomes. By comparing Figure 4.10 with Figure
4.12 it can be seen how reaction moments affect the topology. With rigid bushings
the reaction moments on the left boundaries are larger. The corresponding topology
is adapted by a slightly off centered lower link, where the adjusted angle favours the
vertical support for the system. Similarly the section closest to the left boundary
for the upper link is thicker and points slightly upwards (compare with the perfectly
horizontal in Figure 4.12.)

Figure 4.11: Performance plot: Minimize mass (flexible bushings)

Figure 4.12: Topology evolution: Minimize mass (flexible bushings)

37



4. Results

For minimum compliance (minimizing strain energy) the strain energy, tip displace-
ment and the outcome topology all behaves smoothly with respect to hardpoint
position (see Figures 4.13-4.15).

Figure 4.13: Strain energy Figure 4.14: Tip Displacement

Figure 4.15: Topology evolution: Minimize strain energy (flexible bushings)

Despite that the outcome topologies in Figure 4.15 looks reasonable with respect to
design realization, strain energy may not be suitable as a performance measurement.
It is not easily understood and may thereby be hard to communicate with other
disciplines. However, by comparing Figures 4.13 and 4.14 it can bee seen that the
strain energy correlates perfectly with the tip displacement. This should be expected
for this problem since the strain energy is merely a scaling of the tip displacement,
i.e. 0.5F∆.

It should be noted that displacement measurement may be harder to correlate with
the strain energy for problems that involves multiple loads, acting on different parts
of the boundary. Hence, this requires careful consideration of how load cases and
stiffness performance measurements are combined.

4.2.2 TO of the wheel suspension system

The results of the topology optimization of the wheel suspension system of both cases
defined in Section 3.3.2.3 are presented in this section. The results are compared to
the FE-model presented in Section 4.1, here called the reference model.

38



4. Results

4.2.2.1 Case 1

The results from the optimization in case 1 are summarized in Table 4.5 and the
resulting topology of the components are presented in Figure 4.16. Note that these
are just results from using all concept load cases and that the corresponding results
for optimization against single load cases are presented in Appendix D.

Table 4.5: Results from optimization case 1 with all load cases, BIP, DOC and
SAC

Opt system Ref model Change

BIP 187 187 0.0%
Strain energy DOC 227 283 -19.6%

[Nm] SAC 404 411 -1.6%

sum 819 881 -7.0%

LCA 5.39 5.22 3.2%
Mass UCA 0.94 1.19 -20.7%
[kg] Toelink 0.62 0.59 5.0%

sum 6.95 7.00 -0.7%

| uBIP
x,L | 14.19 13.98 1.5%

| uBIP
x,R | 0.71 0.87 -17.9%

Displacements | uDOC
z,L | 3.21 2.62 22.6%

[mm] | uDOC
z,R | 3.30 2.72 21.6%

| uSAC
y,L | 6.25 6.06 3.2%

| uSAC
y,R | 3.28 3.28 0.1%

In this optimization there are no constraints on displacements but the displacements
are still measured and compared to the reference model. As described in Subsection
3.3.2.3 the displacements are measured at the wheel center and in the governing
direction of each event. As can be seen in Table 4.5 all but one are larger for
the optimized system than for the reference model. However, the strain energy is
reduced for load cases DOC and SAC while it remains the same for BIP. Hence,
the optimized system is overall stiffer than the reference model. The relative mass
distribution between the components is also different, where some of the material
from the UCA is redistributed to the toelink and LCA. The sum of the component
masses shows that the optimized system converge towards a total mass that lies
slightly below the predefined mass constraint. This is discussed further in Section
5.2.2.

39



4. Results

(a) Lower control arm (b) Upper control arm

(c) Toelink (d) System model

Figure 4.16: Topology of the optimized components for case 1 with all load cases,
BIP, DOC and SAC

The topology of the components from the optimization with individual load cases
are presented in Figures D.1-D.3 in Appendix D. These show how the topology is
affected by the different load cases. By comparing these figures it can be noted that
high stiffness in the toelink is not very crucial during the DOC event and the toelink
is thus very slender. In the SAC event it is instead the LCA that becomes slender
while the toelink almost uses all available mass.

4.2.2.2 Case 2

The results from the optimizations in case 2 are summarized in Table 4.6-4.7, where
Table 4.6 corresponds to the mass constraint M ≤ M̄ and the resulting topology
of the components are presented in Figure 4.17. Just as in case 1 (where no dis-
placement constraints are added) the relative mass distribution between the links is
different from the reference model. However, in this case the redistribution is even
more significant, where more mass is redistributed from the UCA to the LCA and
toelink. The strain energy is decreased for DOC and SAC, just as in case 1, but
here it is also reduced for BIP. The displacements at the wheel center (HP9) are less
than for the reference model and thus the constraints are fulfilled.

40



4. Results

Table 4.6: Results from optimization case 2, M ≤ M̄Ref

Opt system Ref model Change

BIP 186 187 -0.4%
Strain energy DOC 233 283 -17.5%

[Nm] SAC 408 411 -0.7%

sum 827 881 -6.0%

LCA 5.55 5.22 6.4%
Mass UCA 0.73 1.19 -38.7%
[kg] Toelink 0.66 0.59 13.0%

sum 6.95 7.00 -0.8%

| uBIP
x,L | 13.97 13.98 -0.0%

| uBIP
x,R | 0.76 0.87 -12.7%

Displacements | uDOC
z,L | 2.60 2.62 -0.7%

[mm] | uDOC
z,R | 2.71 2.72 -0.3%

| uSAC
y,L | 6.06 6.06 -0.1%

| uSAC
y,R | 3.28 3.28 -0.1%

(a) Lower control arm (b) Upper control arm

(c) Toelink (d) System model

Figure 4.17: Topology of the optimized components for case 2 with all load cases,
BIP, DOC and SAC

41



4. Results

In Table 4.7 the results from the iterations of the mass constraint are summarized.
Here the mass constraint ranges from 90 to 50 percent of the reference mass. For
each iteration, percentage changes (with respect to the reference model) for strain
energy, mass and displacements are presented. With less than 90 percent of the
reference mass the sum of the strain energy is above that of the reference model,
i.e. the system has become less stiff over all. However, until a mass of only 50
percent, the displacement constraints are still fulfilled. Hence, it may be hard to
tell whether the optimized system has the same performance as the reference model.
An increased strain energy indicates that the system becomes weaker over all.

Note that the used displacement constraints are just for illustration during this pilot
study and may not be interpreted as proper requirements. Some displacements
in Table 4.7 behaves rather unpredictable as the mass is reduced. Non-physical
deflections of the FE-model may be a contributing factor, but the system in itself
is also rather complex. Weakening of links may for example cause the knuckle to
rotate such that certain displacements in HP9 are reduced. However, it is clear
that the modeled system fails to fulfill the displacements constraints when the mass
constraint is at 50 percent of the reference model’s. The resulting topology of the
components consisting of 60 to 90 percent of the reference model’s mass are presented
in Figures D.4-D.6 in Appendix D. As the mass is reduced, the rib structure for the
LCA becomes more pronounced and the UCA becomes more truss-like.

Table 4.7: Results from optimization case 2, with M ≤ [0.5− 0.9]M̄Ref

0.9M̄Ref 0.8M̄Ref 0.7M̄Ref 0.6M̄Ref 0.5M̄Ref

BIP 0.7% 3.7% 6.8% 19.4% 43.2%
Strain energy DOC -14.0% -4.5% 4.5% 35.0% 180.4%

[Nm] SAC 0.3% 5.4% 9.6% 21.2% 117.1%

sum -4.2% 1.9% 7.4% 25.2% 121.7%

LCA -5.3% -15.1% -25.8% -36.2% -46.6%
Mass UCA -40.7% -49.0% -53.7% -60.2% -70.4%
[kg] Toelink 1.6% -14.2% -26.0% -39.3% -45.3%

sum -10.7% -20.8% -30.6% -40.6% -50.6%

| uBIP
x,L | -0.0% -0.1% -0.0% -0.1% 5.4%

| uBIP
x,R | -2.6% -0.4% -0.2% -0.1% -3.3%

Displacements | uDOC
z,L | -0.7% -3.0% -4.4% -8.2% -31.6%

[mm] | uDOC
z,R | -0.3% -1.2% -1.3% -2.8% -17.4%

| uSAC
y,L | -0.1% -0.2% -0.0% -0.2% -10.5%

| uSAC
y,R | -0.1% -0.2% -0.6% -1.3% -21.4%

42



5
Discussion

This chapter presents a discussion of outcomes from the project and whether ap-
plied methods are suitable for the corresponding methodology. The project aims to
propose a methodology that can be used to combine TO with parametrization of
joint positions as well as providing a prestudy for future project within this area.
In particular the goal is to propose methods for simplified FE-modeling of automo-
tive wheel suspension systems together with concept load cases, as well as studying
different optimization strategies. It also aims to provide a pilot study of how the
material distribution between linkages can be optimized, with respect to stiffness
and weight. Hence, by adopting TO with a system level approach.

5.1 FE-model of the wheel suspension system

The FE-model involves rather many simplifications, e.g. linear geometry and linear
bushing characteristics, whereof the most important are discussed in this section.
It is important to involve load cases that capture key characteristics of the system,
where longitudinal forces are essentially covered by BIP, vertical forces by DOC and
lateral forces by SAC. These are expected to capture the most relevant loads at an
early stage of development. However, additional load cases such as rolling loads may
be introduced as the methods are refined.

5.1.1 Rigid bushing attachments

The proposed FE-model involves rigid behavior between the theoretical centers of
bushing bodies and the corresponding interfaces on the components. Hence the
model is somewhat over stiff. However, it seems like bushing deflections, i.e. de-
formations in connector elements, are dominating system level deflections and that
the proposed simplification for bushing sleeves seem to be suitable. At least within
stiffness analysis.

43



5. Discussion

5.1.2 Quasi-static analysis with linear assumptions

The bushings are modelled with connector elements together with a corresponding
connector behavior for linear elastic characteristics. This seems to work well with
respect to force signals. Although, it should be emphasized that the model is not
validated with respect to displacements and excluding non-linearities may contribute
to non-physical system level deflections, which in turn can affect the load distribution
within the system. Hence, this may well be a contributing factor to why some load
signals shows poor correlation (see Section 4.1.1). However, the deviations of lateral
forces in HP4 during BIP and DOC are likely affected by the damper loads (see
Figure 4.2). DOC is associated with large damper loads and for BIP event these
forces are larger for the left hand side than for the right. In the BIP event, F LH

y

corresponds to the wheel that goes into the pothole (see Figure 4.2) while FRH
y

corresponds to the wheel that stays on the road surface and therefore shows better
correlation with respect to the RLD (see Figure B.3). Also note that HP4 looks
better for the SAC event (see Figure B.23) where the damper forces are smaller.
Hence, modeling the dampers with external forces may contribute to poor correlation
for lateral forces in HP4. Nevertheless, it should be noted that the signals get even
worse if damper forces are excluded.

Forces that corresponds to a dynamic event are applied within a quasi-static analysis
and the peak damping forces that in a dynamic case act to decelerate unsprung mass,
instead act to deform the system statically (see Section 4.1.2). Hence, unreasonable
system level deflections may be the common reason behind the force signals with
high RMS. It is therefore likely that reasonable displacements will reduce these
deviations. Except for non-linearities, Equivalent Static Loads (ESL) [49], i.e. a
static load that compensates for mass inertia, may be introduced in order replicate
system level deflections during quasi-static analysis.

5.1.3 Boundary conditions and bushing stiffness

The proposed boundary conditions are based on an assumption that the car body
can be considered as rigid. However, relative movement between the subframe and
the car body depends on stiffness of car body bushings and flexibility in the car body
may be compensated by choosing a relatively low tangent stifness when linearizing
the corresponding bushing characteristics. The sensitivity analysis with respect to
bushing stiffnesses shows that force signals corresponding to hardpoints for linkages
are not sensitive with respect to car body bushing stiffnesses. Similarly, the force
signals in the car body bushings are not sensitive with respect to stiffnesses in
bushings for linkages. Hence, hardpoints for linkages and hardpoints for car body
bushings may be analyzed separately. At least as long as the stiffness magnitudes
are reasonable, i.e. somewhere between the lowest and highest tangent stiffness of
the corresponding non-linear bushing characteristics.

44



5. Discussion

The sensitivity analysis also shows that the bushing stiffness in HP31-32 can be
tuned further in order to reduce RMS of the deviation. This last fine tuning is
skipped in order to have a sensitivity analysis that corresponds to the presented FE-
model. Hence, only varying one variable at time and having everything else fixed
in accordance with the reference model. Updating the reference means that the
entire sensitivity analysis should be updated in order to be valid. Hence, requiring
a huge amount of computational capacity. Also note that an update would have
had negligible affect to the optimizations of linkages since loads in linkages are not
sensitive with respect to stiffness in car body bushings.

5.2 TO on system level

The proposed scheme (see Figure 1.1) introduces an outer parametric loop that works
around the TO. It may be relatively computationally demanding, but it proves to
work well for industrial level applications, i.e. on 3D-structures such as the two-
component model. With a well thought out DOE, it should be possible to study
trends from repositioning of hardpoints within chassis applications. As a starting
point, it seems reasonable to study repositionings of about ±20 mm in different
directions. This is with respect to packaging restrictions and other attributes. It may
be possible to gain a relatively large mass reduction per unit length of repositioning
within the wheel suspension system. However, the potential mass reduction depends
on how structurally efficient the current setup of hardpoint positions is and this needs
to be investigated further.

Figure 5.1: Straight
UCA

Another essential part is parametrization of the system
FE-model, where intersections between design volumes
must be avoided. A previous thesis work at Volvo Cars
covered a parametric approach for design volumes [14]
and it may be possible to adopt some of their methods.
They found out that the UCA can gain from having a
raised design volume and this should have been expected
since it is preferable with a straight UCA. This design
is not allowed within the current configuration since it
will collide with a body beam. Although, it would be
interesting to study if a better solutions for the UCA can
be achieved by parametrization of hardpoint positions
together with parametrization of design volumes.

Figure 5.2: Bound-
ary for hardpoint posi-
tion during parametric
optimization

Except for a parametric loop that follows a DOE, it is
also relevant to consider parametric optimization. It can
be implemented in a similar manner, with an outer para-
metric optimization loop instead of the predefined DOE.
In order to get reasonable results, additional constraints
should be added in terms of limiting boundaries for each
hardpoint position (see Figure 5.2).

45



5. Discussion

Such a scheme can be useful for fine-tuning of parameters. However, the wheel
suspension system is very complex and requirements from multiple disciplines have
to be considered. Simply optimizing parameters by only considering structural per-
formance may lead to waste of computational power. Hence, as a starting point it
seems more relevant with a well thought out DOE in order to study trends and to
provide knowledge about the system.

The future potential for TO on system level seems promising. Optimizations of the
two-component model show that a broader solution space through parametrization
can lead to decreased system weight. As can be seen for the linkages in Section 4.2.2
the relative mass distribution between different components are treated within one
single optimization. Hence, this may reduce iterations during the component phase.

5.2.1 Load distribution during optimization

The mass of the studied suspension system is small in comparison with the total mass
that is supported by the rear wheels (about 4-5 %) and component stiffness have been
shown to not significantly affect the load distribution. Hence, for given outer loads
it seems reasonable that the load distribution remain relatively unchanged during
TO. However, the load distribution as well as outer loads to the system may still
change with repositioning of hardpoints and this should be investigated further.

5.2.2 Optimization strategies

The over all goal with TO in context with the wheel suspension system is to reduce
mass while fulfilling system requirements that are generally formulated as stiffness
constraints. Hence, minimize mass subject to stiffness constraints is therefore prefer-
able for evaluation of different hardpoint positions. Minimum compliance is thus
not suitable for evaluation of different hardpoint positions when the performance
is to be evaluated in terms of optimized mass. However, it may be used for fixed
hardpoint positions and is then performed through iteration with lower and lower
mass constraint, as for the optimizations in case 2 (see Table 4.7).

Minimum compliance seems to be numerically stable. Numerical difficulties arises
when trying to use minimize mass for the wheel suspension model. However, stability
is reached when damper loads are removed. Hence, solving the issues with the
damper loads is essential for future implementation of the proposed scheme in Figure
1.1. Furthermore, bushing stiffnesses are dominating the strain energy. As a result
the derivative of the strain energy with respect to component mass seems to be small
in comparison with the total strain energy of the system. In turn, convergence of the
optimization problem with respect to strain energy tends to be reached slightly below
the given mass constraint. See for example the results in Tables 4.5- 4.6 where the
mass of the optimizations tends to be slightly lower than the mass constraint. The
difference can be considered as acceptable, but it is still relevant to try alternative
convergence conditions.

46



6
Conclusion

This chapter summarizes the main conclusions of the project and presents recom-
mendations for future follow-up projects. The proposed methodology enables TO to
be introduced earlier in the system phase (see proposed workflow for development
of wheel suspension systems in Appendix E).

This work resulted in the following main conclusions:

• The proposed modeling of bushing attachments seems to be a