Multidisciplinary Design Optimization of Automotive Aluminum Cross-Car Beam Assembly by Mohsen Rahmani A thesis submitted in conformity with the requirements for the degree of Masters of Applied Science Graduate Department of Mechanical and Industrial Engineering University of Toronto © Copyright by Mohsen Rahmani 2013
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Multidisciplinary Design Optimization of
Automotive Aluminum Cross-Car Beam Assembly
by
Mohsen Rahmani
A thesis submitted in conformity with the requirements
for the degree of Masters of Applied Science
Graduate Department of Mechanical and Industrial Engineering
University of Toronto
© Copyright by Mohsen Rahmani 2013
ii
Multidisciplinary Design Optimization of Automotive Aluminum
Cross-Car Beam Assembly
Mohsen Rahmani
Masters of Applied Science
Graduate Department of Mechanical and Industrial Engineering
University of Toronto, 2013
Abstract
Aluminum Cross-Car Beam is significantly lighter than the conventional steel counterpart and
presents superior energy absorption characteristics. The challenge is however, its considerably
higher cost, rendering it difficult for the aluminum one to compete in the automotive market.
In this work, using material distribution techniques and stochastic optimization, a
Multidisciplinary Design Optimization procedure is developed to optimize an existing Cross-
Car Beam model with respect to the cost. Topology, Topography, and gauge optimizations are
employed in the development of the optimization disciplines. Based on a qualitative cost
assessment, penalty functions are designed to penalize costly designs. Noise-Vibration-
Harshness (NVH) performance is the key constraint of the optimization. To fulfill this
requirement, natural frequencies are obtained using modal analysis. Undesirable designs with
respect to the NVH criteria are gradually eliminated from the optimization cycles. The new
design is verified by static loading scenario and evaluated in terms of the cost saving it offers.
iii
Acknowledgements
Completing this work would not have been possible without the help and support of several
wonderful people. First and foremost, I would like to present my deep gratitude to my
principal supervisor Dr. Kamran Behdinan for providing me with the opportunity to work on
this project and to learn from him. His guidance, patience, and encouragement through the
duration of my Master’s program are most appreciated. Apart from the academic side, I
always see him as an inspiring person in my personal life.
My sincere appreciation goes to my supportive co-supervisor Dr. Jean Zu. She was serving as
the chair of the department in the period of my program and I cannot forget the time and
energy she devoted to our work considering her busy schedule and limited time.
This project is sponsored by Van-Rob Inc. and I have learned a lot through this collaboration.
I would like to acknowledge the help from Van-Rob specialists. My special thanks are
expressed to Dr. Sacheen Bekah from Van-Rob for the useful discussions and for providing
me with creative suggestion and comments.
I should also express my appreciation for the members of the Advanced Research Laboratory
for Multifunctional Lightweight Structures at the Department of Mechanical and Industrial
Engineering. Being among these bright and warm people has made my experience more
enjoyable. I have had the pleasure of friendship of many beautiful people at the University of
Toronto. I wish to thank them for their kindliness and I consider our companionship an
inordinate gift I received during the time span of this program.
Last but definitely not least, I owe the completion of this work to my family. Even though we
have been miles away in the past two years, my father, my mother and my sisters have been
the greatest sources of motivation for me. Their deep love, understanding and continuous
support has certainly made me to be better in what I am doing. It is to them that this thesis is
dedicated.
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Contents
Abstract................................................................................................................................ ii
Acknowledgements ............................................................................................................. iii
List of Tables ..................................................................................................................... vii
List of Figures ................................................................................................................... viii
List of Appendices ................................................................................................................x
Nomenclature...................................................................................................................... xi
CHAPTER 1: INTRODUCTION ........................................................................................1
1.1 Aluminum in Automotive Industry ...............................................................................1
1.2 Advantages of Aluminum .............................................................................................3
1.3 Instrument Panel (IP) System and Cross-Car Beam (CCB) Assembly...........................6
1.4 Objectives .................................................................................................................. 10
1.5 Thesis Outline ............................................................................................................ 11
CHAPTER 2: THEORETICAL FOUNDATIONS ........................................................... 13
2.1 Noise-Vibration-Harshness (NVH) Analysis .............................................................. 13
2.1.1 Modal Analysis ................................................................................................. 13
2.1.2 Noise-Vibration-Harshness (NVH) Performance Evaluation and Testing .......... 15
2.2 Multidisciplinary Design Optimization (MDO) .......................................................... 16
2.2.1 Aeroelastic Optimization of a Wing .................................................................. 18
2.2.2 Multidisciplinary Design Optimization (MDO) Architectures ........................... 20
2.2.3 Multi-Disciplinary Feasible (MDF) Algorithm .................................................. 22
2.2.4 Individual Disciplinary Feasible (IDF) Algorithm ............................................. 23
2.2.5 All-at-Once (AAO) Algorithm .......................................................................... 25
2.3 Material Distribution Method for Design .................................................................... 27
2.3.1 Topology Optimization ..................................................................................... 28
2.3.2 Topography Optimization ................................................................................. 31
2.4 Cross-Car Beam (CCB) Assembly Optimization ........................................................ 33
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CHAPTER 3: MULTIDISCIPLINARY DESIGN OPTIMIZATION ............................. 37
3.1 Finite Element Model ................................................................................................. 37
3.2 Optimization Objective and Constraints ..................................................................... 41
3.3 Modal Analysis of the Cross-Car Beam Assembly ..................................................... 42
3.4 Sensitivity Analysis of the Parts ................................................................................. 43
3.5 Multidisciplinary Design Optimization Architecture................................................... 46
3.5.1 Gauge Discipline .............................................................................................. 47
3.5.2 Shape Discipline ............................................................................................... 47
3.5.3 Part Discipline .................................................................................................. 52
3.5.4 Integration of Disciplines .................................................................................. 57
3.6 Cost Estimation .......................................................................................................... 61
3.7 Particle Swarm Optimization (PSO) ........................................................................... 62
CHAPTER 4: IMPLEMENTATION OF MULTIDISCIPLINARY DESIGN
OPTIMIZATION ............................................................................................................... 66
4.1 Implementation Process ............................................................................................. 66
4.1.1 Preparing the Finite Element Model .................................................................. 66
4.1.2 Calling the RADIOSS Solver ............................................................................ 69
4.1.3 Collecting the Responses .................................................................................. 69
4.2 Fitness Function and Constraints Handling ................................................................. 69
4.3 Algorithm Stopping Criteria ....................................................................................... 72
CAHPTER 5: RESULTS AND DISCUSSION .................................................................. 75
5.1 Optimum Design ........................................................................................................ 75
5.1.1 Gauge Thicknesses ........................................................................................... 79
5.1.2 Shape Morphing Optimum Values .................................................................... 80
5.1.3 Part Selection .................................................................................................... 81
5.2 Statistics of the Results............................................................................................... 83
5.3 Mass Reduction and Cost Saving ............................................................................... 84
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5.3.1 Mass Reduction ................................................................................................ 84
5.3.1 Estimated Cost Reduction ................................................................................. 86
5.4 Static Analysis of the Cross-Car Beam (CCB) Assembly ........................................... 87
CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS ..................................... 90
6.1 Concluding Remarks .................................................................................................. 90
6.2 Future Directions........................................................................................................ 91
REFERENCES ................................................................................................................... 93
vii
List of Tables
1-1 Aluminum content of several aluminum intensive vehicles (2012 models) [4] 2
3-1 Mechanical and physical properties of the Aluminum alloys used in the CCB 37
3-2 FE details of the entire model including the CCB 40
3-3 Sensitivity data of the main parts with regard to changes in the thickness 45
4-1 Part configurations based on the corresponding design variable values 68
4-2 Different types of stopping criteria. Based on Ref. [57] 73
5-1 Summary of optimization results for 10 trials 76
5-2 Optimum gauge thickness for 10 trials (all in millimeters) 79
5-3 Optimum Shape Coefficients for 10 trials 81
5-4 Optimum part selections for 10 trials 82
5-5 Statistical analysis of the fitness values for different population sizes 83
5-6 Statistical analysis of the CCB mass values for different population sizes 84
5-7 Passenger cars sold in US and Canada. Obtained from Ref. [39] 87
5-8 Tube static load analysis results 89
viii
List of Figures
1-1 Aluminum content in pounds per light vehicles in North America, history and
forecast [5]
3
1-2 Passenger vehicles’ average fuel economy [7] 4
1-3 Passenger vehicles’ GHG emissions regulations [7] 5
1-4 Summary of aluminum-intensive automotive advantages. Inspired by Ref. [6] 6
1-5 Sample IP complex with the steering column mounted on it [8] 7
1-6 The CCB (shaded area) as the skeleton of the IP system [9] 8
2-1 Pendulum in free vibration 13
2-2 Multidisciplinary problem consisting of N subsystem. Adopted from Ref. [12] 18
2-3 Multidisciplinary Optimization of the Aeroelastic system 19
2-4 MDF architecture. Adopted from Ref. [12] 23
2-5 IDF architecture. Adopted from Ref. [12] 24
2-6 AAO architecture. Adopted from Ref. [12] 26
2-7 Material distribution method 27
2-8 Common numerical problems pertaining to the topology optimization 31
2-9 Bead patterns on a suspension support created by topography optimization.
Obtained from Ref. [30]
33
2-10 Design Sensitivity Analysis method for the CCB design. Adopted from Ref. [35] 35
ix
3-1 Finite Element model of the CCB assembly 38
3-2 CCB main parts and leading load path of the assembly 44
3-3 Bead characteristics 48
3-4 Topography optimization results for (a) DS VB and (b) PS VB 49
3-5 DS Tube (a) before shape morphing and (b) after shape morphing 50
3-6 PS Tube (a) before shape morphing and (b) after shape morphing 50
3-7 DS Vertical Brace (a) before shape morphing and (b) after shape morphing 51
3-8 PS Vertical Brace (a) before shape morphing and (b) after shape morphing 51
3-9 Topology optimization of Cowl Top part 53
3-10 Two distinct designs of the Cowl Top 54
3-11 Three distinct design of the DS End Bracket 55
3-12 Two distinct designs of the DS Tube 56
3-13 Two distinct designs of the DS Vertical Brace 57
3-14 The MDO procedure for CCB optimization 60
5-1 Convergence history for optimization trials using population size of 15 77
5-2 Convergence history for optimization trials using population size of 10 78
5-3 Mass reduction associated with various design trials 85
5-4 Static load analysis of the CCB tube 88
5-5 Stress distribution of the CCB under the static load 89
x
List of Appendices
Appendix A: Element Specifications 99
Appendix B: Mass Reduction History Diagrams 101
Appendix C: Matlab Codes 103
xi
Nomenclature
OEM Original Equipment Manufacturer
AIV Aluminum Intensive Vehicle
MPG Mile per Gallon
GHG Green House Gas
CAFE Corporate Average Fuel Economy
CCB Cross-Car beam
MIG Metal Inert Gas
NVH Noise, Vibration and Harshness
DOF Degree of Freedom
PDE Partial Differential Equation
FE Finite Element
MDO Multidisciplinary Design Optimization
MDF Multidisciplinary Feasible
IDF Individual Discipline Feasible
AAO All At Once
xii
FPI Fixed Point Iteration
SIMP Solid Isotropic Material with Penalization
Y Young Modulus
Poisson’s Ratio
CAD Computer Aided Design
AMLS Automatic Multi-level Sub-structuring
DS Driver Side
PS Passenger Side
C.V. Coefficient of Variation
1
CHAPTER 1: INTRODUCTION
1.1 Aluminum in Automotive Industry
The idea of using light weight structures in the automotive industry is nourished by the need
for enhancing the vehicle performances. Simply said, a lighter vehicle needs less power to
move and consumes less energy per unit distance traveled. Traditionally, steel and cast iron
were used to produce the majority of the vehicle components. However, today’s cars are
vastly reliant on non-ferrous materials as well, with the most common metal among them
being aluminum alloys.
Car manufacturers began using aluminum in the cars over a century ago when people in the
Berlin International Motor Show observed for the first time a sport car featuring a body made
of aluminum in 1899. At that time, the technology for extracting and forming aluminum was
still rudimentary. Therefore, it was not cost-efficient to fundamentally include aluminum in
the designs. However, the light weight and excellent corrosion resistance properties of the
metal helped it to make its way into the automotive industry [1].
Upon proclaiming an oil embargo in 1973 by members of OAPEC known as the Oil Crisis,
car manufacturers started to search for novel ways of fuel saving. It turned out that the most
effective action to lower the fuel consumption is to reduce the weight of the cars since
according to rough calculations, reducing the weight of a medium-sized vehicle by 100 Kg
causes a 700 liters reduction in fuel consumption during a typical vehicle’s lifespan.
Motivated by such facts, using lighter designs experienced a significant growth amongst the
manufacturers. Nowadays a typical car contains an average of 110-145 Kg of aluminum [1].
Original Equipment Manufacturers (OEM) now strive for higher fuel economy and lower
harmful emissions by investing more in the so called Aluminum Intensive Vehicle (AIV).
2
AIVs comprise of considerable amounts of aluminum sheets and extruded wrought alloys in
their structure [2]. Aluminum is used in a variety of components in a vehicle including the
wheels, radiators, condensers, and body. Audi played the role of pioneer for OEMs by
releasing its A8 model with complete aluminum body in 1994. The company developed Audi
Space Frame principle to support creating large aluminum sheet elements to become
integrated as load-bearing structures. The Audi A8 Space Frame weighs only 249 Kg which is
almost 200 kg less than a steel counterpart of the same class. Mass production of an all-
aluminum body vehicle was truly a milestone in the automotive industry. Later on Audi
launched Audi A2 as the next generation of all-aluminum cars with a body weighing 156 Kg.
That is to say, 43 percent lighter than conventional steel body. Audi A2 consumes only 3
liters of diesel oil per 100 Km. So far, more than 150,000 aluminum car have been produced
by Audi consisting of both A8 and A2 [3].
Table 1-1 presents some of the 2012 models with a minimum of 10% aluminum curb weight.
The total weight of a vehicle including all the necessary equipment and a full tank of oil and
excluding any load and passenger is referred to as the curb weight.
Table 1-1 Aluminum content of several aluminum intensive vehicles (2012 models) [4]
Vehicle Aluminum Percentage Aluminum Pounds
Mercedes-Benz ML-Class 11.60 581
Saab 9-4X 11.60 406
Chrysler/Fiat C-Sedan 11.30 351
Lincoln MKZ 11.30 397
Chevrolet Malibu 11.20 385
Nissan Altima 11.10 355
Ford Escape 10.90 355
Honda CR-V 10.90 290
Dodge Viper (ZD) 10.70 369
3
Ducker Worldwide has been collecting records on the growth of aluminum usage in light
vehicle applications on an annual basis since 1991. Figure 1-1 represents the aluminum
content in light vehicles in North America from 1975 to 2011 and the anticipated amount up
to 2025. The forecasted aluminum content for 2025 is approximately 16% of curb weight
which can be achieved by 3.5% aluminum growth rate in the period of 1990 to 2025.
However, this prediction is considered to be conservative [5].
Figure 1-1 Aluminum content in pounds per light vehicles in North America, history and forecast [5]
1.2 Advantages of Aluminum
Using aluminum components in cars and trucks is persistently growing. This is primarily due
to light weight, high strength, and environment friendly characteristics of aluminum alloys.
Employing the light metal in the vehicles allows for reducing the weight without downsizing
the vehicle, i.e. having the same car with a lower weight. This is specifically an important
merit and goal in electric cars. The most challenging concern in the performance of electric
cars is the large and heavy batteries necessary to empower the car to travel reasonably long
distances without demanding a battery recharge. It is estimated that an electric car can roughly
travel the same extra percentage as the percentage of the reduced weight, e.g. if the vehicle is
4
lightened by 20%, it is expected to travel an extra 20% distance without additional power
supply [6].
The weight reduction of vehicles can be recognized as being primary or secondary. The
primary weight reduction is the direct reduction of the body weight as a consequence of
utilizing a lighter alloy. A secondary weight reduction can then be attained in the engine,
drive train, and chassis since the lightened vehicle needs less power to propel.
A main driver for further usage of light materials in the cars is coming from the ever-
toughening regulations on Mile per Gallon (MPG) fuel economy and Green House Gas
(GHG) emission for vehicles. Every year the vehicles are expected to have higher MPG
values and lower GHG amounts. The following Corporate Average Fuel Economy (CAFE)
chart represents the fuel economy regulations in different regions, both history and the
forecasted extents. It is inferred from Figure 1-2 that European and Japanese cars are required
to have significantly higher MPG values compared to the rest of the regions. This is mainly
due to the smaller sizes of these cars. The Canadian fuel economy is more or less similar to
the US requirements as the North America market is fed mainly by the same car companies.
Figure 1-2 Passenger vehicles’ average fuel economy [7]
5
A lighter car needs less fuel and produces a smaller amount of CO2 gas, therefore preserving
the environment from being polluted by harmful emissions. Figure 1-3 represents the
requirement on the GHG emission for car manufacturers of different regions. Considering this
diagram, one can realize that the GHG emissions in all regions are expected to shrink
drastically from the current state to the 2030 predictions.
Figure 1-3 Passenger vehicles’ GHG emissions regulations [7]
Aluminum can be easily and efficiently recovered and recycled forever. From all aluminum
produced since 1888, almost 75% of it is still in use [6]. When the AIVs life is ended, they
will be dismantled and recycled in recycling infrastructures capable to recycle more than 75%
of the materials in the vehicle [2].
Another key feature of this light metal is its great performance in the crash. Intelligent designs
have been made possible using high-strength aluminum alloys that can absorb the destructive
crash force by permanent deformation into the expected patterns. Furthermore, a larger body
is generally safer in crash. Using aluminum, the vehicle weight can be reduced while
maintaining its size; hence it provides a safer crash experience [6].
6
There are more and more advantages which come with utilizing aluminum alloys. Figure 1-4
emphasizes some of the most significant advantages that follow aluminum-intensive
automotive designs.
Figure 1-4 Summary of aluminum-intensive automotive advantages. Inspired by Ref. [6]
1.3 Instrument Panel (IP) System and Cross-Car Beam (CCB) Assembly
The Instrument Panel (IP) is a collection of different modules each responsible for an aspect
of the performance, control, and safety of the vehicle. It is referred to the front part of the
vehicle interior where the steering column, airbags, glove-box, etc. are located. A typical IP of
a car is depicted in Figure 1-5. There are many essential sub-systems that are tightly packed in
the IP complex; hence the IP as a whole is a compound multi-functional system. Since it is
located immediately in front of the driver and the passenger in the front seat, there are certain
safety concerns that need to be accounted for in the design of the IP. Furthermore, each of the
• A 5-7% fuel saving can be realized for each 10% of weight reduction
• Using the same battery, the range of electric vehicles can be increased Efficiency
• More than 75% of the aluminum used in automotive is recovered
• Recycling aluminum requires only 5% of the primary production energy Sustainability
• High-strength aluminum shows superior impact energy absorbsion
• The structures can be designed to fold as predicted during the crash Safety
• Aluminum products are long-lasting and rust-resistant
• They are strong enough to be used even in rough military applications Durability
• Lighter vehicles have a faster acceleration and shorter break distances
• Its design flexibility enables the engineer to design for optimum shapes High-Performance
• It is impossible to achieve 50+ mpg target without reducing the weight
• Up to $3000 per unit can be saved in electric cars using fewer batteries Cost-Effective
7
main sub-systems requires its special design considerations and its local design constraints
need to be satisfied.
Figure 1-5 Sample IP complex with the steering column mounted on it [8]
The IP exterior is an enclosure typically made from light composites. Being light weight and
durable is a key factor in the design of the IP system. Furthermore, it is also critical to
consider the event of crash; the material used in the IP should cause the minimum harm
possible when crashed under the forces of a collision. Esthetic considerations affect the
material selection and design as well.
In the heart of the IP there is a metallic support structure to bear the loads and partly respond
to crash forces. This metallic frame is designated as the Cross-Car Beam (CCB) assembly.
Figure 1-6 depicts an IP system with the CCB assembly represented as a shaded structure
inside the IP. The CCB constitutes of many parts, each of them serving a special requisite in
the structure. Steering column support brackets, knee bolster supports, end brackets, airbag
supports, and cowl top are among the main CCB parts [9].
8
The main functions of the CCB can be described in a number of points:
The CCB is the load path that transfers the weight of the steering column to the body
through its junctions with the car body on sides, top and the bottom. The two ends of
the CCB meet the A-Pillars on the sides.
It acts as a skeleton for the IP complex and integrates various parts into the assembly.
There are several supplementary components attached to the CCB (not shown in
Figure 1-6) each serving as the support for a sub-system, e.g. the airbag.
On the occurrence of a crash it is the responsibility of the CCB and its energy
absorbers to deform under the crash force and prevent the occupants from
experiencing the severe impact force.
It is a main component of the car body frame contributing to its overall integrity and
stiffness and affecting the frequency response of the entire vehicle
Figure 1-6 The CCB (shaded area) as the skeleton of the IP system [9]
9
Traditionally, mild steel is the material used to build the parts of the CCB assembly. The
majority of the components are produced by stamping sheets; however, there are a few parts
that are manufactured by extrusion. The parts are eventually attached together by laser
welding and Metal Inert Gas (MIG) welding to form the CCB assembly. Efforts have been
made to fabricate CCB supports using injection molding and one-piece die casting
technologies. Although the cost of welding can be saved using these technologies, the
material and tooling cost of such designs may be prohibitive for their volume production [9].
Similar to other components of vehicles, the need for weight reduction which is supported by
the demand for higher fuel efficiency has made its impact on the design of CCB. Aluminum
CCBs tend to substitute the steel ones thank to the excellent energy absorbance and light
weight of aluminum alloys. Although aluminum CCB weighs roughly half of a steel one and
hence offers great mass reduction, fulfilling the performance requirements of the CCB can be
more challenging for the aluminum CCB. The main constraints in the design of CCB are
recognized as the Noise-Vibration-Harshness (NVH) and the crashworthiness performances.
It is essential to meet the specific criteria describing each of these constraints.
The NVH performance is concerned about the natural frequencies of the structure and keeping
them securely detached from the frequencies of prevailing excitement sources. The engine,
power transmission system, and the road surface are common resources for vibration. If the
natural frequencies of the CCB structure overlap with one of the excitement frequencies a
phenomenon known as resonance occurs. In resonance, the amplitude of the vibration can go
very large (theoretically it can go to infinity). Such high-amplitude vibrations can produce
significant annoying noise inside and outside of the car. It can also be transmitted to the
steering column and cause discomfort and distraction to the driver too [9]. To avoid this
destructive phenomenon the modal response of the CCB assembly should be studied and
modified as required.
The second constraint deals with the ability of the structure to absorb undesirable forces in the
event of a crash. The airbag absorbs a significant portion of the energy and plays a vital key in
protecting the driver. However, the energy absorbers are also critical to protect the driver’s
knees when the collision happens. If they don’t absorb adequate energy it becomes likely for
10
the knee bolster to move upward and come in contact with the steering column. This causes
the steering column to rotate and remove the airbag from its proper location and orientation,
therefore jeopardizing the safety of the occupants [9].
To guarantee the safety of the driver and passenger in the front seat, regulations have been
developed by OEMs to examine the performance of the vehicle in crash. There are different
crash tests such as frontal crash and side crash that need to be performed on every vehicle
before its mass production.
Apart from the tests that evaluates the crash performance of the car as a whole, the parts and
assemblies that have a prominent role in the safety of the occupants need to be tested and
evaluated individually in the design and development stages. The CCB employs the energy
absorbers and knee bolsters for controlling the impact energy and damping the force
transmitted to the occupants. In the crashworthiness evaluation the full CCB assembly is
subjected to various simulated impacts and the maximum deformation of the components is
assessed as a measure for crash performance.
1.4 Objectives
The aluminum CCB can overtake the steel counterparts considering the continuously
toughening fuel economy and emission regulations. That is, the essential direction to reach
superior fuel efficiency is weight reduction and it can be achieved by substituting the
conventional designs with modern ones using light weight alloys. However, employing the
aluminum CCB in the vehicles is still cost-intensive. It typically costs about twice the mild
steel CCB.
The common treatment in automotive engineering to reduce the cost of components is to
lighten them. It is well appreciated that the weight contributes greatly to the cost of the
structure. However, the cost is not merely a function of the weight. The manufacturing
complication of the parts and the attachment tools are other causative aspects to be
considered. Considering the complexity of the CCB assembly and various ways to affect its
cost, a robust algorithm is required to handle optimization of this structure. In this thesis,
11
weight reduction is considered as the main goal. Furthermore, effects of structural
modifications on the CCB weight is projected into the objective function to account for
manufacturing costs based on the available cost data.
The major objectives of this project are as follows:
Determines the best Multidisciplinary Design Optimization (MDO) architecture for
optimization of the Cross-Car Beam (CCB) assembly
Employs Material Distribution methods to develop optimization disciplines of the
MDO
Performs finite element based modal analysis on the CCB assembly to obtain the NVH
performance of the assembly in each design cycle
Integrates the optimization disciplines into the MDO architecture and executes the
procedure using an evolutionary approach
Validates the design and investigates the impact of optimization on the cost
1.5 Thesis Outline
The main body of the thesis consists of five chapters. These chapters present the thesis
material in an organized way as the following paragraphs briefly explain.
Chapter 2 presents the theoretical foundation required for understanding the methodology of
the optimization. Multidisciplinary Design Optimization (MDO) architecture and material
distribution method are reviewed in detail. Previous works on the optimization of CCB
assembly are gathered and reported as well.
Chapter 3 is the core chapter of the thesis focusing on the MDO architecture developed. It
goes through details of optimization disciplines as well as sensitivity analysis of the parts.
Optimization sub-problems related to each discipline are presented and solved. The
evolutionary optimization method and is introduced.
12
The implementation process is presented in Chapter 4. Fitness function and constraint
handling method are also discussed.
Chapter 5 presents the optimization results. The outcome of each discipline is presented and
discussed. Using statistics, the reliability of the procedure is validated. The impact of the
optimization on the CCB cost is investigated.
Chapter 6 summarizes the contribution of the thesis and proposes directions for further
research on the subject.
13
CHAPTER 2: THEORETICAL
FOUNDATIONS
2.1 Noise-Vibration-Harshness (NVH) Analysis
In this section, modal analysis is briefly introduced as the tool for NVH analysis. NVH
evaluation and testing methods are discussed as well.
2.1.1 Modal Analysis
A body vibrating in the absence of external force and due to an initial excitation is doing free
vibration. The frequencies under which the body in free vibration moves are designated as
natural frequencies. The number of natural frequencies an object possesses depends on the
number of Degrees of Freedom (DOFs) it have. Figure 2-1 illustrates a pendulum with mass
hanging from a string of length in free vibration condition.
Figure 2-1 Pendulum in free vibration
14
This pendulum has only one DOF and its natural frequency can be found using
√ ⁄ relationship in which is the gravitational acceleration. The natural frequency
of a simple pendulum is independent of its mass.
Most of the real world systems are continues ones rather than discrete assemblies of lumped
masses. However, it is possible to model them as Multi-DOF discrete systems which are
governed by ordinary differential equations. Continues systems are more challenging to
model without making simplifying assumptions. The resultant model will be governed by
partial differential equations (PDEs) [10]. The analytical solution of the PDEs if possible, is
not always straightforward; hence the PDEs are usually solved using numerical techniques,
e.g. Finite Element (FE) method.
The technique of finding the natural frequencies and mode shapes of a body is known as
Eigenvalue analysis or Modal Analysis. The governing equation of a vibrating multi-DOF
system neglecting all the damping effects reads [10]:
, - ( ) , - ( ) (2.1)
where is a constant and is a function of time. The vector is describing the mode shapes
of the system while function governs the behavior of the system in time. , - and , - are
system mass and stiffness matrices which depend on the inherent characteristics of the
system. Further manipulation of the above relationship leads to the characteristic equation for
finding the natural frequencies:
[, - , -] (2.2)
Different values of are distinct natural frequencies of the system. Each of the natural
frequencies (also known as Eigen Value) corresponds to a mode shape (also known as the
Eigen Vector) .Considering continues systems such as the CCB assembly, FE technique can
be utilized to discretize the system to small elements that can be treated as single or multi-
DOF objects. More details and description on the FE aspect of the problem will follow in the
upcoming chapters.
15
2.1.2 Noise-Vibration-Harshness (NVH) Performance Evaluation and Testing
NVH analysis is an inevitable practice in today’s automotive industry. Analysis of the noises
produced by the systems in a vehicle is performed for a number of purposes. The most
prominent cause is to reduce the undesirable sounds. However, not all the noises are
undesirable. While a luxury car is expected to have minimum level of the sounds, an
enthusiastic sport car owner may wish to have certain characteristic noise and vibrations [11].
The vibration characteristics of the various vehicle parts and systems should be examined in
order to prevent the incidence of resonance. Apart from making the occupants uncomfortable,
high-amplitude vibrations will diminish the lifespan of the components. Superposition of
various vibrations may lead to complex vibrations in the vehicle which is harder to control.
The Harshness evaluation is however more of a qualitative nature. There is no globally agreed
scale for measuring the harshness performance of a vibrating body. It depends on the
judgment of the costumers and manufacturers to rate the harshness of a prototype as satisfying
or in need of improvement.
In order to keep the noise and vibration of vehicles in a standard range, OEMs have specific
NVH regulations which should be met before a new product can make it to the market. The
NVH requirements are often expressed in terms of bounds for the natural frequency of the
parts and assemblies under consideration. In the case of CCB design, it is required to preserve
natural frequencies of the structure in an interval which is sufficiently away from those if the
vibrations resources. That is, to ensure that resonance is avoided.
The NVH behavior of a vehicle can be examined through standard tests designed for this
purpose. Most of the automotive companies are subjected to ISO9000 and QS9000
regulations. These standards oblige them to formally write the tests procedure and reports in
required order and details. This makes it easier to maintain and improve the test procedures.
Furthermore, the time and burden of teaching new staff to do the documented procedures are
significantly less and it can be done almost without special training [11].
To test the full vehicle NVH performance, a 4-Poster arrangement is commonly used
consisting of four low-noise hydraulic actuators. Each of the vehicle wheels is placed on one
16
of the actuators and the actuators are moved in a pre-defined manner to simulate the force and
vibrations of the road surface. Component-level tests are usually accomplished on a shake
table. Although it is not possible to completely remove the actual testing of the parts and
systems, the advanced computational capabilities available can minimize the number of tests
needed to develop a new product.
It is required for development of a new assembly that the NVH performance be carefully
examined using the computer simulation and analysis tools. The components are modeled in
the virtual environment and the modal analysis of the full assembly is performed. FE method
is typically utilized to solve the characteristic equations of modal analysis. Obtained mode
shapes are needed to be carefully examined to distinguish between the mode shapes that are
possible to happen (when the assembly is mounted on the main body, i.e. the real working
condition) and those that are unlikely to occur. If a mode shape is dominated by vibration of a
single part which is going to be securely constrained when mounted on the body, that mode
shape will not happen in the actual case and should be neglected as it is not present. Mode
shapes that are global to the assembly are the main mode shapes; hence they need to be
accounted for in the design. Optimizing the assembly can modify the mode shapes chiefly.
Therefore, the NVH requirements are one of the imperative optimization constraints.
2.2 Multidisciplinary Design Optimization (MDO)
Design and optimization of complex systems is a challenging task demanding specialized
solution strategies. A complex system is one that its detailed function including the
collaboration between various subsystems cannot be easily perceived. The complexity of a
system may arise from having many subsystems (hence many distinct system variables) or
from possessing many interactions between subsystems rendering the whole system difficult
to understand. An interaction between subsystems exist when some aspects of one subsystem
is affected by deviations in another subsystem. Having interaction between members
complicates the design and optimization process. At the same time, the synergy among the
system members can be fully exploited [12].
17
In the traditional design process also recognized as Block Coordinate Descent [13], the
subsystems are allowed to independently seek their optimal state while overlooking the
interactions between the subsystems. This process is done successively, i.e. when a subsystem
reaches its optimum, that aspect of the system design is frozen and the next subsystem
perform its search while some of the design variables of the system are already decided in the
previous subsystems and cannot be altered. This approach prevents exploring the full
capabilities of the design space.
Since the analysis of a complex system as a whole can be intractable or ineffective, such
systems are divided to subsystems to ease the analysis and optimization procedures.
According to Wagner [14], systems can be partitioned by aspect, object, sequentially, or
matrix. Dividing a system based on different aspects (also referred to as disciplines) of the
design results in multidisciplinary system architecture. Multidisciplinary Design Optimization
(MDO) is a methodology to design and optimize complex systems by splitting the system into
a number of subsystems while exploiting and managing the interactions between the
subsystems in a systematized fashion.
Figure 2-2 illustrates the general multidisciplinary problem setting [12]. For a system
consisting of subsystems, a total of ( ) interaction between the subsystems can be
realized. The optimization algorithm provides the system with design variable vector
* + consisting of the design variables common between all the subsystems ( )
and local design variables associated with each of the subsystems ( ). Upon
completion of the system analysis, the fitness function ( ) and constraint functions ( ) are
returned back to the optimizer.
A piece of information sent from subspace and received by subspace (if such interaction
exist) is represented by . Each of the subspaces may generate its own response set
( ) that can be utilized by other subspaces or by the optimizer in construction of the
system fitness or constraint functions. As the interconnections between the subsystems
increases and become more complicated, MDO technique tends to be more helpful in the
design process.
18
Figure 2-2 Multidisciplinary problem consisting of subsystem. Adopted from Ref. [12]
The field of MDO is concerned about efficient analysis and optimal design of a system that is
governed by several coupled disciplines [15]. It is an imperative part of current engineering
practice enabling it to develop highly sophisticated systems in shorter design cycles and
shorter design times. To provide some intuition into the MDO concepts, an Aeroelastic
Optimization example inspired by the example in Ref. [16] is presented here. This sample
case helps to better clarify the disciplines and inter-disciplinary interactions.
2.2.1 Aeroelastic Optimization of a Wing
In this example, a flexible wing in steady flight is considered. The air is rushing over the wing
causing a pressure pattern to surround the wing. The imposed pressure force applied to the
wing surface deforms the shape of the wing accordingly. Considering that the wing takes a
new shape, the distribution of the pressure around the wing changes and consequently, new
forces are applied to the wing structure. Altering the forces on the wing leads to form a new
wing shape. This procedure is supposed to reach equilibrium in some point.
19
A thematic illustration of the Aeroelastic optimization is given in Figure 2-3. The problem
involves two analysis disciplines, namely the Aerodynamic ( ) and the Structure ( ). The
computational problem associated with each of these problems is commonly solved using a
numerical code. For instance, a Finite Element code and a Computational Fluid Dynamics
code (CFD) may be utilized as the Aerodynamic ( ) and Structure ( ) solvers respectively.
The Structure code is given the input parameters describing the wing structure (e.g. material
properties) and the wing initial shape data is provided to both of the disciplines. The
Aerodynamic discipline receives the deflection of the wing as an additional input ( ) and
uses it to calculate the pressure data ( ). The Structure discipline takes the Aerodynamic
forces ( ) calculated based on the pressure data and using structural data inputs, calculates
the deflection of the wing ( ). Any mappings needed to convert data from one discipline to
the other is done in and modules. Given the input shape, Discipline Feasibility is
achieved for the Aerodynamic module when the CFD code has been executed successfully
and solution of the pressure data is produced. Similarly, the Structure discipline has reached
discipline feasibility when the FE code has solved the structural problem and the deflection of
the wing is found under the given force (pressure) data.
Figure 2-3 Multidisciplinary Optimization of the Aeroelastic system
20
When the following conditions are satisfied, a Multidisciplinary Analysis is accomplished:
Single discipline feasibility is achieved in both Structure and Aerodynamic disciplines
The output of each disciplines corresponds to the input to the other one
Under the multidisciplinary feasibility, the system is in equilibrium, i.e. executing all of the
models will not result in any change in the value of the variables. It is also possible to have
both disciplines reached individual discipline feasibility while the input of one discipline does
not match the output of the other one, hence no multidisciplinary feasibility. The optimization
problem is completed only when the multidisciplinary feasibility has been achieved.
2.2.2 Multidisciplinary Design Optimization (MDO) Architectures
Using MDO strategies, a multidisciplinary problem can be transformed into one or a number
of problems forming a specific system hierarchy. The emerged problems can then be solved
using specialized analytical or numerical techniques. There are quite a few MDO architectures
with some of them being more popular among practitioners. Every MDO architecture has its
own inherent advantages and disadvantages; however, the performance of an MDO
architecture is problem-dependent as well [17].
The MDO architectures can be categorized as single-level and multi-level algorithms. In a
single level algorithm, the design decisions are made solely by a single central optimizer.
They use a nonhierarchical structure to manage the interactions between disciplines [19].
Multiple Discipline Feasible (MDF), Individual Discipline Feasible (IDF), All-At-Once
(AAO), and Multidisciplinary optimization based on independent subspaces (MDOIS)
[12,18,20] are among single level algorithms.
Multilevel methods transfer the nonhierarchical relationship between disciplines into a
hierarchical structure. Each level of the optimization will have its own optimizer, with usually
having one system level optimizer to manage the responses of the disciplines. Concurrent
Subspace Optimization (CSSO) [18], Bi-level Integrated System Synthesis (BLISS) [18] and
Collaborative Optimization (CO) [18] are among well-known multilevel MDO methods.
21
An important question present in every MDO practice is how to choose the best MDO
algorithm for a given problem. There are a number of metrics used by researchers to compare
different architectures; however, the selection of the MDO method is commonly done in an ad
hoc manner. There are several comparison studies available in the literature. A portion of
these studies consider the efficiency of the optimization algorithm using metrics such as
number of iterations, design variables, coupling variables, and accuracy [17,18]. To name a
few, the study by Hulme and Bloebaum [21] presents comparisons between MDF, IDF, and
AAO algorithms by applying them to simulated multidisciplinary test systems created by
CASCADE, a computer tool which generates user-specialized coupled systems. Yi et al. [19]
studied the performances of seven MDO algorithms, including MDF, IDF, AAO, CO, CSSO,
BLISS, and MDOIS. They examined the number of the function calls that each algorithm
requires when employed to solve several mathematical examples.
An alternative view in comparing MDO methods is given by Kodiyalam and Yuan [22,23].
They reminded the importance of the formulation evaluation over just conventional metrics
used to evaluate optimization algorithms. They examined MDF, IDF, and CO methods based
on formulation-oriented metrics such as generality, robustness, and performance. Perez et al.
[18] presented and extended comparison of five MDO algorithms while focusing on both
optimization-oriented and formulation-oriented metrics. They solved an analytical example
and an aircraft conceptual design problem to evaluate the algorithms. In another study, Brown
[24] compares BLISS, CO, and modified collaborative optimization (MCO) through applying
them to the optimization of next generation Reusable Launch Vehicle (RLV). The algorithms
were rated qualitatively based on formulation and implementation difficulty, optimization
adeptness, and convergence errors.
In the following sections, three single-level MDO methods namely MDF, IDF, and AAO are
introduced. Due to the nature of the optimization problem of this project, only single-level
algorithms are considered and examined. Based on the disciplines used in the problem and
based on reports from the literature describing application of different MDO methods, it is
inferred that multi-level algorithms are not the best candidate for this problem and most
probably will lead to redundancy if employed. Multi-level algorithms are more suitable for
more sophisticated design problems such as the conceptual design of aircraft systems.
22
2.2.3 Multi-Disciplinary Feasible (MDF) Algorithm
MDF is one of the firstborn and most basic MDO algorithms. It has several alternative names
including Nested Analysis and Design (NAND), All-in-One (AIO), and One-at-a-Time. In
this method, all of the system analyses are managed by a system analyzer. A complete system
analysis is performed in each iteration and communication with the optimizer is done after the
system responses are obtained. The entire problem is seen by the optimizer as a normal
optimization task while inside the system analyzer the subsystems are required to reach
consensus. The MDF problem statement can be described as [12]:
( ) , - (2.3a)
subjected to
( ) , - (2.3b)
( ) , - (2.3c)
The Fixed Point Iteration (PFI) method is commonly used to solve the MDF system. In this
method initial guesses are used to obtain unknown values. Based on the assumed values,
dependent values are calculated and using the relationship between the subsystems the next
value (a better estimate) for the unknowns are obtained. This iterative process is continued
until the unknowns become stationary within the desired tolerance interval, i.e. further
execution of the FPI will not improve the solution.
MDF is nonhierarchical in nature and best suits the problems where there are little coupling
between the subsystems. Since in each iteration the whole system should be analyzed, the
algorithm is not efficient when the entire system evaluation is computationally expensive. The
prominent reason for using MDF is that it completely supports using legacy analysis tools, i.e.
the same solvers and analyzers used to solve each subsystem in the conventional analysis and
design can be used without modification.
23
Figure 2-4 MDF architecture. Adopted from Ref. [12]
There are a number of shortcomings associated with using MDF method. Since the system
analyzer communicates with the optimizer only when a complete system analysis is
accomplished, the algorithm cannot be used in parallel manner. That is, it is possible to have a
subsystem spending a significant amount of time idle waiting for other subdomains to finish
their analysis and proceed to the next iteration. The efficiency of the MDF depends on that of
the optimizer greatly. If a gradient-based optimizer is used, it may be necessary to perform
several more complete system analyses to provide the sensitivity information [12].
2.2.4 Individual Disciplinary Feasible (IDF) Algorithm
Similar to MDF, in IDF method each discipline is analyzed by its own analyzer and one
central optimizer governs the entire design process. In this architecture, disciplines are linked
to the optimizer independently, i.e. they send the disciplinary responses directly to the
optimizer regardless of what other disciplines have accomplished. It provides the grounds to
parallelize the design process and it offers more robustness compared to MDF architecture.
24
Figure 2-5 illustrates a sample IDF structure consisting of two disciplines. The optimizer is
responsible to send the subsystems all the required inputs, including design variables ( )
and coupling variables ( ). The subsystems do not interconnect directly and all the
interactions are made internally within the optimizer. Due to this fact, IDF is known as
Simultaneous Analysis and Design (SAND) as well, implying that there is no distinct borders
between the analysis and design (aka optimization) phases.
Figure 2-5 IDF architecture. Adopted from Ref. [12]
A general IDF problem can be formulated as follows:
( ) , - (2.4a)
subjected to
( ) , - (2.4b)
( ) , - (2.4c)
( ) ( ) (2.4d)
25
The auxiliary constraints ( ) are introduced to ensure the consistency of the solution when
the convergence is achieved. The values of the coupling variables set by the optimizer ( )
should match those calculated by the disciplines ( ( )) in order to acquire consistency. In
each iteration of the optimization process, feasibility in the disciplines is guaranteed (hence
Individual Feasible, IDF) but the system feasibility (Multidisciplinary Feasible) follows only
when the optimization converges. Therefore, if the IDF process is interrupted in some point
prior to convergence, the design may be inconsistent. Contrary to this, when the MDF is
stopped prematurely the system consistency is ensured while it may not be feasible [12].
Due to viewing the coupling variables independent variables similar to the design variables,
the dimension of the problem can significantly increase in IDF method. When the possibility
of parallelization exists, the IDF can lead to substantial time saving in the design procedure.
2.2.5 All-at-Once (AAO) Algorithm
The All-at-Once optimization algorithm is more centralized compared to the previously
discussed MDF and IDF algorithms. No analysis task is done within the disciplines.
Evaluators are being implemented in place of analyzers in the disciplines, i.e. the equations in
each discipline are only evaluated and the residuals ( ) are reported to the optimizer to deal
with. As observed in Figure 2-6, the optimizer provides the subsystem evaluators with three
types of variables: design variables ( ), coupling variables ( ), and state variables ( )
[12]. State variables are required in order to evaluate the discipline equations without solving
them. For instance, a stress field can be the state variable in a structural discipline evaluation.
In AAO algorithm no type of feasibility (disciplinary, system, or even for the equations in
each discipline) is sought before convergence is achieved. That is, the time is not spent in
disciplines to solve the equations and obtain feasibility [16]. Instead, a set of new auxiliary
constraints are added to ensure that the residuals vanish by the end of the process, i.e. at the
moment of convergence. Having the residuals equal to zero indicates that the disciplines have
reached feasibility.
The AAO formulation reads:
26
( ) , - (2.5a)
subjected to
( ) , - (2.5b)
( ) , - (2.5c)
( ) { ( )
( ) (2.5d)
The major drawback of the AAO formulation is that it is often impossible to map this
structure to the legacy analysis tools already available for various analyses. Therefore, one
may need to develop new analysis infrastructures to handle the problem in the new way it has
been formulated. This is the consequence of the centralization and the special structure of the
AAO architecture.
Figure 2-6 AAO architecture. Adopted from Ref. [12]
27
2.3 Material Distribution Method for Design
Material distribution method is the scheme for finding the optimal layout of the structure. The
optimal layout consists of information of topology, shape, and sizing of the structure. Each of
these three problems addresses distinct design problems while all of them are sub-problems of
the material distribution problem [25].
In the sizing optimization, the optimal values of the thickness of plates, as well as cross-
sections of the members of trusses are determined. An optimal material distribution is one that
minimizes or maximizes a certain response of the given structure while maintaining the
essential requirements. Shape optimization deals with finding the optimal shape of the
structure (e.g. bead patterns, member angles) on the design domain with the structure’s
topology being preserved. Determining the optimal layout of the structure (e.g. connectivity,
holes) is the purpose of topology optimization. For a topology optimization problem, the only
known conditions are typically some design restriction and the volume of the structure. Figure
2-7 illustrates a simple example in which all of these methods are applied separately to a
design problem. The initial structures (i.e. the baseline configuration) are shown in the left
and the designed ones are depicted in the right hand side.
Figure 2-7 Material distribution method; (a) size optimization, (b) shape optimization, (c) topology
optimization. Obtained from Ref. [25]
28
2.3.1 Topology Optimization
Topology optimization can be perceived as an optimization problem in which the topology of
the object varies until the desired performance is obtained. The volume of the object can
either play the role of objective function or be considered as a constraint. The topology of the
object (e.g. the connectivity pattern and number of holes) is the design variables while a
number of responses (e.g. compliance, frequencies) are sought to satisfy given conditions. In
the past two decades, a lot of research is devoted to the topology optimization problem. The
developed theories are fairly mature and topology optimization technology is currently being
actively used in various sectors of industry for the product development including automotive
and aerospace.
Given a design domain , the objective is to find a subdomain in the design domain with
limited volume that optimizes a given objective function , e.g. compliance of a structure.
This is to say, the aim is to find the connectedness, shape, and holes in such a way that
minimizes the objective function. The approach is to introduce the density function taking 1
on and zero elsewhere on . Therefore, the optimization problem can be formulated as
follows [26]:
( ) (2.6a)
subjected to
∫
(2.6b)
( ) (2.6c)
Numerical methods for topology optimization began to grow starting from the paper by
Bendsøe and Kikuchi [27] in 1988. Typical approaches to deal with topology optimization of
continuum structures are based on material distribution method. In the so called
homogenization approach, the material properties of design cells are obtained using
29
homogenization theory. Solution of the material distribution problem provides the optimal
topology of the structure. This approach implies the following formulation for the material
properties [25]:
{
(2.7a)
∫
(2.7b)
in which and are the original and modified stiffness tensor for the given isotropic
material respectively. The major drawback of this approach is the manufacturability of the
designs which can be impossible due to the infinitesimal pores in the material [28].
An alternative approach comes from the engineering prospective which is known as Variable
Density method as well as Solid Isotropic Material with Penalization (SIMP). In the SIMP
method the integer variable is replaced with a continuous variable and a penalty approach is
introduced to render the solution toward a discrete 0-1 pattern. In this model the stiffness
function depends on a continuous penalty function interpreted as the density ( ) in each
point. The density ( ) is then the design variable to play with until the optimized layout is
obtained. The formulation in this case reads:
( ) ( )
( ) (2.8a)
∫ ( )
( ) (2.8b)
The power is chosen to be greater than unity so that intermediate densities are
explicitly penalized and the possibility of having intermediate densities in the design is
diminished. It is shown that for problems with the constraint on the volume being active,
30
choosing a sufficiently big value of (ususally ) results in a fairly 0-1 distribution
pattern [25]. The power law formulation is applied to the stiffness matrix when the problem is
discretized and finite element formulation is employed. Representing the original stiffness
matrix by , The modified stiffness matrix is defined as ( ) .
Common numerical problems pertaining to the topology optimization can be divided to the
three following types [26]:
Checkerboard: refers to the regions of the design where solid and void elements are
formed in a checkerboard-like fashion. Contrary to the early hypothesis relating the
checkerboard patterns to some sort of microstructures, further studies confirmed that
checkerboards form due to errors in the finite element formulation and are numerical
noise [29].
Mesh Dependence: this problem happens when qualitatively different solutions are
obtained for different mesh size and discretization patterns.
Local minima: refers to the situation when different solutions are obtained using
different starting solutions (algorithmic parameters) while the same discretization
pattern is employed.
Figure 2-8 illustrates different numerical problems discussed above. Part (b) of the figure
represents the checkerboard pattern obtained from the topology optimization. Parts (c) and (d)
together describe the mesh dependency issue. Part (e) shows three distinct solutions for one
single problem obtained by topology optimization.
Much research has been done on finding ways to prevent these numerical flaws. Generally,
techniques dealing with mesh dependency overcome the checkerboard problem as well.
Common methods introduced for this purpose in the literature are Perimeter Control Method,
Mesh Independent Filtering, and Density Slope Control. One can refer to Ref. [29] for further
information and comparison of these methods.
31
Figure 2-8 Common numerical problems pertaining to the topology optimization; (a) Design space of
a simple beam, (b) checkerboard pattern, (c) solution using 600 elements, (d) solution using 5400
elements, (e) non-unique solution. Obtained from Ref. [25]
2.3.2 Topography Optimization
Topography optimization can be viewed as a type of shape optimization. In shape
optimization, similar to size (i.e. gauge, thickness) optimization, the topology of the structure
will not be manipulated. For instance, new holes will not be created. The focus is then on the
various changes that can be applied to the pre-defined topology, including changing the hole
shapes, introducing and modifying bead patterns, ribs, fillets, flanges, etc. Using FE
discretization, the design variable in a shape optimization usually affects more than one
element or node of the model, e.g. changing radii of fillets becomes possible by modifying
several nodes and elements.
32
Topography optimization deals in particular with the bead patterns. It creates protrusion or
corrugations in the direction perpendicular to the surfaces; hence it is shape optimization in
the third dimension [30]. These corrugations are known as bead patterns. The advantage of
introducing beads is that they increase the moment of inertial of the part. Therefore the
stiffness of the part is increased and the structure is reinforced. Considering the FE
discretization of the model, only the nodal positions of the shell elements are modified as
design variables. A common approach to define design variables in a shape (topography)
optimization is to use perturbation vectors as follows:
∑
(2.9)
The perturbed location vector is derived from the original one ( ) by adding a set of
distinct perturbation vectors . Each perturbation vector is accompanied by a shape
coefficient to determine the magnitude of the perturbation. The vectors are predefined
based on the possible and useful changes that the designer wishes to try on the structure. The
shape coefficients ( ) however are the design variables to be determined.
It has been showed in practice that topography optimization can effectively increase the
structural performance. Du et al. [31] report successful usage of topography optimization for
prediction and optimization of vibration response of engines. According to them, not only the
vibration response is reduced effectively, but also the repetition of design and optimization
cycles is decreased.
Kilian et al. [30] used topography optimization along with topology optimization to optimize
the suspensions in hard disk drives. They were able to increase the natural frequencies of the
hard disk suspension by more than 25% using the combined topography and topology
optimization. Figure 2-9 shows a sample of topography results they obtained.
Together with other material distribution methods, topography optimization can accelerate the
design procedure significantly. It helps the designer to figure out the optimal layout of
material distribution with fewer design iterations. Ref. [32] presents a good example of how
these tools can be creatively utilized in the concept development stage of the design.
33
Figure 2-9 Bead patterns on a suspension support created by topography optimization. Obtained from
Ref. [30]
2.4 Cross-Car Beam (CCB) Assembly Optimization
The IP system is an essential part of the vehicle, performing several demanding functions
such as safety of the occupants, increasing HVAC and air flow performance, and aesthetic
features of the vehicle design. The design of the IP system is regulated by some legal
requirements [33] concerning its core functions. The optimal design of the CCB assembly as a
key part of the IP is then a key assignment to be accomplished. To meet the NVH and
crashworthiness criteria and reduce the weight of the CCB as much as possible, various
design optimization approaches have been utilized. In the following paragraphs some of the
recent and important works on this subject are summarized.
Lam et al. [34] performed material and gauge thickness sensitivity analysis on a mild steel
CCB. Trying to maintain same structural performance, they pursued to check if materials
other than mild steel can be used for the CCB design. They used aluminum and magnesium
alloys as new materials to replace mild steel. Then, they pursued by changing the thickness of
the shell structure by 10% increments and up to 40% increment. NVH analysis is performed
using MSC/NASTRAN package to track the normal mode frequencies of various material and
thicknesses configurations. The crash performance of the CCB was also investigated through
34
side and knee impact analysis. The report confirms that if the thicknesses of the components
are enhanced by 40%, the aluminum CCB offers the same crashworthiness responses as the
baseline mild steel model. Increasing the thickness of the CCB parts also results in higher
frequencies of the first few natural modes. The 10% increment in the model thickness is a
simple and crude approach to find the optimum gauge values. The NVH and crash analyses
are performed independently and hence the full capacity of the design space is not exploited.
In other words, traditional sequential approach of designing each aspect of the problem
independent of the others prevents to create the most efficient design (i.e. the global optimum)
within the given design space. The natural frequencies are not used as a constraint and rather
they act as a check to verify the NVH performance.
Another sensitivity analyses and weight optimization of CCB is reported by Tawde et al. [35].
According to them, although the main problem in optimizing a structure like CCB is often the
cost of the product, what is optimized in practice is its weight. Estimating the cost of
production is a very rigorous and often impossible task since it involves many costs unknown
even to the manufacturer prior to the manufacturing. The cost of production can be affected
by the tooling cost, manufacturing methods, production run size, attachments and many other
contributing factors specific to a given production task.
They have described a useful methodology to help the engineer identify potential components
that can be improved upon by the optimization. Figure 2-10 summarizes their proposed
methodology. Elimination of insignificant parts from the analysis and reducing the number of
design variables can be achieved by Design Sensitivity Analysis (DSA). They have tried to
specify components that contribute to the crash performance and excluded them in the NVH
analysis in order to prevent weakening the baseline crash performance. The reason behind it is
that only NVH analysis is performed and it is simply assumed that the crash performance
remains unaffected if such NVH analysis is performed. This methodology significantly
reduces the design and development time.
35
Figure 2-10 Design Sensitivity Analysis method for the CCB design. Adopted from Ref. [35]
A genetic algorithm based multidisciplinary design optimization of the IP system is described
in the work of Ping and Guangqiang [36]. They have formulated a multi-objective
optimization problem and used Collaborative Optimization (CO) architecture to deal with the
IP design task. Knee bolster and head impact tests as well as NVH analysis form the
disciplines of the MDO algorithm. A response surface method approximation is used with the
MDO procedure which is based on the Kriging method.
36
Finally, an analysis and optimization of magnesium-based CCB assembly is carried out by Iei
and Zhi-yong [37] that considers only NVH criteria of the CCB with bounds on maximum
stress and displacement. They substituted the steel material in the baseline design with
magnesium and thickened the CCB parts to acquire the desired NVH and structural
performances. Optimization of the Mg-based model has led to a design that fulfills the NVH
criteria on the natural frequencies while the maximum displacement of the CCB in X, Y, and
Z direction are greater than the steel counterpart. However, the maximum stress of the
designed sample is smaller than the baseline model.
37
CHAPTER 3: MULTIDISCIPLINARY
DESIGN OPTIMIZATION
3.1 Finite Element Model
This CCB belongs to a midsize car and it is re-constructed by Van-Rob based on the original
CAD model provided by the client to them. Van-Rob Inc. is a Tier One supplier to the major
automobile manufacturers, specialized in the design and manufacturing of metal stampings,
modular welded assemblies, structural welded assemblies, mechanical assemblies and heat
shields for thermal applications [38]. The FE model of the CCB is developed in the
HyperMesh virtual environment as observed in Figure 3-1. The CCB itself consist of 40 parts
attached together by welding. The full model being used in the entire work is the CCB plus
the steering column assembly. However, the steering column is not illustrated in the figures
due to confidentiality matters. Having the steering column mounted on the CCB is necessary
to obtain realistic responses as required.
The CCB components are built from two different aluminum alloys for extruded and stamped
parts with the material properties mentioned in Table 3-1.
Table 3-1 Mechanical and physical properties of the Aluminum alloys used in the CCB
Purpose Aluminum
Alloy
Tensile Yield
Stress (MPa)
Tensile Ultimate
Stress (MPa)
Young Modulus
(GPa)
Density
(g/cm3)
Poisson’s
Ratio
Extrusion AA6082 T4 170 260 70 2.70 0.33
Stamping AA6061 T4 145 241 69 2.70 0.33
38
Figure 3-1 Finite Element model of the CCB assembly (a) attachment points, (b) FE discretization
No extra loads such as point or distributed loads are applied to the model. The behavior of the
structure is studied solely under the weight of the steering column and the parts which is not a
significant load and can be neglected without losing the generality of the analysis. Only when
the strength of the final model is evaluated, a linear static analysis is performed under
uniformly distributed load. The purpose of the linear static analysis is to ensure displacements
and stresses do not exceed the maximum allowable values supported by the material.
Therefore, the maximum stress and displacement are not a constraint for the optimization
problem and act only as checks to validate the design. Preliminary tests (not reported here)
39
confirm that maximum deflection and stress will be inactive constraints if included in the
optimization.
The locations of the attachment of the CCB to the body are designated in Figure 3-1 using
straight lines. Since the complete vehicle body model was not available, the true conditions at
external attachment points remain unknown. External attachment points here refer to the
points in which the model being examined (the CCB) shall be attached to a larger assembly,
in this case the vehicle body. The most realistic results can be achieved if the CCB is analyzed
while it is attached to the full body model. In the absence of such information there are two
approaches commonly used to run the analysis. In the first approach one replaces every
external attachment point with a rigid constraint. This is by far the simplest way to go and not
surprisingly the most inaccurate one. The reason is that the actual joints do not behave like
rigid points and rather resemble a spring and damper system whose characteristics depend on
the whole model structural features.
The second approach stands in between the real conditions and the rigid attachment modeling.
In this approach each attachment point is modeled as a spring with zero length and specific
spring constant. The spring constants can be determined by performing simulated tests on the
full vehicle model. Having access to the entire model, this is typically done by the client and
calculated equivalent spring constants are revealed to the designer as design parameters. The
equivalent spring constants are values that produce same condition as the actual attachment
points. In the majority of cases the client is not willing to reveal the entire model to the
designer due to confidentiality of the designs and only the estimated equivalent properties are
passed to the designer. In this work, spring constants as reported to Van-Rob are utilized.
Analysis of the CCB is carried out using FE method. The entire model is built in Altair
Hypermesh environment. The CCB assembly consists of several parts. For each single part a
single property card is defined which includes information on the material and section
specifications of the corresponding part. For solid parts, the property card specifies only the
material specification, i.e. Young Modulus (Y) and Poisson’s Ratio ( ). A thin-shell property
card includes material specification plus the thickness of the shell.
40
The mesh quality and size are chosen to be uniform throughout the model with Quadrilateral
Plate elements of average edge size equal to 5 mm being used for all the CCB components
except for the steering column bracket which is modeled using Six-sided Solid Elements.
More specifications on the element types can be found in Appendix A. The baseline model is
developed by Van-Rob based on the Computer Aided Design (CAD) model from the client.
Throughout the passage, whenever the term “baseline design” is stated it refers to the model
as received from Van-Rob.
Table 3-2 represents the detailed number of various FE entities used in the CCB model. It is a
fairly large FE model having 219349 nodes, 137465 elements, and 808974 DOFs which
necessitates using a powerful computing machine to obtain the desired solutions in a
reasonable amount of time. The Steering Column and Steering Shroud exist in the model but
they are not depicted throughout this report due to confidentiality considerations.
Table 3-2 FE details of the entire model including the CCB
Number of
Nodes
Number of
elements
Number of
parts
Number of
DOFs
CCB 44780 41416 40
Steering Column 169379 91115 21
Steering Shroud 5190 4934 17
Total 219349 137465 78 808974
Altair’s FE solver known as RADIOSS is utilized to solve the FE model. RADIOSS is a finite
element solver utilizing implicit and explicit integration schemes applicable to a wide variety
of engineering problems. Linear static, linear buckling, non-linear implicit quasi-static,
normal modes, frequency response, fatigue, etc. are among the analyses effectively handled
by this solver. The Altair Hypermesh serves as a pre-processing tool to define the problem
inputs and the required responses. The entire model is then passed to RADIOSS as a single
file and the output files are produces based on user’s requisites.
41
3.2 Optimization Objective and Constraints
The optimization procedure in this work tries to reduce the final production cost per CCB to
make it suitable for the competitive automotive market. It is chiefly believed in the
automotive industry that even striving for saving pennies is effective in making profit as the
production rate is quite high. For instance, more than 7,241,900 passenger cars have been
produced in US in 2012 [39]. The situation can be compared to the aerospace industry where
the production quantity is much lower (a few thousands per year) and only significant cost
savings can really boost the profit.
The main objective of this study is to decrease the cost of the provided CCB model as much
as possible. The method used to estimate the cost and make improvement upon it is discussed
in great details in the following chapters. Estimating the cost of various components of the
CCB is not straightforward. In order to build a cost model capable of predicting the cost of the
CCB or at least the cost variations, it is necessary to understand the consequences of various
structural modifications on the cost. Since the cost assessment of the components in Van-Rob
is done based on the personal judgment and experience of professionals, little documented
data on the variations of cost under the design variations is available. The main contributor to
the cost which is easily measurable is the mass of the structure; hence in this study it has been
used as the main indicator of cost reduction. Minimizing mass in order to achieve cost
reduction is a common trend in the structural design.
The constraints posed on the optimization problem come from the OEM regulations (which
are under the influence of federal standards) regarding the NVH performance of the vehicle.
The first two modes of the CCB structure are subject to lower bound constraints. It is required
that the first and second natural frequencies of the CCB should be always above 38 Hz and 40
Hz respectively. Being aware of the complete frequency response of the vehicle and the
vibration source characteristics, such bounds are determined by the client itself and dictated to
the designer to be met in the final product.
The stiffness and integrity of the structure is not used as a constraint for the optimization.
Instead, the static analysis of the CCB assembly is carried out for the baseline and final design
and the maximum deflection and stresses are assessed. Knowing that the baseline design is
42
safe with regard to the stiffness concerns (based on Van-Rob tests on the baseline model) one
can verify the final model through comparing the responses and by checking the maximum
deflection and stress with the allowed values. A distributed load equivalent to a total 450 N
(approximately 100 lb) force is applied to the model to check the maximum deflection and
stresses. The value of the force is a somehow rough estimation of the full IP system weight.
According to Van-Rob, the maximum allowable deflection is 5 mm and the maximum stress
can be verified versus the material properties.
The MDO technique should be employed to build an integrated optimization framework to
handle the task described above. One needs to choose a suitable MDO architecture and tailor
it to the problem if required. It is expected that an automatic procedure is developed
accounting for a number of design aspects. The computation time is an imperative factor
showing the effectiveness of the procedure. Therefore the procedure speed should be
considered when choosing the optimizing engine. Upon completion of the optimization
process, the outcomes need to be discussed and interpreted in terms of the mass and cost
saving achieved.
3.3 Modal Analysis of the Cross-Car Beam Assembly
Optimization of the CCB assembly includes NVH constraints. To prevent the CCB from
experiencing destructive resonance phenomenon, natural frequencies of the assembly should
be well parted from those of the excitation resources that may affect the vehicle, i.e. the
engine, power train and road surface. That is, the first two natural frequencies are required to
have a lower bound in order to meet the NVH criteria requested by the OEMs. Considering
this fact, Eigenvalue Analysis of the whole structure (including the steering column) is
performed to detect its first few modes. RADIOSS FE solver offers two algorithms to solve a
modal analysis problem, namely Lanczos algorithm and Automatic Multi-Level Sub-
structuring (AMLS) method. The Lanczos method is more suitable for small models and
especially when calculation of the exact mode shapes is sought. It has the drawback of long
computation times for models with large number of DOFs [40].
43
The AMLS method which is developed by Bennighof and co-workers in the past two decades
[41,42] is an algorithm for solving huge eigenvalue problems. It is basically a projection
method where a large problem is projected into a subspace spanned by a number of
eigenmodes of the structure [43]. Dissimilar to Lanczos method, AMLS algorithm is much
faster for large models with millions of DOFs and when hundreds of mode shapes are to be
calculated. In this method only calculation of a portion of the eigenvector is required; hence
the disk space and disk input/output are critically decreased [40]. As an example of
application of AMLS algorithm, one can refer to vibration and acoustic analysis of huge FE
models of car bodies which have been successfully handled using AMLS [44].
Due to the large number of DOFs in the CCB model and based on preliminary tests with both
algorithms offered by RADIOSS, the AMLS algorithm is selected to perform the modal
analysis in the given frequency range. It is noted that there is no necessity to obtain the
precise mode shapes of the CCB in each analysis during the optimization process. The mode
shapes are carefully investigated once before the analysis to make sure that only mode shapes
corresponding to the vibration of entire CCB assembly are picked, i.e. the actual mode shapes
of the structure. Therefore, AMLS method can be implemented without any concern about the
accuracy of mode shapes.
3.4 Sensitivity Analysis of the Parts
The CCB model consists of 40 parts. However, many of them are much smaller in size and
weight compared to the main tube (the largest part of the assembly) and do not contribute
significantly to the weight of the CCB. The majority of such parts play the role of connector
between major parts. There are a number of parts which form the main load path of the
structure as represented in Figure 3-2. These parts are greater in size and seem to contribute
more to both of weight and frequency response of the CCB. In order to realize the relative
contribution of the parts to the mass and frequency response of the structure, sensitivity of the
CCB natural frequencies and the CCB mass to modifications of these parts is assessed.
The main load on the CCB comes from the steering column and the IP complex weight which
is transferred mainly by the DS End bracket, DS Tube, Cowl Top, and DS Vertical Brace to
44
the vehicle body frame. The Driver Side (DS) components are generally experiencing greater
loads comparing to the Passenger Side (PS) components and therefore they are considered
primary components of the beam. It worth noting that due to this fact, the main tube is
typically designed as a beam with varying cross section. It can be simply shown by stress and
modal analysis that using the same cross section of the driver side in the passenger side will
be redundant.
Figure 3-2 CCB main parts and leading load path of the assembly (represented by straight lines)
The analysis here is not supposed to be comprehensive in all the senses. Instead, it only serves
to give a sensible idea of the relative importance of the major parts to assist in choosing parts
for modification. In order to obtain the sensitivities, the mass and first two natural frequencies
are measured for two different values of the part thicknesses, one of them being the baseline
configuration. To obtain the sensitivity values, the ratio of change in the quantity to its
original value (here the quantities are natural frequencies and mass) is divided by the same
ratio for the thickness, i.e. it is defined as:
45
⁄
⁄
(3.1)
Table 3-3 presents the sensitivity data of the main parts. According to this table, the DS Tube
contributes the most to all the responses hence it can be taken as a base and the rest of
sensitivities can be compared to it. Furthermore, it can be clearly observed that natural
frequencies are much more sensitive to the changes in driver side components rather than
those of passenger side.
Table 3-3 Sensitivity data of the main parts with regard to changes in the thickness
Part Part Mass Baseline
Thickness (mm)
F1 Sensitivity
( )
F2 Sensitivity
( )
Total Mass
Sensitivity ( )
DS tube 1.426 3.0 16.594 6.129 20.423
PS tube 0.564 2.5 0.512 2.884 8.080
DS VB 0.350 3.5 3.764 0.392 5.005
PS VB 0.158 2.0 0.717 0.384 2.288
DS EB 0.295 2.8 2.599 0.420 4.254
Cowl Top 0.427 3.0 2.688 1.153 6.092
Looking at the vertical braces as an example, one can readily realize that the first natural
frequency can be altered much more effectively by changing DS vertical brace (DS VB)
thickness while PS VB has a meaningfully lower effect on the same matter (almost 5 time less
than the DS VB). The sensitivity analysis clearly shows that the parts in Table 3-3 can be
viewed as potential candidates for the cost reduction purpose. Since the mass sensitivities of
all of them are more or less in the same order of magnitude, one can rightfully hope that by
modifying them some weight and subsequently cost reduction can be achieved.
46
To conclude this section, the following components (shown in Figure 3-2) are considered the
most important ones in terms of the system response. Therefore they are the parts which are
subject to adjustment in different disciplines. Detailed description of such adjustments is
presented in the following sections.
Driver Side tube (DS Tube)
Passenger Side tube (PS Tube)
Driver Side Vertical Brace (DS VB)
Passenger Side Vertical Brace (PS VB)
Driver Side End Bracket (DS EB)
Cowl Top
3.5 Multidisciplinary Design Optimization Architecture
As indicated in the background section, MDO is beneficial for designing highly complex
systems with lots of design variables and numerous interactions between different agents or
disciplines of the problem. Given the number of distinct analysis of the design problem and
the hierarchy of them, one or more MDO structures may fit to the problem. Before choosing
an MDO architecture for the problem, one needs to carefully consider all the options available
based on the disciplines and the coupling among them. In this section, the disciplinary design
variables are introduced then an MDO structure is chosen based on problem properties and
discipline interactions.
To begin with, it is reminded that the problem being considered is a cost optimization
problem. One can readily infer that cost should be somehow present in the fitness function
and the design variables should have an effect on the cost of the CCB. The CCB assembly
consists of many parts which are mainly manufactured by stamping. The only parts that are
produced differently are driver side and passenger side tubes which are extruded product.
The design variables are bundled in distinct disciplines. Three different disciplines are used to
modify different aspects of the design and consequently affect the cost of the CCB. Each of
47
these disciplines includes a family of design variables that can alter the geometry of the CCB
structure. Since all of the disciplines can manipulate a component simultaneously, the
response of the system will be a function of the simultaneous disciplinary tasks. The three
disciplines are introduced in the following sub-sections.
3.5.1 Gauge Discipline
The Gauge Discipline is developed to handle the thickness changes of the main parts. All of
the six parts designated in the Figure 3-2 are modeled as shell profiles with constant
thicknesses. The thickness of each of these parts is a design variable to be determined by the
optimization procedure. Due to the welding limitations, the thickness of aluminum
components should be between 2.00 and 5.00 millimeters. The Gauge routine is responsible to
change the thickness of these parts according to optimizer’s decision. Some of the main parts
experience other types of modifications as well (i.e. Shape and Part modifications). Other
modifications are presented in the following sections.
3.5.2 Shape Discipline
The Shape routine morphs the geometry of some of the parts. Two types of shape changes are
used in this discipline. The first one which is applied to the driver side and passenger side
tubes tries to shrink the cross section of the beams. Therefore, a lighter and cheaper tube can
be built. On the other hand, slimmer tubes make the CCB prone to have a lower NVH
performance. This clarifies the trade-off present in the design of the tubes.
Other shape modifications introduce bead patterns into the design of vertical braces. These
beads will not change the amount of material used while they can significantly change the
structural performance of the parts. They are capable of enhancing bending and torsional
rigidity. Within the Shape discipline, the vertical braces are granted the capacity of having
beads with pre-defined location. The height of the beads is the design variable regulated by
the optimization procedure. The location and shape of the beads are defined based on
topography optimization performed on the parts separately. The topography optimization
problem solved for the DS vertical brace can be stated as follows:
48
( ) * + (3.2a)
Subjected to:
(3.2b)
(3.2c)
(3.2d)
in which is the first natural frequency of the CCB structure and is the position vector of
the point on the DS VB. The bead characteristics are depicted in Figure 3-3 for better
understanding. Same problem has been solved for the PS VB component.
Figure 3-3 Bead characteristics
Figure 3-4 illustrates the topography optimization results for the vertical braces. The regions
with the highest shape changes offer potential places for placing the beads. In both of the
vertical braces, there are deep beads located along the braces pointed by lines. The areas in
the vicinity of the ending tale are neglected as the dimensions are smaller over there and there
are holes located in that area; therefore no beads can be introduced.
49
Figure 3-4 Topography optimization results for (a) DS VB and (b) PS VB
Based on the above discussion and results, the two types of shape changes are defined as a list
of node perturbation vectors. For each of the nodes, there is a specific vector describing the
direction of the movement of that node. The magnitude of the shape morphings (i.e. the
magnitude of the vectors) are chosen as design variables in this discipline. That is to say, the
optimizer chooses a real number in the [0, 1] interval. Each perturbation vector is multiplied
by this number known as the Shape Change Coefficient and the resultant vector is added to
the current coordinates of the nodes. Here are the four parts whose shapes are modified during
the optimization procedure:
Driver-side Tube: The DS Tube is shrunk only along its height as observed in Figure 3-5.
The right hand side image represents the tube when the maximum shape change is applied to
it i.e. when the Shape Change Coefficient is equal to 1.
50
Figure 3-5 DS Tube (a) before shape morphing and (b) after shape morphing
Passenger-side Tube: The PS tube is shrunk in both dimensions. Furthermore, the tube
corners are rounded.
Figure 3-6 PS Tube (a) before shape morphing and (b) after shape morphing
Driver Side Vertical Brace: The DS VB width and height dimensions are slightly decreased
(barely visible in the figure). Two longitudinal beads are introduced as well. The location of
the beads is chosen based on topography optimization results performed on this component.
The bead height is the design variable.
51
Figure 3-7 DS Vertical Brace (a) before shape morphing and (b) after shape morphing
Passenger-side Vertical Brace: Similar to DS VB, this part also receives some minor
shrinkage in width and height dimensions. The bead pattern applied to this part is shown in
Figure 3-8. Similarly, the design variable is the bead height.
Figure 3-8 PS Vertical Brace (a) before shape morphing and (b) after shape morphing
52
3.5.3 Part Discipline
Looking at the main parts of the CCB assembly, one can think about re-designing some of
these parts in order to enhance the performance and reduce the cost. Re-designing of the parts
can lead to a completely different design which still maintains the same constraint and
attachment requirements. When a component is going to be re-designed, emphasis should be
placed on the aspects of the design which contribute to the optimization cause. Therefore in
the CCB optimization problem transferring the design toward a lighter and cheaper one is the
objective to be sought.
Two different ways of part modification are introduced in this work. In the first one, new
designs for some components are provided which are going to take the place of the old ones.
The second type of part modification happens by adding or removing parts. For instance, one
can think about eliminating a part which is present in the assembly only to reinforce another
component or joint. If the same purpose can be fulfilled in some other way, that part can be
removed which contribute to both mass and cost reduction.
Obviously, weight reduction is a contributing factor that can drop the cost greatly. Other than
weight, one should attempt to create designs that need less stamping and/or machining
process. Based on discussion with Van-Rob specialists (hence employing developed
experience in the company) and browsing various possibilities with the aid of topology
optimization tool, new components which are lighter and easier to manufacture are proposed.
Cowl Top and the DS End Bracket parts are subjected to such re-designing.
Topology optimization is used for the conceptual design of the parts which are candidate for
re-designing. Figure 3-9 represents a sample topology optimization outcome for the Cowl Top
part. Given the desired frequency response and geometrical constraints of the assembly and
setting mass as the objective to be minimized, the topology optimization introduces the
element pattern which is required to satisfy the problem as observed in the Figure 3-9b.
Realization of the element pattern obtained from topology optimization leads to the
approximate shape of the part. Using this pattern as a rough idea, the rest of the design
process relies greatly on experience to finalize the part. The purpose of redesigning (cost
53
reduction in this case) and the manufacturing limitations are also considered for the design to
be feasible.
The topology optimization problem in this case tries to minimize the CCB mass and can be
formulated as:
( ) (3.3a)
subjected to
(3.3b)
(3.3c)
in which is the density of elements with and representing first and second natural
frequencies of the CCB.
Figure 3-9 Topology optimization of Cowl Top part. (a) Full design space, (b) elements with densities
greater than 0.25
54
Using the SIMP method, topology optimization assigns a density value in the [0,1] interval to
each element. The designer has the freedom to filter out elements with densities below a
critical value and use the rest of elements to form the component. Typically, elements with
densities less than 0.3 do not need to be included. In Figure 3-9 the 0.25 threshold is chosen
and the remaining elements (those who possess densities more than 0.25) are represented. The
next step is to create a CAD model based on the topology optimization outcome. The CAD
model is then implemented in the assembly and tested to verify its performance. Commonly a
number of iterations are required until the final design which meets the performance and
satisfies constraints is produced.
Figure 3-10 presents two designs for the Cowl Top component. Design I is the original layout.
Based on the topology results shown in Figure 3-9, Design II is proposed which has a smaller
surface area than Design I and is relatively easier to manufacture. Therefore, Design II
contributes to cost and mass reduction.
Figure 3-10 Two distinct designs of the Cowl Top
Using a similar process described, different designs for the DS End Bracket are created. In
Figure 3-11 three different designs of this component are presented. Two fixed constraint
points can be observed in all of the designs. These are among the points where the CCB is
55
attached to the vehicle body. A rectangular hole is common between all of them as well. This
hole is the place where the driver-side tube is attached to the End Bracket by means of
welding. These are the geometrical constraints that need to be met in every designs proposed
for this component. The new layouts for the DS End Bracket are more based on the existing
experience at Van-Rob as the topology optimization reveals the minimum material needed for
the part to be feasible. That is, layouts similar to Design II.
Design I (baseline design) and Design III have a flange on the border of the surface to
increases the stiffness of the part and make it harder to twist and deform. Having a smaller
surface area, Design II is superior to the others in terms of mass reduction and ease of
manufacturing (i.e. cost reduction). However, it may be not as rigid as them and it can be
more easily twisted, therefore it advocates tendency toward decreasing the natural frequencies
of the whole structure. That is to say, the trade-off in choosing the parts is about having a
stiffer component or reducing the cost.
Figure 3-11 Three distinct design of the DS End Bracket
As mentioned in the beginning of this section, apart from substituting a part with another
design for that purpose, it is possible to add or remove some parts to tailor the CCB
56
performance and cost. It is common to have some additional parts as supports for a main part
of the assembly, e.g. to increase the torsional rigidity of a tube. Such support parts may be
added to or eliminated from the baseline design if the performance of the assembly is
preserved. In the part discipline, two of such part modifications are present.
The first case is about an open-section part (designated as DS Tube Support hereafter) to
reinforce the DS Tube which is the main part of the CCB load path by making it a compound
tube. As it can be observed in the Figure 3-12, the Tube Support is there only to enhance the
torsional and bending rigidity of the DS tube. Since the CCB becomes lighter and the number
of parts decreases, removing the supporting part may lead to cost reduction. However, it is
highly probable that a thicker tube will be compulsory when there is no extra support for it.
Therefore Design I is more rigid while Design II is less expensive.
Figure 3-12 Two distinct designs of the DS Tube; Compound Tube including the Tube Support
(Design I) and Simple Tube without the Tube Support (Design II)
The second case is about an additional part (designated as DS VB Support hereafter) which
converts the open section of the DS Vertical Brace to a close one as shown in Figure 3-13.
57
Adding this supporting part increases the torsional and bending rigidity of the DS Vertical
Brace and distributes the load over a greater cross section area. It comes with the expense of
more material and excess manufacturing effort. The optimization procedure determines
whether or not combination of this component with other disciplinary modification can result
in the optimum design. Since the PS VB contributes significantly less than the DS VB to the
NVH performance, a similar scenario is perceived to be redundant for that.
Figure 3-13 Two distinct designs of the DS Vertical Brace; Open section DS Vertical Brace (Design I)
and Closed section DS Vertical Brace (Design II)
3.5.4 Integration of Disciplines
The complexity of a design problem is what directs practitioners toward using MDO
algorithms. The more complex the design case, the more sophisticated MDO approaches
58
might suit the problem. Due to the interactions between the disciplines of the CCB cost
optimization problem, it is inferred that it can be effectively handled by single-level MDO
algorithms. Recognizing the proper MDO architecture for a given design case depends on a
number of factors. The following discussion aims to assist in selecting the best architecture
for the given problem among the single level MDO algorithms.
The most important factor is the coupling between distinct disciplines of the problem. If the
disciplines are weekly coupled it will be relatively easier to find a feasible full-system
solution at a reasonable computation cost. Therefore MDF structure may be a good candidate.
Especially if the disciplines can be handled using similar tools and solvers, attempting to
solve them altogether (full-system solution) may be more efficient. When the disciplines are
highly coupled, it is commonly preferred to reach feasibility in each discipline and then seek
the entire system consistency, hence using IDF architecture. AAO is even more centralized
than IDF, leaving all the consistency (both system level and discipline level) to happen at the
moment of system convergence [12].
Another key factor for choosing the proper MDO structure is the legacy tools available. While
MDF and IDF structures allow using existing analysis tools, AAO demands re-arranging the
problem to a significant extent and will not allow to fully use existing tools. When there is an
effective analyzer to handle a specific discipline, it is often unreasonable to re-arrange the
problem in a way that needs a new and special analyzer to solve the same aspect of the
problem combined with other disciplines. This concept is also referred to as portability [18],
which deals with measuring the feasibility to integrate the method into an existing
organizational structure.
The simplicity and transparency of the algorithm being utilized is also important. The natural
choice is the method with more ease of implementation if the required performance can be
delivered by the algorithm. Transparency is about understanding and ability to extend the
mathematical model behind the method [18]. That is to say when several models can be
applied to the problem, one naturally goes for the simpler model which provides better
understanding of the problem.
59
The efficiency of different architectures varies as well. The number of model or sub-model
evaluations needed until the convergence is the key factor in the overall computation cost
associated with the algorithm. In the CCB optimization case, all of the analysis is done using
a single solver, i.e. RADIOSS. Therefore, it is more efficient to allow all the disciplinary
effects to take place and then solve it rather than applying them separately and invest multiple
solver runs to get a single design point.
Based on the above points, MDF structure is chosen as the architecture for CCB cost
optimization. The disciplines in this work are not highly coupled. Assuming a specific part
such as the DS Vertical Brace, the gauge modulus changes the thickness of the part while the
shape discipline moves some of the elements to shape the beads. These two procedures are
done in series and the result of both changes will be reflected when the full CCB model is
analyzed. Furthermore, Altair’s RADIOSS solver is a powerful analysis tool already available
and MDF architecture makes it possible to fully benefit from this tool. Each of the disciplines
contributes to the FE model based on the design variables chosen by the optimizer. The
RASIOSS solves the FE model then the required responses are gathered and sent back to the
optimizer.
Figure 3-14 specifies the complete optimization procedure followed to find the optimum CCB
design with NVH constraints. As observed from the diagram, the topology and topography
optimizations contribute to part and shape disciplines respectively. The design variables
chosen by the optimizer are divided to three groups each corresponding to a discipline. The
local design variables are thickness values, Shape Coefficients, and part numbers
respectively. Modified by various disciplines, the FE model is then solved by RADIOSS.
The mass and frequency responses are utilized to develop the penalties and calculating the
fitness function value. Detailed description of the penalty functions and fitness function will
be presented in following chapter. The optimizer checks for the convergence and stall criteria
and halts the procedure if required.
The proposed algorithm is in fact an integrated optimization scheme which employs material
distribution algorithms (topology and topography optimization) and a powerful non-gradient
optimization engine in the form of MDF method. The optimizer used is a population based
60
method which progressively transforms the solution toward the optimum value. Details of the
optimizer will be presented in the final subsection of this chapter.
Figure 3-14 The MDO procedure for CCB optimization
61
3.6 Cost Estimation
Information regarding the influence of possible changes in the parts and their contribution to
the cost of manufacturing has been gathered through discussions with Van-Rob specialists. In
the beginning, the intent was to collect quantitative data on how much each design feature
will increase or reduce the cost of the components. For instance, considering the parts
manufactured by stamping, it is common to use flanges, introduce holes and beads, corrugate
surfaces, taper edges, etc. Ideally, it would be great if a data sheet relating each of the above
features to an estimated cost effect existed. In that case, it would be doable – although
complicated – to build a mathematical model capable of estimating the cost for each part. The
cost model could be implemented as a distinct discipline of the MDO. Such information was
not available at Van-Rob. Instead, the specialists estimate the cost of different designs based
on experience and by comparing it to the similar designs whose cost is known to them. For
the extrusion parts, the cost remains unknown to Van-Rob until it is manufactured by the third
party manufacturer and the final cost is revealed to them.
These limitations in the cost data suggests more qualitative cost assessment as the solution.
Qualitative cost assessment will not estimate a number as the change in the cost due to adding
/eliminating a specific feature to/from the components. Alternatively, it will predict whether
or not case A is better than case B in terms of the cost. For example, one can imagine two
different designs for a part, one of them being lighter but more complicated in terms of
manufacturing. Obviously lighter designs consume less material and reduce the mass. In such
occasion the cost assessment decides which one results in a more cost-efficient CCB design
when seen together with the rest of components.
There are a number of cases that can be considered as a desirable change, meaning a change
that will cause the cost to decrease. Similarly, undesirable changes are those that add to the
total cost of the CCB. The following list includes scenarios that have been considered in the
CCB optimization problem:
As the mass decreases, so does the cost. A lighter design consumes less material and
possibly requires smaller shell thickness which is in turn a reduction in the cost.
62
An easier-to-manufacture part offers cost saving over a complex design which needs extra
manufacturing effort and time.
Adding a part to the CCB assembly will add cost. This is not only due to the excess
material used but also due to the extra manufacturing effort and the attachment costs
required to connect the part to the assembly.
Eliminating a part from the existing design means a reduction in cost. The reason can be
inferred from the previous point.
When one of these cases happens to the CCB model, its effect is projected in the fitness
function by introducing corresponding terms accompanied by their chosen multipliers.
Therefore, a general expression for the fitness function looks like
∑
(3.4)
in which ( ) is a function of the design variables ( ) and represents the effect of the i-
th contributing agent, e.g. the weight. The relative significance of different agents are decided
by which have been chosen by considering the merits of each of the contributing agents. A
detailed discussion on the value of these coefficients is presented in the implementation
chapter.
3.7 Particle Swarm Optimization (PSO)
Using the MDF structure, the optimizer sees the whole CCB design as a regular optimization
problem. Generally, one should choose between two broad classes of optimizers, namely
deterministic and stochastic optimization algorithms. The former are usually known as
gradient-based methods as well. They need the first and possibly higher derivatives of the
fitness and constraint functions known as the sensitivity data. Acquiring this information can
be very costly and time-consuming and even impossible sometimes. Furthermore, many of the
gradient-based methods need a starting point to initiate the search which should be selected
carefully since the quality of the solution depends on the starting point used. These algorithms
can usually find the local optimum in a small number of steps as long as the sensitivity data is
63
provided and the design spaces are continues. When the problem is a convex optimization
one, the local optimum is same as the global one; otherwise these algorithms are not suitable
for global optimization tasks. Dealing with large complex systems such as automotive and
aerospace structures, the common situation is to confront problems having discrete, multi-
modal, non-convex, and non-differentiable design spaces. Global stochastic methods are
usually the right choice in such situations.
Stochastic search algorithms are generally known as metaheuristics or heuristics. Although
there is a small difference between these two, some researchers use them interchangeably. In
fact a heuristic is trial and error method to find quality solutions to a tough problem in a
practical amount of time. However, metaheuristics generally perform better than heuristics.
They all use a certain trade-off between global and local search. The main components of
such search method are exploration and exploitation [45]. Exploration means to search
globally via randomization for optimal solutions while exploitation refers to exploiting the
currently best solution’s information which promotes local search. Balanced combination of
these components and the quality of global and local search techniques regulates the
performance of the metaheuristics as a tool [46].
Almost all metaheuristics are nature-inspired algorithms which mimic the behavior of a
natural phenomenon to redirect the current solution toward one with a lower cost while
sustaining constrains. Simulated Annealing (SA) [47], Genetic Algorithms (GAs) [48], and
Particle Swarm Optimization (PSO) [49] are among the most renowned ones. There are a
number of studies dedicated to comparing the performance of various metaheuristics (see Ref.
[50][51][52] as some recent examples). They usually tackle some benchmark problems using
these algorithms and report the key factors such as convergence speed and scattering of the
solutions. A study on the application of metaheuristic algorithms on structural optimization
problems is also carried out by the author which is not included in this report. The
summarized version of that study can be found in Ref. [53].
In this study, the well-known Particle Swarm Optimization (PSO) method is selected as the
optimizer for the MDO problem. It is the most popular algorithms from the swarm
intelligence family. Swarm intelligence is a broad area that offers a new paradigm for solving
64
problems by mimicking behavior of natural swarms such as ants, fishes, birds, flies, etc. Such
an algorithm is based on the premise of social sharing of information between the individuals
of a swarm leading to an evolutionary approach [54][55]. There are multiple agents in the
swarm that exchange information about the best found solution’s location and promising
directions for swarm to travel. In the same time, the randomly generated coefficients used in
computing new directions for species ensure that a global search is performed.
PSO was proposed by Kennedy and Eberhart [49] back in 1995. The approach utilizes
multiple agents called particles to move the whole swarm toward optimality. Each particle in
the swarm possesses a location and a velocity. The velocity is employed to update the
positions of the particles while the velocity vector itself is updated based on the knowledge
gained by the particle and also by the swarm as a whole. The adaptation of the swarm to the
environment guides the particle to return to the promising regions found and search for better
solutions over time. Mathematically, the position of the particle in iteration can be
expressed as [54]:
(3.5)
with being the updated velocity vector and being the time step. The time step is
usually taken as unity. The velocity vector is determined using the following relationship:
(
)
(
)
(3.6)
In the above relationship, and
are locations of best solutions found by the particle and
by the swarm so far. The inertia weight makes it possible to control the exploration
behavior of the algorithm. Large inertia weights produce large velocities leading to better
global search of the space while smaller values promote faster convergence. and are two
random numbers in , - interval that are responsible for the stochastic aspect of the search.
Finally, and are coefficients to be tuned based on the problem and they play a prominent
role in the convergence behavior. The former is known as Cognitive parameter determining
the amount of confidence in the best solution found by the particle. The later which is
65
designated as Social parameter regulates the confidence in the best solution found by the
swarm.
Unlike the majority of the metaheuristics that the algorithm convergence cannot be
guaranteed, there is a set of simple conditions that can ensure the convergence of PSO.
According to Perez and Behdinan [54], satisfying the following conditions will assure PSO
algorithm convergence:
(3.7)
( )
(3.8)
Perez and Behdinan [54] studied the dependency of the algorithm performance to the
cognitive and social parameter as well as the inertia weight. Based on the convergence rate
and the optimum values found, they concluded , , and results in the best
performance in their work. The selection of the algorithm parameters is problem-dependent.
However PSO shows a great deal of robustness with respect to the algorithm parameters, i.e.
even if the perfect algorithm parameters for a given problem are not used the algorithm still
performs fairly well. Usually it is beneficial to try a number of configurations of parameters
affecting the algorithm performance. This can help the designer to understand the trends of
the algorithm performance versus the changes in each of the parameters.
The cognitive and social parameters of the PSO used for all the trials are 0.9 and 1.1 and the
inertial weight ( ) is set to zero. These values satisfy Eq. 4.7 and 4.8, hence the convergence
of the algorithm is guaranteed. The values of the PSO parameters used in the current work are
selected mainly based on the parametric study of algorithms by the author [53] and based on
preliminary parameter testing on the CCB optimization procedure. The PSO code employed is
an open source toolbox developed for Matlab [56].
66
CHAPTER 4: IMPLEMENTATION OF
MULTIDISCIPLINARY DESIGN
OPTIMIZATION
4.1 Implementation Process
Various aspects of implementation of the MDO are covered in this section. One can follow
the details presented to realize how disciplines are implemented using RADIOSS and Matlab
codes and to reproduce similar setups.
4.1.1 Preparing the Finite Element Model
The computing machine used in this work is a Gateway computer equipped with an Intel®
Core™ i7-2600 3.4 GH processor and 8.00 GB of RAM on a 64 bit Windows 7 platform.
This machine is capable of running various Altair softwares used during different stages of
the work. The Matlab software is also executed on this computer.
The RADIOSS solver is in charge of solving the FE problem each time the optimizer calls it.
Before calling the RADIOSS solver, it is required to prepare the FE model according to
optimizer’s choice of design variables. Each of the disciplines is responsible to inject into the
model the piece of information they have analyzed and prepared based on the local design
variables passed to them.
The FE information is written in a “.fem” formatted file which is based on standard UTF-8
encoding. The data is written in the Nastran Bulk Data format, one of the formats that
RADIOSS recognizes and processes. Matlab is used to develop all the required codes
67
including the MDF procedure. Since it is required to modify portions of the baseline .fem file
according to optimizer’s design decisions, an efficient way of modifying such a large file
(approximately 438,000 lines of data) is desirable.
Using the flexibility of the RADIOSS’s script language known as Bulk Data Format re-
writing the full data into a new file can be prevented, hence acquiring a huge time saving.
There is a multi-file setup option available through use of “INCLUDE file_address”
command enabling the user to include information from different files without copying and
pasting it. It is possible to place the INCLUDE line anywhere in the main file as long as the
information that it contains is relevant to that section. This is the approach taken in this work;
whenever there is a need to modify a section, an external file is prepared and called by an
INCLUDE command already located at that position in the main file. Here is what happens in
each discipline in more details:
Gauge Discipline: The thicknesses of the shell parts are addressed in the Bulk Data Format
by PSHELL cards. The original PSHELL cards of the six parts that are subject to
modifications are stored in a text file. To introduce the new gauge thicknesses based on the
optimizer’s decision, this text file is modified based on the original data of PSHELL cards and
the design variable values. This file is included in the PSHELL DATA section of the baseline
“.fem” file using the INCLUDE command and the modified text file’s address. Therefore,
when the RADIOSS solver scans the “.fem” file for the part thicknesses, it will be referred to
the text file containing the information.
Shape Discipline: All the shape changes are made using the Hypermesh GUI and saved in
text files as a list of node perturbation vectors. That is, for each node that is going to be
moved there is a vector describing its movement. All the vectors of a specific shape change
are saved in a single file. Upon calling the Shape discipline, the module starts to calculate the
new coordinates of the nodes based on the following relationship:
(4.1)
where and represent the current and original position vector of the i-th node in
the global coordinate system respectively. is the design variable designated as Shape
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Coefficient which is independently chosen for each shape change by the optimizer. The is
the perturbation vector applied to i-th node. The new coordinate information is saved in
separate text files and is addressed by the INCLUDE command in the GRID section of main
Bulk Data file.
Part Discipline: The selection of the parts is different in nature from the previous modules.
Since changing a part or a number of parts will affect the attachment points and weld
elements, it is required to use pre-modified FE solver decks (i.e. .fem files) instead of
modifying some sections of one baseline FE model with respect to parts. For each
combination of part changes there is a pre-modified FE file containing those parts. The
following table represents the choice available for various parts with their corresponding
design variable values.
Table 4-1 Part configurations based on the corresponding design variable values
Design Variable Name Discrete Value Associated Part Change
Cowl_no
1 Design I of Cowl Top is used in the model
2 Design II of Cowl Top is used in the model
Endbkt_no
1 Design I of End Bracket is used in the model
2 Design II of End Bracket is used in the model
3 Design III of End Bracket is used in the model
Cover_no
1 Compound DS Tube (tube plus the DS Tube Support) is used
in the model
2 Simple DS Tube is used in the model
Sup_no
1 Open section DS Vertical Brace is used in the model
2 Closed section DS Vertical Brace (DS VB Support is added)
is used in the model
69
Based on Table 4-1, in total there are distinct possibilities for the part
combinations. Upon the optimizer’s call, the appropriate file is selected and it is copied to the
active run directory for execution. All of the “.fem” files contain the same INCLUDE
information and use the same modified text files when external information is needed.
4.1.2 Calling the RADIOSS Solver
The RADIOSS can be executed through Hypermesh GUI or by calling it in batch mode. Due
to the nature of the optimization procedure, the FE model should be solved many times and
the results should be collected after each execution. All of the pre-processing, processing and
post-processing of the FE model should be accomplished automatically. Since there will be no
manual manipulation of the FE model, the RADIOSS is called in batch mode through a
Matlab command. Each evaluation of the FE model takes approximately 3 minutes and the
results are exported into the .out files. At the end of each complete optimization process there
will be a number of “.out” files equal to the number of times the RADIOSS has been
executed.
4.1.3 Collecting the Responses
A detailed report of the iterative solution of the FE system is written in the “.out” file.
Collecting the necessary responses is facilitated by a Matlab routine. It will collect the first
two natural frequencies of the system, as well as the total mass and volume of the CCB
structure and pass them back to the fitness function and penalty function evaluation modules.
The mass is calculated based on the geometrical properties and material data already written
in the FE model. As mentioned earlier, the modal analysis using AMLS method is employed
to find the natural frequencies and their mode shapes. All of the “.out” files remain untouched
in the active execution folder for further reference when it is required.
4.2 Fitness Function and Constraints Handling
As mentioned earlier, the cost evaluation is achieved qualitatively and by comparing the
relative merit of different configurations of parts and shapes and thicknesses. The fitness
70
function then contains representative terms of two major agents contributing to the cost of
CCB; the mass of the CCB and the cost penalty.
The CCB mass is calculated by RADIOSS solver and collected after each model evaluation.
The original mass of the baseline CCB model is 6.992 kg. The cost penalty reflects the effect
of choosing different parts by the part discipline. Considering various Cowl Top and DS End
Bracket designs, the merit they can have over each other will be mainly in terms of mass
reduction which eventually leads to cost reduction. This is verified through consulting with
Van-Rob specialists. The situation is different for the remaining part variables. Either of the
DS Tube Support and the DS VB Support design variables can result is adding or eliminating
a part to/from the CCB assembly. Considering this fact, the following penalties are defined
for these design variables to reflect the cost effect of the decision made. For the DS Tube
Support the penalty function used is:
( ) (4.2)
in which is the area of the aluminum plate in square millimeters needed
to build the cover from. The “cover_no” is the discrete design variables that can take values of
1 or 2. If 1 is the value, it is interpreted as the DS Tube Support exists in the assembly; hence
the area is multiplies by 2. Otherwise, the value is 2 meaning there is no DS Tube Support and
the area is multiplied only by 1. Therefore the penalty is twice big if the costly Tube Support
is part of the model.
Using the same approach, a penalty is defined for the DS Vertical Brace Support component.
This additional part makes the vertical brace closed-section which is stiffer compared to the
open profile. The penalty function for this component reads:
(4.3)
If the “sup_no” takes the value of 1, it is interpreted as the DS Vertical Brace has an open
section while sup_no=2 means the section is closed, hence one more part is added to the CCB
assembly. Accordingly, the penalty is twice bigger when the section is closed. In this
expression the value of corresponding to the consumed plate area in square
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millimeters is used. The above two penalties are accumulated to form the Cost Penalty as
follows:
( ) (4.4)
Using the areas of the required plate for each of the potential parts is a rational coefficient to
rate the importance of them within the Cost Penalty, considering that manufacturing process
of both of them is fairly simple and is mostly folding plates to get the shape.
The constraint of the optimization problem is NVH performance of the CCB. It is required
that the first two natural frequencies of the CCB remain greater or equal to 38 Hz and 40 Hz
respectively. The baseline model has natural frequencies equal to 39.05 Hz and 41.60 Hz
respectively. The constraint handling method used is additive penalty functions. A penalty
function is defined as follows to bypass designs that fail to meet the NVH constraints:
, ( )- , ( )-
(4.5)
In this penalty function the deviation from the desired values of natural frequencies are
multiplied by coefficients of equal magnitude. More precisely, only the CCB designs with
natural frequencies smaller than the lower bounds are penalized. Technically, the designs with
natural frequencies greater than the lower bound values have no constraint issue. However,
using the optimization algorithm, such solutions are systematically transferred toward designs
that possess natural frequencies that are very close to constraint values since such designs are
lighter and have a smaller fitness value. That is, using the complete capacity of the
components to get the desired NVH response while keeping the mass as low as possible. It is
possible to choose different values for the two terms in the . However, in this case
equal values work well since achieving either of the constraint goals is equally worthy.
After introducing the penalty functions developed for the optimization procedure, it is time to
present the fitness function. The method of additive penalty functions is utilized to reflect the
effect of unfeasible and/or expensive designs into the fitness function. The following function
is passed to the optimizer as the fitness function to rate different CCB designs:
72
(4.6)
The coefficients in the fitness function are chosen in a way that unfeasible designs get filtered
gradually from the population of possible designs. This is in first place due to the large
coefficient of the compared to other terms present in the fitness equation. For
instance, if each of the natural frequencies is off by 2 Hz, a value of 800 is added to the fitness
function with is a large penalization considering the values of the other two terms. The mass
of the CCB is approximately (and less than) 7 Kg hence multiplying the mass by a factor of
20 results in a value of approximately 140. The least value of the is about 71,000 and
its largest value is twice this value. Therefore a desired design will possess a fitness function
of order of magnitude of 200. The selection of the coefficients has been done empirically and
by trying different values to understand the changes in behavior of the algorithm using
different penalty coefficients. The current configuration is able to successfully lead to designs
that meet the NVH criteria and are less expensive.
4.3 Algorithm Stopping Criteria
Stopping criteria need to be incorporated in any optimization scheme. Using maximum
number of evaluation (or generations) and/or maximum run time are not reasonable choices
since they are liable to result in pre-mature termination. There are different stopping criteria
used in the literature with various algorithms and for various problems. What is common
between the stopping criteria is the need to introduce a few parameters to recognize the event
of convergence. The parameter values depend on the given optimization problem [57].
Table 4-2 briefly describes various types of stopping criteria used by practitioners. One can
refer to the work by Zielinski and Rainer [57] to obtain more detailed information and
comparison of the criteria. In this study, two improvement-based criteria are used together to
determine the algorithm termination more effectively.
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Table 4-2 Different types of stopping criteria. Based on Ref. [57]
Improvement-based Criteria Movement-based Criteria Distribution-based Criteria
Detects the convergence based on
the fitness improvement. It is noted
that large improvements are likely
to happen in the beginning of the
optimization while at the end only
small improvements occur.
The movement of the individuals is
monitored. If their position is
changed less than a given threshold
in a specific number of generations,
the optimization procedure will be
terminated.
The diversity (e.g. standard
deviation) of the individuals is
assessed. It is assumed that the
convergence moment is reached if
the particles are closer than a given
parameter to each other.
Distinct criteria to detect the instances of convergence and stall of the optimization algorithms
are implemented in the PSO optimizer’s code. When one of these criteria is observed, the
algorithm stops and the proper messages are printed. There is neither a time limit for the
procedure nor a limit for the fitness function value and its number of evaluations. The
convergence happens as soon as the following inequality reads true:
( ) ( )
(4.7)
where ( ) represents the best fitness value found among all the particles since the
starting generation and up to -th generation. The is an integer parameter. This
parameter is used in both convergence and stall criteria and its definition is more perceivable
in the stall criterion context. According to the above convergence criterion, if compared to
generations prior to the current one, the value of the fitness function represents an
improvement less than the value, the convergence is achieved. The deviation between the
two best values is divided by the starting generation number to decrease the chance of
continuation of the procedure when the number of generations grows. The is a real
number which is commonly selected between and 1 depending on the application and
depending on the order of magnitude of the fitness value.
The parameter designates the largest number of generations that can happen without an
improvement. The algorithm monitors if there has been improvement in successive
74
generations. If no improvement is observed in the last generations the procedure will be
ended.
The algorithm checks for both convergence and stall criteria after evaluation of each
generation of particles and acts accordingly. In this work the values of and
are used. The selection of these values is based on previous experiences with the
PSO algorithm (one can refer to Ref. [53] for an excerpt of the study carried out on the
application of metaheuristics in structural optimization) and trying a number of values to
make sure that the criteria are not chosen too loose or tight. As mentioned before, selection of
the right parameters is chiefly problem-dependent. Therefore one need to try a few values and
realizes how the algorithm performance can be affected and set to its best by changing these
parameters.
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CAHPTER 5: RESULTS AND DISCUSSION
5.1 Optimum Design
The MDO procedure commences by the optimizer when the initial swarm of solutions is
generated and sent to the analyzing routine. Subsequently, future generations are created
based on previous ones and some randomness to explore the design space effectively.
Considering the fact that PSO is a stochastic algorithm, one can rightfully expect to obtain
different results out of multiple execution of the algorithm. That is, depending on how the
random parameters are selected by the algorithm in each step, various paths toward
convergence can be traveled by the swarm.
Table 5-1 presents the convergence and optimum results of the MDO procedure for 10 of the
best executions of the algorithm using the same parameters and settings. The value of the
NVH constraints and the time spent on each try is included as well. The best fitness value
obtained is 187.382 corresponding to Run 1 with the total CCB mass of 5.784 kg.
It is observed from Table 5-1 that distinct runs reached convergence by different numbers of
generations. The runtimes are a function of the number of generations. Two different
population sizes of 10 and 15 per generation are used and the algorithm successfully
converges in both cases. However, in general the runtimes are smaller for smaller number of
generations due to fewer execution of the analyzer. Table 5-1 is sorted based on the value of
the fitness function which necessarily does not need to be the same order of the CCB mass
values. The reason is that the mass is only one of the contributors to the overall fitness value.
It is possible to have a lighter design for which the NVH constraints are violated and thus its
overall fitness can be higher than of a heavier design.
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Table 5-1 Summary of optimization results for 10 trials
Generations Population Fitness Mass (Kg) F1 (Hz) F2 (Hz) Time (h)
Baseline - - - 6.992 39.05 41.60 -
Run 1 78 15 187.382 5.784 37.78 42.13 31.3
Run 2 63 10 188.882 5.859 37.71 42.19 18.3
Run 3 72 15 189.636 5.897 37.77 42.27 29.3
Run 4 80 10 190.178 5.924 37.63 42.16 22.4
Run 5 66 15 190.898 5.960 38.11 41.81 28.1
Run 6 64 15 191.360 5.983 38.25 42.19 25.8
Run 7 57 15 191.804 6.005 37.96 41.90 23.5
Run 8 59 10 193.202 6.075 38.15 41.68 11.4
Run 9 65 10 195.226 6.176 38.26 42.10 17.8
Run 10 72 10 196.070 6.217 38.18 41.97 19.5
The convergence history of various runs with the population size of 15 is displayed in Figure
5-1. Different convergence paths can be traced on the figure. In some of trials there are big
jumps (usually in the beginning of the procedure) in the best value of the fitness function
representing very successful movement of the swarm at that moment. The final values are
different which depends on the selection of the design variables by the procedure. However,
all of the designs have met the NVH criteria within an acceptable tolerance. The greatest
deviation from the NVH criteria corresponds to the first mode of Run 3 (37.77 Hz compared
to the target of 38 Hz) which is off by only 0.61 percent.
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Figure 5-1 Convergence history for optimization trials using population size of 15
Similar to Figure 5-1, the convergence history of the runs using the population size of 10 is
represented in Figure 5-2. The key difference between the two figures is the speed of reducing
the fitness function. When the population of 10 is used, it takes longer (more generations) for
the algorithm to significantly reduce the fitness value, i.e. the big jumps happen later. It can
be justified by noting that the more search agents (particles) present in the search, the chance
of finding better solution increases. The greatest violation of the NVH criteria among the runs
with population size of 10 happened in Run 4 which is off only by 0.97 percent.
78
Figure 5-2 Convergence history for optimization trials using population size of 10
Similar to the previous figures on the overall fitness value history, Appendix B includes
figures representing the history of mass reduction over the course of generations for trials
with 15 and 10 particles. As observed from the mass history trends, it is possible to have
increase in the CCB mass over the span of the optimization task. In other senses, the mass
reduction trends are very similar to the fitness reduction ones. While the fitness is
continuously decreasing, there are designs with heavier weight that possess smaller (hence
better) fitness values. That is, the mass in not necessarily decreasing during the entire
79
optimization procedure. This situation is due to having lower penalty values (less constraint
violation). The following sections include detailed information of design variables found by
the MDF optimization procedure.
5.1.1 Gauge Thicknesses
Table 5-2 presents the thickness of parts as suggested by the optimization procedure. The
mass of each CCB design is also included. For the best solution found, the minimum
thickness is 2 mm and the maximum one is 2.5 mm. The gauge thicknesses are allowed to
vary between 2 mm to 5 mm. Although the majority of the thickness values found are in the
[2,3] interval, there are design which make use of thicknesses as large as 4.9 mm.
Table 5-2 Optimum gauge thickness for 10 trials (all in millimeters)
Mass (kg) DS Tube PS Tube DS VB PS VB DS EB Cowl Top
Baseline 6.992 3.0 2.5 3.5 2.0 2.8 3.0
Run 1 5.784 2.5 2.0 2.0 2.0 2.1 2.5
Run 2 5.859 2.7 2.1 2.0 2.0 2.1 2.3
Run 3 5.897 2.3 2.0 2.1 3.7 2.1 3.1
Run 4 5.924 2.3 2.2 2.0 2.0 4.0 2.8
Run 5 5.960 2.2 2.0 3.6 2.9 2.5 2.8
Run 6 5.983 2.3 2.0 2.2 2.7 4.7 2.9
Run 7 6.005 2.6 2.1 2.4 3.1 3.5 2.5
Run 8 6.075 2.2 2.0 4.9 2.3 3.0 3.3
Run 9 6.176 2.4 2.0 4.7 4.9 2.2 2.1
Run 10 6.217 2.9 2.1 3.2 2.0 3.9 2.6
C.V. 0.0902 0.0327 0.3706 0.3266 0.3021 0.1282
80
Values of Coefficient of Variation (C.V.) for each of the design variables can be observed in
the last row of the table. This value for the tubes thicknesses is meaningfully smaller than the
rest of the parts. It can be interpreted that the tube thickness values are more strictly
constrained in a small interval to obtain the desired NVH behavior while the rest of parts have
taken very different combinations of thickness and still remained close to the optimum. This
is in accordance with the sensitivity analysis and confirms that the tubes play a more
prominent role in the natural frequency response of the structure while at the same time they
contribute greatly to the mass of the CCB.
5.1.2 Shape Morphing Optimum Values
The amount that the designs benefit from the shape morphings is different among distinct
trials. As observed from Table 5-3, the first two shape morphings which are bead patterns
applied to vertical braces have values of more than 40% in most of the cases (the shape
coefficients are in the [0,1] interval). These bead patterns are chosen based on topography
optimization on the corresponding parts and tend to increase the stiffness of the parts and
compensate for thinning the part. In the majority of the designs, the DS vertical tube possesses
a greater shape coefficient compared to the PS vertical brace. This is due to the fact that the
main load path of the CCB travels through merely the driver side parts, including the vertical
brace. The passenger side contributes significantly less to the load transmission to the A-
pillars and the body.
The other two shape coefficient variables correspond to DS Tube and PS Tube cross section
modifications. It can be observed in the methodology chapter that these two morphings serve
the material reduction by shrinking the cross section of the beams. Therefore, the optimizer
tries to shift the values of these variables up to save mass as much as possible. The greater
these two shape coefficients become, the smaller cross section will be used in the design,
hence saving more mass.
On the other hand, as the tubes become slimmer, the natural frequencies are witnessed to
happen on lower frequency values. For example, in the best design found, only 32% of the DS
Tube morphing capacity is employed. However, it can be observed that this design has the
81
lowest thickness of parts. Therefore, its mass is even more reduced compared to other trials
with larger DS Tube shape coefficients. Referring to the above discussion on the DS and PS
vertical brace shape changes, the PS Tube usually possesses larger shape change coefficient
than the DS Tube.
Table 5-3 Optimum Shape Coefficients for 10 trials
DS Vertical Brace
Shape Coefficient
PS Vertical Brace
Shape Coefficient
DS Tube Shape
Coefficient
PS Tube Shape
Coefficient
Run 1 0.68 0.06 0.32 0.65
Run 2 0.42 0.66 0.60 0.28
Run 3 0.78 0.48 0.37 0.62
Run 4 0.99 0.94 0.50 0.11
Run 5 0.11 0.47 0.39 0.29
Run 6 0.33 0.85 0.65 0.38
Run 7 0.53 0.46 0.85 0.67
Run 8 0.77 0.21 0.56 0.77
Run 9 0.46 0.38 0.31 0.94
Run 10 0.46 0.03 0.99 0.78
5.1.3 Part Selection
Considering the parts variables, one can observe form Table 5-4 that except for the case of
Run 10, the same selection of part variables is present in all of the runs. The Cowl Top and
DS End Bracket do not contribute to the cost penalty. However, since the CCB mass depends
on them, during the optimization process the designs with smaller mass tend to survive and
form the new populations. The Design II for both of the Cowl Top and DS End Bracket has a
smaller surface area hence it results in a smaller mass if the same thickness is maintained.
82
The DS Tube Support and DS VB Support parts are those present in the cost penalty while
they can affect the total CCB mass as well. Design II for DS Tube Support suggests that there
is no need to have the supporting part on top of the DS Tube; hence the simple DS tube is stiff
enough in the presence of the configuration nominated by the optimizer. The selection of
thicknesses, shape morphings, and other part variables can keep the design firm enough to
fulfill the NVH criteria.
Table 5-4 Optimum part selections for 10 trials
Cowl Top part
number
DS End Bracket
part number
DS Tube Support
part number
DS VB Support
part number
Baseline 1 1 1 1
Run 1 2 2 2 1
Run 2 2 2 2 1
Run 3 2 2 2 1
Run 4 2 2 2 1
Run 5 2 2 2 1
Run 6 2 2 2 1
Run 7 2 2 2 1
Run 8 2 2 2 1
Run 9 2 2 2 1
Run 10 1 2 2 1
The same point is true about the DS VB Support. The DS Vertical Brace performs well
enough without a support to make the section a closed one. This selection means less number
of parts and lower mass of the assembly at the same time. The MDF procedure explores all
the possibilities and suggests the most efficient one based on the fitness function introduced.
83
As observed in Table 5-4, the only design with a different part selection has the highest fitness
value. There are several other configurations of the parts which are not reported here. They all
have a greater fitness value (and often greater mass values) and hence they are not cost-
efficient designs.
5.2 Statistics of the Results
It is insightful to analyze the results of various runs from the statistical perspective to
understand how reliable the procedure is. Table 5-5 is dedicated to the mean value and
standard deviation of the fitness data for the runs with a population of 10 and 15. The
coefficient of variation (C.V.) which is the ratio of standard deviation over the mean value is a
measure to evaluate the amount of scatter in the obtained data.
It is observed that the set of optimization trials with the population size of 15 have a smaller
mean value. The standard deviation of these runs is also smaller compared to the runs using a
population size of 10, resulting in a coefficient of deviation of 0.84% versus 1.45%. Both of
the deviations are small enough to gain confidence in the procedure. Most probably, further
execution of the algorithm produces fitness values very similar to the obtained ones.
However, the results advocate that using more particles (more search agents) makes the
search more thorough. That is so say one can generally expect obtaining results of the same
quality in a shorter time or obtaining better results by using more populated swarms.
Table 5-5 Statistical analysis of the fitness values for different population sizes
Population size Mean Standard Deviation ( ) Coefficient of Variation (%)
15 190.216 1.591 0.84
10 192.711 2.790 1.45
84
A similar analysis is accomplished for the mass values obtained from various designs.
According to the Table 5-6, the same pattern exists among the two set of design, i.e. the first
set (population=15) offers less scattered designs. It is interesting that the values of coefficient
of variations of the mass data are larger than that of the fitness data. One can understand this
point by recalling that the optimization is performed with respect to the fitness value. The
mass is only one of the three main contributors to the overall fitness and an optimum design
obtained in this procedure will not necessarily have similar mass values. Different scattering
pattern may be obtained if the mass is considered as the fitness and the constraints were
handled separately by the algorithm.
Table 5-6 Statistical analysis of the CCB mass values for different population sizes
Population Size Mean Standard Deviation ( ) Coefficient of Variation (%)
15 5.926 0.080 1.34
10 6.050 0.139 2.30
5.3 Mass Reduction and Cost Saving
The main objective of the optimization of the CCB assembly is to modify the existing
aluminum design to reduce its cost as much as possible. The amount of mass reduction and
consequently the reduction of the cost are explored in this section.
5.3.1 Mass Reduction
The lowest mass obtained (i.e. the best design) corresponds to Run 1 which weighs 5.784 kg
in total. The original mass of the assembly was 6.992 kg. Therefore the mass reduction per
CCB is 1.208 kg. This is more than 17% of material saving per CCB:
85
(5.1)
This result offers significant reduction in the mass which is due to all the structural
modification suggested by the optimization procedure. Heavier parts are substituted with
lighter ones, and the thicknesses are decreased to an extent that the NVH and strength criteria
are still satisfied. The introduction of shape changes helps greatly to maintain the constraints
while lightening the assembly.
The design with the highest fitness value is Run 10 with mass of 6.217 kg. The reduction of
the mass for this design is assessed as follows:
(5.2)
The amount of mass reduction for the rest of the trials is presented in Figure 5-3. As observed
in Figure 5-3, all the sample designs obtained show a mass reduction percentage above 10%.
This mass reduction will eventually affects the total cost of the CCB.
Figure 5-3 Mass reduction associated with various design trials
0 5 10 15 20
1
2
3
4
5
6
7
8
9
10
Percentage of Mass Reduction
Ru
n N
um
be
r
86
Using lighter structures in the vehicles is a core sponsor for lowering the fuel consumption
and preventing the hazardous gases to enter the atmosphere. The currently used steel CCBs
that are used in the medium sized vehicles weigh approximately 10-12 kg. Switching to
aluminum designs is a big step toward lightening the cars and a legitimate illustration of
facilitating AIV technology flourishing. The cost challenge of the aluminum CCB is what
makes it hard to compete with the steel counterparts. The following section looks at the
impact of the optimization of the cost of the CCB assembly.
5.3.1 Estimated Cost Reduction
Cost estimation and assessment is not always straightforward. It is commonly completed with
some extent of estimations. Especially in the automotive industry, collaborating companies
need to exchange the cost of the services they offer to each other. Since each company has
some competitors in the market, they are not willing to release the exact pricing data of the
services and usually propose a final number as their bid for the service.
In the case of this project, as it is mentioned previously, the extrusion parts are manufactured
by a third party company which only provides the cost of the submitted model once they built
the component. The stamping parts of the CCB are produced at the home company (Van-Rob)
but still there is not documented data on how different designs can alter the total cost. They
assess the cost of the design based on personal experience and judgment. As it can be
expected, there are usually different estimated costs suggested by different people for one
specific part.
To overcome this challenge, the effect of mass reduction is interpreted as cost reduction using
an approximate rule commonly used within the company. As a rule of thumb, a $5 saving in
the cost per each saved pound of aluminum is expected. The best mass saving obtained here is
2.66 lb, therefore and approximate value of $13 saving per CCB is expected. Considering the
baseline model cost to be $90, this will result in almost 15% reduction in the cost.
87
In this cost estimation, the effect of using one less part is not taken into account, i.e. the DS
Tube Support existing in the baseline design is not present in the final design. This can also be
interpreted as a saving in the manufacturing and welding cost. That is, the total cost saved can
be more than the estimated one due to less number of parts and fewer welds. Although this
method of cost optimization is fairly rough, crude cost models seems to be inevitable for an
optimization task since the fitness value passed to the optimizer should be easy and fast to
calculate. The quality cost assessment accomplished by the company specialists offers more
insight to the optimization procedure; however it cannot be integrated into an automatic
procedure.
The importance of the cost reduction in automotive assemblies can be better understood by
reviewing the large number of vehicles produced. The following table presents some data on
the number of passenger cars sold in USA and Canada in the past years.
Table 5-7 Passenger cars sold in US and Canada. Obtained from Ref. [39]
2009 2010 2011 2012
Canada 729,023 694,349 681,956 748,530
USA 5,400,890 5,635,432 6,089,403 7,241,900
Saving a small portion of the production cost per vehicle can then be considered a big gain for
the OEMs. The economic influence of the new technologies constantly introduced to the
industry sectors cannot be neglected. The aluminum intensive vehicle design illustrates this
influence excellently as discussed in various parts of the current work.
5.4 Static Analysis of the Cross-Car Beam (CCB) Assembly
In order to check the stiffness of the optimized CCB assembly after the application of
modifications, a static loading scenario is used. In this analysis, the weight of the IP system
88
carried by the CCB is estimated at 450 N based on the data from similar models studied by
Van-Rob. It is assumed that the 450 N load is evenly distributed on the span of the CCB tubes
consisting both DS and PS tubes. The direction of the load is in the negative Z direction, i.e.
in the direction of gravity acceleration.
To obtain a realistic response of the CCB under the simulated weight of the IP structure, only
the two ends of the tube is constrained. In one end, all DOFs are constrained while at the other
end the DOF in the direction of the tube is set free. This allows for the sagging of the CCB
under the load. In addition, one degree of freedom of the Cowl Top is also constrained to
prevent the CCB from rotating under the torque of steering column weight. The Z
displacement DOF of the node on the Cowl Top attachment is held fixed for this purpose.
Figure 5-4 represents the loading configuration for the static load analysis.
Figure 5-4 Static load analysis of the CCB tube
Table 5-8 presents the deflection at the mid-span for baseline and optimized models as well as
the maximum stress found on the CCB. The mid-span deflection of the baseline model is
taken as the reference state. Compared to the baseline model, the maximum deflection is
increased by 0.84 mm. However, both deflections are less than the allowable deflection which
is 5 millimeter according to Van-Rob. It should be noted that in practice, although the weight
of the IP system still applies to the CCB structure, the CCB uses more constraint and joints to
bear the load.
89
Table 5-8 Tube static load analysis results
Deflection at Mid-span (mm) Maximum Stress (MPa)
Baseline Design 2.04 146.3
Optimized Design (Run 1) 2.88 151.8
As observed in Figure 5-5, the maximum stress occurs at the attachment point of the two
tubes, where the two sections are connected by welding. The maximum stress is below 170
MPa which is the yield stress of the aluminum alloy used for extruded parts. Furthermore, the
stress in the attachment point greatly depends on the stress concentration factor and can be
effectively reduced by proper welding and tapering the DS tube to smoothly match the cross
section of the PS tube.
Figure 5-5 Stress distribution of the CCB under the static load
The static loading analysis then confirms that there is no concern regarding the strength of the
structure and its maximum deflection under the IP load. A more accurate assessment of the
structure will be possible when the CCB is simulated together with the entire IP system and
the real boundary conditions are applied to the model. However, for the current stage of the
design when only the CCB assembly is considered, the above crude analysis is sufficient.
90
CHAPTER 6: CONCLUSIONS AND
RECOMMENDATIONS
6.1 Concluding Remarks
The cost optimization of a CCB assembly made from aluminum is pursued in this work.
Multidisciplinary design optimization architecture is employed to manage the design task and
coordinate all the modifications made to the baseline model. The following points conclude
the outcome of this research project:
The CCB is a complicated structure. For this reason, a sophisticated optimization
procedure needs to be devised to handle all the aspects of the CCB design effectively
and concurrently. The MDF method is used in this work to deal with the various
design disciplines and manage their interactions.
A gauge thickness module is developed to handle the changes made to the thickness of
selected parts of the CCB. Implementing the new thicknesses of the parts is achieved
by this discipline while other part specifications (shape and topology) can be altered as
the same time.
The shape morphing disciplines utilizes perturbation vectors to define possible
movements of each of the part’s nodes. Using the shape coefficients as the design
variables provides the procedure with the flexibility to change the shapes of selected
parts toward the desired shapes while staying consistent with modifications by other
design aspects of the problem. Topography optimization technology plays the central
role in defining the allowable shape changes, i.e. the shape design variables.
The part discipline is a useful module to substitute heavier and more costly part
configurations with lighter and less expensive alternative components designed for the
CCB. It is capable of replacing some parts, as well as eliminating or adding parts to
91
the design. The new components are designed with the aid of topology optimization
method offered by Altair.
Particle swarm optimization is employed as the optimization engine of the MDO
procedure. It brings a lot of flexibility to the procedure rendering the control of
algorithm behavior fairly user-friendly. One can try various parameters setting in the
algorithm and figure out the best set of control and convergence parameters. This has
been done and the optimization task is carried out successfully.
The final design of the CCB is investigated and studied to reveal the essential effects
of the modifications. It has been showed that the NVH criteria are fully satisfied and
the final design is tested to make sure that the structural integrity and stiffness of the
assembly is maintained.
The mass reduction corresponding to the CCB design is calculated to be
approximately 18% and then it is interpreted as reduction in the cost. Using a crude
cost estimation based on the mass reduction, a total cost saving of $13 per CCB is
predicted.
6.2 Future Directions
Crashworthiness of the IP structure is a critical aspect of the vehicle design which is highly
involved with the safety performance of the vehicle. In order to make the CCB design more
reliable, one can think about an integrated NVH and crash optimization procedure. Each of
these criteria can be handled by one or a few distinct disciplines.
More sophisticated MDO techniques can be implemented in the design if new disciplines are
added to the procedure. IDF architecture can be a suitable tool to deal with the problem
involving both NVH and the crash analysis. It provides more centralization to the algorithm as
well as enables using parallel computation architecture. Using parallel computation
architecture is going to become essential for large and complex systems like the CCB in
which one complete analysis of the assembly takes a few minutes (if not hours, depending on
the solver) to be done. Considering the numerous analysis of the subsystems that are required
for the optimization procedure, the drive for reducing the computation time can be better
understood.
92
Other aspects of the design may be introduced into the optimization procedure. Material
properties are one of these aspects than can be utilized to obtain lighter designs with superior
performance. It is possible to add alternative light metals such as magnesium to the materials
used in the IP system. This of course depends on the welding technologies required to attach
parts made from different materials. Different grades of aluminum may be used for different
parts of the assembly and the possibility of weight and cost reduction can be checked
automatically by the MDO procedure.
Finally, one can consider introducing more design variables corresponding to almost all the
parts present in the CCB structure. It enables the designer to benefit from all the capacities of
weight and cost reduction inherent in the baseline design. At the same time, increasing the
number of design variables generally causes the procedure to last longer and the convergence
to be harder to achieve. Such challenges can be faced with more advanced MDO structures
and superior computational power, especially by using parallel computing architecture.
93
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Appendix A: Element Specifications
(Adopted from Ref. [40])
Element Description Coordinate System
Quadrilateral
plate element
(QUAD4)
This element uses a 6 degree-of-
freedom per node formulation. All of
the interior angles must be less than
180 degrees. The elemental coordinate
system is a bisection definition as
depicted in the figure.
Six-sided Solid
Element
(CHEXA)
This element can have 8 or 20 nodes.
The latter case happens when nodes
are placed on the middle of each edge
too. Each node has six degrees of
freedom. To define the element
coordinate system, three intermediate
vectors R, S, and T are chosen as
follows:
R: Joins the centroids of the faces
described by the grid points G4, G1,
G5, G8 and the grid points G3, G2,
G6, G7.
S: Joins the centroids of the faces
described by the grid points G1, G2,
G6, G5 and the grid points G4, G3,
G7, G8
T: Joins the centroids of the faces
described by the grid points G1, G2,
G3, G4 and the grid pints G5, G6, G7,
G8.
The origin of the element coordinate
100
system is at the intersection of these
vectors. If the vectors do not all
intersect at one point, the average
location of the intersection points is
used. The element z-axis corresponds
to the T vector. The element y-axis is
the cross product of the T and R
vectors. The element x-axis is the cross
product of the element y-axis and the
element z-axis.
101
Appendix B: Mass Reduction History Diagrams
B-1: Mass reduction history for optimization trials using a population size of 15
102
B-2: Mass reduction history for optimization trials using a population size of 10
103
Appendix C: Matlab Codes
C-1: Optimization Procedure (myMDO.m)
This Matlab script initializes the optimization procedure. It calls the PSO solver with the
MDF script being its fitness function. The values of upper and lower bounds are defined and
the total execution time is measured.
%% Multidisciplinary Design Optimization Procedure time_i=clock; % Counting the executions number fid=fopen('C:\Users\UofT\Desktop\CCB02\batchrun04\exe_number.txt', 'w'); exe_num=0; fprintf(fid,'%u', exe_num); fclose(fid); % Connecting the Multidisciplinary Design Analyser to the PSO optimizer option=psooptimset('PopulationSize', 15, 'Generations', 99, 'CognitiveAttraction', 0.9,
'SocialAttraction', 1.1, 'TolFun', 0.1, 'StallGenLimit', 30); Lb=[2 2 2 2 2 2 0 0 0 0 0 0 0 0]; Ub=[5 5 5 5 5 5 1 1 1 1 2 3 2 2]; [v, fval]=pso(@MDF,14,[],[],[],[],Lb,Ub,[],option) %% Reporting the Total Time Spent time_f=clock; t_difft=time_f-time_i; t_hours=t_difft(1,4)+t_difft(1,3)*24; t_minutes=t_difft(1,5); t_seconds=t_difft(1,6); if t_minutes<0 t_hours=t_hours-1; t_minutes=60+t_minutes; end; if t_seconds<0 t_minutes=t_minutes-1; t_seconds=60+total_seconds; end; % Printing fprintf('Total MDO Time: %2.0d hr., %2.0d min., %6.4f sec.\n\n\n\n', t_hours, t_minutes,
t_seconds);
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C-2: Multidiscipline Feasible Analysis (MDF.m)
This script is responsible to carry out the multidiscipline feasible analysis and return the
fitness function value to the PSO optimizer. Various disciplinary effects are applied and the
solver deck is sent to RADIOSS for the NVH analysis. Upon completion of the analysis, the
responses are gathered and penalty functions are calculated. The fitness value is obtained and
details of the variable selection and responses are recorded in a text file.
function f=MDF(X) % runs the MultiDiscipline Feasible Analysis
th=X(1,1:6); % gauge variables
shape=X(1,7:10); % shape variables
part=X(1,11:14); % part variables
%% counting the cycles
fid=fopen('C:\Users\UofT\Desktop\CCB02\batchrun04\exe_number.txt', 'r');
tline=fgetl(fid);
exe_num=round(str2num(char(tline)));
fclose(fid);
fid=fopen('C:\Users\UofT\Desktop\CCB02\batchrun04\exe_number.txt', 'w');
exe_num=exe_num+1;
fprintf(fid,'%u', exe_num);
fclose(fid);
%% solver deck addresses
sdeck='ccb_MDF2';
addr1='C:\Users\UofT\Desktop\CCB02\batchrun04\';
addr2='C:\\Users\\UofT\\Desktop\\CCB02\\batchrun04\\';
ex_in='.fem';
file_in=sprintf('%s%s', sdeck, ex_in);
fid=fopen('C:\Users\UofT\Desktop\CCB02\batchrun04\gbatch.bat', 'wt');
fprintf(fid, '"C:\\Program Files\\Altair\\11.0\\hwsolvers\\bin\\win64\\radioss.bat" ');
fprintf(fid, addr2);
fprintf(fid, file_in);
fclose(fid);
%% Handling part variables (Part Discipline)
cowl_no=ceil(part(1));
endbkt_no=ceil(part(2));
cover_no=ceil(part(3));
VBsup_no=ceil(part(4));
part_deck=sprintf('%s%u%s%u%s%u%s%u%s', 'ccb_MDF2_cowl_', cowl_no, '_endb_',
endbkt_no, '_cover_', cover_no, '_sup_', VBsup_no, '.fem');
source_file_deck=sprintf('%s%s', 'C:\Users\UofT\Desktop\CCB02\PARTvars\', part_deck);
destination_file_deck=sprintf('%s%s', addr1, file_in);
copyfile(source_file_deck, destination_file_deck,'f');
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%% Handling shape modifications (Shapes Disciplines)
shape_num=length(shape); %number of shape variables
for i=1:shape_num
shape_cof(i)=shape(i);
end
for sh=1:shape_num
% reading from shape files
file_shape=sprintf('%s%s%u%s', addr1, 'shape', sh , '.txt');
fid=fopen(file_shape, 'r');
i=0;
while ~feof(fid)
i=i+1;
temp1=textscan(fid, '%*s %*s %u %*s %*s',1);
fseek(fid, ftell(fid)+2, 'bof');
grid_num(i)=temp1[1];
temp2=char(fgetl(fid));
grid_x(i)=str2num(temp2(1,9:24)) ;
grid_y(i)=str2num(temp2(1,25:40));
grid_z(i)=str2num(temp2(1,41:56));
end
fclose(fid);
% calculating the new grid data
file_grid_new=sprintf('%s%s%u%s', addr1, 'newgrid', sh, '.txt');
fid2=fopen(file_grid_new, 'w');
file_grid_org_c=sprintf('%s%s%u%s', addr1, 'shape', sh, '_org_corr.txt');
fid3=fopen(file_grid_org_c, 'r');
for i=1:length(grid_num)
tline = fgetl(fid3);
temp3=char(tline);
gn_char=temp3(1,9:16);
new_x=str2double(temp3(1,25:32))+shape_cof(sh)*grid_x(i) ;
new_y=str2double(temp3(1,33:40))+shape_cof(sh)*grid_y(i) ;
new_z=str2double(temp3(1,41:48))+shape_cof(sh)*grid_z(i) ;
remaining=temp3(1,49:72);
tline2=sprintf('%s%s%s%f%s%f%s%f%s%s','GRID, ', gn_char, ', ,', new_x, ' , ',
new_y,' , ', new_z,' , ', remaining);
fprintf(fid2,'%s\r\n', tline2);
end
fclose(fid3);
fclose(fid2);
clear grid_num grid_x grid_y grid_z;
end
%% Handling thickness modification (Gauge Discipline)
gauge_num=length(th); %number of thickness variables
for i=1:gauge_num
thickness(i)=th(i);
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end
file_prop_org=sprintf('%s%s', addr1, 'org_prop.txt');
fid=fopen(file_prop_org, 'r');
file_prop_new=sprintf('%s%s', addr1, 'newprop.txt');
fid2=fopen(file_prop_new, 'w');
for i=1:gauge_num
tline=fgetl(fid);
temp1=char(tline);
lable=temp1(1,1:24);
tale=temp1(1,28:72);
tline2=sprintf('%s%3.1f%s', lable, thickness(i), tale);
fprintf(fid2, '%s\n\r', tline2);
fprintf(fid2, '\n');
end
fclose(fid);
fclose(fid2);
%% Calling the RADIOSS solver to solve the system for the responces
! C:\Users\UofT\Desktop\CCB02\batchrun04\gbatch.bat
%% Collecting the system responces (objective and constraints values)
ex_out='.out';
file_out=sprintf('%s%s%s', addr1, sdeck, ex_out);
fid=fopen(file_out, 'r');
tline = fgetl(fid);
pattern1='Volume =';
while ischar(tline)
if ~isempty(strfind(tline, pattern1))
[vol_total, mass_total]=strread(tline, '%*s %*s %f %*s %*s %f');
for i=1:4
tline = fgetl(fid);
end
c=textscan(fid, '%u %u %f %f %f %f');
break;
end
tline = fgetl(fid);
end
freq=c[3];
fclose(fid);
mass=1000*(mass_total - 7.537e-3); % subtructing the steering column mass and converting
to Kg
vol= vol_total - 1.499e6 ; % subtructing the steering column volume
%% NVH Constraints Penalties
NVH_cons_1=(min((freq(1)-37), 0))^2 ;
NVH_cons_2=(min((freq(2)-39), 0))^2 ;
NVH_pen=100*NVH_cons_1+100*NVH_cons_2;
%% Cost Penalties
cost_pen= (390*130)*[3-cover_no] + 420*50*[VBsup_no];
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%% Fitness Value Calculation
f=20*mass + NVH_pen + 0.001*cost_pen;
%% Saving the all the data to a text file
fid=fopen('C:\Users\UofT\Desktop\CCB02\batchrun04\1_opti_history.txt', 'a+');
fprintf(fid,
'%u\t%f\t%f\t%f\t%f\t%f\t%f\t%f\t%f\t%f\t%f\t%f\t%u\t%u\t%u\t%u\t%f\t%f\t%f\t%f\t%f\t
%f\n', exe_num, f, X(1), X(2), X(3), X(4), X(5), X(6), X(7), X(8), X(9), X(10), cowl_no,
endbkt_no, cover_no, VBsup_no, X(11), X(12), X(13), X(14), mass, vol );
fclose(fid);
end
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