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Design and Optimization of Aluminum Cross-Car Beam
Assemblies Considering Uncertainties
by
Mehran Ebrahimi
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Graduate Department of Mechanical and Industrial Engineering
University of Toronto
© Copyright by Mehran Ebrahimi 2015
ii
Design and Optimization of Aluminum Cross-Car Beam
Assemblies Considering Uncertainties
Mehran Ebrahimi
Master of Applied Science
Graduate Department of Mechanical and Industrial Engineering
University of Toronto, 2015
Abstract
Designing real-world structures with small failure probabilities has been always a burdensome
issue due to high computational efforts demanded for structural reliability assessment. In this
thesis, a new reliability-based design optimization (RBDO) framework is proposed to hasten
reliability design of practical structures. Exploiting artificial neural networks along with subset
simulation, the developed strategy, significantly diminishes the sampling load of probability
evaluations compared to conventional Monte Carlo simulation, so the design process can be carried
out within a realistic time frame. To explore the superiority of this approach, it is applied to a 25-
bar truss structure. In this study, also, a new framework for designing automotive aluminum cross-
car beam (CCB) assemblies from the ground up is developed by implementing various structural
optimization techniques. To assay the capability of this approach and the proposed RBDO strategy,
an aluminum CCB, for replacing its steel counterpart, was designed considering deterministic and
probabilistic constraints.
iii
Acknowledgments
This dissertation, composed of about hundred pages of words and countless pages of valuable
memories and lessons I gained in these two years, would not have been possible without the
support of so many people in so many ways.
First of all, I would like to express my deepest appreciation to my supervisor, Prof. Kamran
Behdinan, for the precious advice and encouragement he provided all the way long. His broad
knowledge, thorough insight, and primarily his great personality, have set an example to me for
my future career.
I also thank Natural Sciences and Engineering Research Council of Canada, Ontario Centres of
Excellence, and Van-Rob Kirchhoff Inc. for their support during this study. The thought-leading
advice of Dr. Sacheen Bekah from Van-Rob Kirchhoff Inc. in this research is so much
acknowledged.
I appreciate my fellow researchers, specially Nima Bahrani, in the Advanced Research Laboratory
for Multifunctional Lightweight Structures at the Department of Mechanical and Industrial
Engineering for the great atmosphere they made in lab, and the delightful memories they gifted to
me in these two years.
To Neda – wife, companion, friend, and colleague – whose profound understanding helped me to
find myself, to stand on my own feet again, and to never let any fear penetrate into my heart.
Having you smiling before my eyes suffices me.
To my beloved parents, brother, and sister by whom I lived every single second of these two years.
I see the brighter days; the days that we are all around a dinner table, and I am no longer afraid
that you are not beside me tomorrow night.
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Table of Contents
Acknowledgments .......................................................................................................................... iii
Table of Contents ........................................................................................................................... iv
List of Tables ................................................................................................................................ vii
List of Figures ................................................................................................................................ ix
List of Appendices ......................................................................................................................... xi
Nomenclature ................................................................................................................................ xii
CHAPTER 1: Introduction .......................................................................................................... 1
1.1 Reliability-Based Design Optimization of Engineering Structures .................................... 1
1.2 Design and Optimization of Automotive Aluminum Cross-Car Beam Assemblies ........... 2
1.3 Thesis Objectives ................................................................................................................ 3
1.4 Thesis Outline ..................................................................................................................... 5
CHAPTER 2: Evaluation of Metaheuristic Optimization Algorithms .................................... 6
2.1 Overview ............................................................................................................................. 6
2.2 Introduction ......................................................................................................................... 6
2.2 Metaheuristic Optimization Algorithms ............................................................................. 7
2.2.1 Particle Swarm Optimization (PSO) ....................................................................... 7
2.2.2 Genetic Algorithm (GA) ......................................................................................... 8
2.2.3 Evolutionary Strategy (ES) ..................................................................................... 9
2.2.4 Firefly Algorithm (FA) ........................................................................................... 9
v
2.2.5 Harmony Search (HS) ........................................................................................... 10
2.2.6 Simulated Annealing (SA) .................................................................................... 11
2.3 A Comparative Study on Metaheuristic Optimization Algorithms .................................. 13
2.3.1 Mathematical Benchmarks .................................................................................... 14
2.3.2 Structural Benchmarks .......................................................................................... 17
2.3.2.1 Welded Beam Design ............................................................................. 18
2.3.2.2 Pressure Vessel Design ........................................................................... 20
2.3.2.3 Compression Spring Design ................................................................... 21
2.3.2.4 Stepped Cantilevered Beam Design ....................................................... 23
2.4 Closing Remarks ............................................................................................................... 26
CHAPTER 3: Reliability-Based Design Optimization of Practical Structure Using Subset
Simulation and Artificial Neural Networks ......................................................................... 27
3.1 Overview ........................................................................................................................... 27
3.2 Introduction ....................................................................................................................... 27
3.3 Problem Definition ............................................................................................................ 29
3.4 Subset Simulation Method ................................................................................................ 30
3.4.1 Subset Simulation Algorithm ................................................................................ 31
3.4.2 Markov Chain Monte Carlo Simulation ............................................................... 33
3.4.2.1 Markov Chains ....................................................................................... 34
3.4.2.2 Modified Metropolis Algorithm ............................................................. 35
3.5 Artificial Neural Networks ............................................................................................... 36
3.5.1 Feed-Forward Back-Propagation Network ........................................................... 37
3.5.1.1 Feed-Forward Phase ............................................................................... 38
3.5.1.2 Back-Propagation of Errors Phase .......................................................... 39
3.6 Proposed Reliability-Based Design Optimization Framework ......................................... 41
3.7 RBDO of a 25-bar Structure ............................................................................................. 43
3.8 Closing Remarks ............................................................................................................... 49
vi
CHAPTER 4: Reliability-Based Design Optimization of Aluminum Cross-car Beam
Assemblies ............................................................................................................................... 50
4.1 Overview ........................................................................................................................... 50
4.2 Introduction ....................................................................................................................... 50
4.3 Structural Optimization ..................................................................................................... 51
4.3.1 Size Optimization .................................................................................................. 51
4.3.2 Topology Optimization ......................................................................................... 52
4.3.3 Shape Optimization ............................................................................................... 54
4.4 Methodology ..................................................................................................................... 56
4.4.1 Conceptual Design Stage ...................................................................................... 57
4.4.2 Detailed Design Stage ........................................................................................... 57
4.5 Case Study ........................................................................................................................ 59
4.5.1 Conceptual Design ................................................................................................... 59
4.5.2 Detailed Design ..................................................................................................... 65
4.6 Reliability–Based Design of the CCB .............................................................................. 72
4.7 Closing Remarks ............................................................................................................... 75
CHAPTER 5: Conclusion........................................................................................................... 76
5.1 Thesis Summary ................................................................................................................ 76
5.2 Future Work ...................................................................................................................... 77
REFERENCES ............................................................................................................................ 79
APPENDICES ............................................................................................................................. 88
Appendix A. Mathematical Benchmarks of Chapter 2 ............................................................ 88
vii
List of Tables
Table 2- 1 Summary of the main properties of test functions ................................................ 14
Table 2- 2 Performance results of six algorithms ................................................................... 15
Table 2- 3 Performance summary of all approaches .............................................................. 17
Table 2- 4 Optimization results for the welded beam design ................................................. 19
Table 2- 5 Optimization results for the pressure vessel design .............................................. 21
Table 2- 6 Allowable wire diameters for compression spring design .................................... 22
Table 2- 7 Optimization results for compression spring design ............................................. 23
Table 2- 8 Optimization results for stepped cantilevered beam design ................................. 25
Table 2- 9 Performance summary of all approaches .............................................................. 25
Table 3- 1 The members of the 8 groups ................................................................................ 44
Table 3- 2 The discrete values of bar areas ............................................................................ 44
Table 3- 3 The loading condition of the structure .................................................................. 44
Table 3- 4 The characteristics of the random variables .......................................................... 45
Table 3- 5 The performance summary of optimization approaches ....................................... 46
Table 3- 6 Truss stresses in deterministic optimality ............................................................. 47
Table 3- 7 Nodal displacements in deterministic optimality .................................................. 48
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Table 3- 8 Truss stresses in reliability optimality .................................................................. 48
Table 3- 9 Nodal displacements in reliability optimality ....................................................... 48
Table 4- 1 Mechanical and physical properties of design aluminum ..................................... 60
Table 4- 2 Range of shape and size optimization design variables ........................................ 66
Table 4- 3 Optimal value of design variables ......................................................................... 67
Table 4- 4 The discrete values of thicknesses ........................................................................ 68
Table 4- 5 Optimal value of thicknesses ................................................................................ 69
Table 4- 6 The characteristics of the random variables .......................................................... 72
Table 4- 7 The results of optimization approaches ................................................................ 73
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List of Figures
Figure 1- 1 The CCB (shaded area) as the skeleton of the IP assembly (from Ref. [12]) ......... 3
Figure 2- 1 The welded beam (from Ref. [42]) ....................................................................... 18
Figure 2- 2 The pressure vessel (from Ref. [42]) .................................................................... 20
Figure 2- 3 The compression spring (from Ref. [42]) ............................................................. 21
Figure 2- 4 The stepped cantilevered beam (from Ref. [42]) .................................................. 23
Figure 3- 1 A four-level subset simulation .............................................................................. 33
Figure 3- 2 Fully connected ANN configuration ..................................................................... 37
Figure 3- 3 The proposed optimization framework ................................................................. 42
Figure 3- 4 The 25-bar truss structure (from Ref. [72]) .......................................................... 43
Figure 4- 1 Size optimization of a structure: a) Initial design b) Optimum design ................. 52
Figure 4- 2 Topology optimization of a structure: a) Initial domain b) Optimum design ....... 52
Figure 4- 3 Shape (topography) optimization of a structure: a) Initial design b) Optimum
design .................................................................................................................... 55
Figure 4- 4 A cross-car beam assembly and its main components .......................................... 56
Figure 4- 5 The proposed optimization framework ................................................................. 58
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Figure 4- 6 Design space of the CCB meshed by solid elements ............................................ 60
Figure 4- 7 Topology of the CCB after the first optimization ................................................. 61
Figure 4- 8 Topology results of the main beams ..................................................................... 61
Figure 4- 9 Input model of the second topology optimization ................................................ 62
Figure 4- 10 Second topology optimization results ................................................................... 62
Figure 4- 11 Input model for the third phase of topology study ................................................ 63
Figure 4- 12 Finalized conceptual design of the CCB ............................................................... 64
Figure 4- 13 Conceptual design modeled by shell meshes ........................................................ 65
Figure 4- 14 The CCB and its components ............................................................................... 66
Figure 4- 15 Potential locations of the beads ............................................................................. 67
Figure 4- 16 Input CCB model for the last step optimization ................................................... 68
Figure 4- 17 Ultimate Optimum Design of the CCB ................................................................. 70
Figure 4- 18 Convergence history of the final step optimization .............................................. 71
Figure 4- 19 Convergence history of the RBDO optimization .................................................. 74
xi
List of Appendices
Appendix A: Mathematical Benchmarks of Chapter 2
xii
Nomenclature
PSO Particle Swarm Optimization
GA Genetic Algorithm
ES Evolutionary Strategy
FA Firefly Algorithm
HS Harmony Search
SA Simulated Annealing
HM Harmony Memory
HMCR Harmony Memory Considering Rate
PAR Pitch Adjustment Rate
LI Linear Inequality
NE Nonlinear Equality
NI Nonlinear Inequality
RBDO Reliability-Based Design Optimization
ANN Artificial Neural Network
SS Subset Simulation
PDF Probability Distribution Function
xiii
MCS Monte Carlo Simulation
MCMCS Markov Chain Monte Carlo Simulation
DANN Deterministic Artificial Neural Network
PANN Probabilistic Artificial Neural Network
DBO Deterministic-Based Optimization
CCB Cross-Car Beam
IP Instrument Panel
NVH Noise, Vibration, and Harshness
ESO Evolutionary Structural Optimization
SIMP Solid Isotropic Material with Penalization
FE Finite Element
DS Driver Side
PS Passenger Side
1
CHAPTER 1: Introduction
1.1 Reliability-Based Design Optimization of Engineering
Structures
The design of real-world structures has been always subjected to uncertainties of design
parameters. Probabilistic nature of design variables, material properties, and loading conditions
can all greatly influence the performance of structures and not taking into account these factors
may lead to their failure. All airplane crashes, car accidents, and other collapses happening
everyday are due to inaccurate prediction of operating conditions in design stages or operator
incapability of handling unforeseen circumstances. On the other hand, precise anticipation of
parameters, such as environmental conditions, human errors, or manufacturing tolerances, is
almost an impossible task, and hence instead of encountering deterministic constraints or
performance functions in the design processes, appropriate reliability analysis techniques should
be employed to devise the structures under probabilistic constraints. Reliability-based design
optimization (RBDO) problems can be viewed as double-loop procedures; the inner loop addresses
the reliability analysis and examines the feasibility of the design points and the outer loop performs
the optimization process. The main focus of this study is on the inner loop to develop a new RBDO
approach, so reliability design of engineering structures can be accomplished within a realistic
time frame.
Within the past three decades, structural reliability has been extensively improved [1], and has
facilitated the implementation of RBDO procedures in the practical applications. However, due to
extremely high computational costs of reliability analysis of large-scale engineering systems, just
few accomplished real projects are reported in the literature by considering non-deterministic
performance functions [2]–[5]. In the real-world problems, the design variables are normally
restricted to a set of discrete values selected from available commercial standards [6]. Objective
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and performance functions are also mostly highly nonlinear, discontinuous, and non-differentiable,
and are not defined as explicit closed-form functions of input design variables [7]–[9]. Thus, to
optimize such structures, conventional gradient-based optimization methods become inefficient
and in the majority of cases impotent to converge [10]. Therefore, the optimization task has to be
conducted by using gradient-free optimizers that are, in essence, stochastic techniques requiring
considerable number of samples from the design space. Furthermore, to evaluate the value of
fitness and constraint functions at any design point, a finite element analysis process has to be
carried out which again burdens an extra computation cost on the process. On top of these, the
accuracy of reliability approximation methods, such as first-order and second-order methods,
reduces remarkably when dealing with nonlinear and discontinuous performance functions. On the
other hand, the crude Monte Carlo simulation (MCS) demands a huge number of samples, and is
incapable of estimating small failure probabilities, which again imposes another level of
complexity to the situation [11].
1.2 Design and Optimization of Automotive Aluminum Cross-Car
Beam Assemblies
Energy is one of the most significant matters in the world today, and has been the source of many
struggles at the international level. The future of human generations is under serious threat due to
energy-related issues such as resource shortage and global warming. Hence, committed
governments have set up a number of policies to reduce fuel consumption in energy-dependent
industries and their products. Automobile industries, as the manufacturers of principal fuel
consumers are not exempt from these rules, and are always seeking more lightweight industrial
designs.
In order to achieve more lightweight solutions, changing the material of an existing design to a
lower density one is the first option in many applications, and aluminum, as an accessible material
with acceptable mechanical properties could be a suitable replacement for steel in the majority of
industrial purposes. Yet due to its relatively lower yield strength and different mechanical
specifications, such as Young’s modulus and density, existing designs should be modified, so they
can fulfil desired expectations.
3
In the automotive area, one of the main sections which could be focused on to achieve more
lightweight alternatives is the cross-car beam assembly. A cross-car beam (CCB) or an instrument
panel (IP) support integrates the cross-car structure, steering column, air conditioning module and
airbag system, electrical components, and plastic enclosure into one beam, Figure 1-1. Moreover,
it plays a vital role in absorbing the energy of accidents [12]. Currently, the material used to
manufacture this assembly is mostly steel, and the companies would rather to switch to aluminum.
By changing the material, the assembly may no longer meet the design requirements, and should
be redesigned.
Figure 1- 1 The CCB (shaded area) as the skeleton of the IP assembly (from Ref. [12])
1.3 Thesis Objectives
The RBDO process of cross-car beam assemblies, similar to those of other engineering structures,
involves discrete random design variables, uncertainty in the design condition, and non-
differentiable constraint functions. Therefore, this process is a hefty computational task which is
almost impossible to be performed by conventional RBDO approaches. Despite all advances in
the field of RBDO of real-world problems, the literature still lacks a robust RBDO strategy which
can handle reliability design of large-scale structures with small failure probabilities within a
4
reasonable time frame. This encouraged us to seek a new methodology for RBDO procedure by
leveraging well-developed variance reduction techniques and surrogate models. If the goal is
achieved, then this will be precious progress toward more reliable structures which leads to fewer
human tragedies occurring every day in the world.
The proposed RBDO approach is then applied to develop a new framework for design and
optimization of automotive aluminum cross-car beam assemblies considering deterministic and
probabilistic constraints. Currently, in industry, to modify a CCB for new requirements, an existing
CCB is taken and a few components on it are altered to fit the expected performance. Although,
this strategy, in some cases, results in an acceptable solution, it sometimes leads to an infeasible
or oversized design. Also, the whole process may get trapped in a design loop and never
convergences to an industrial result. Thus, proposing a new framework which does not have the
aforementioned shortcomings and delivers the optimized replacement in a shorter time can
considerably reduce the fabrication and engineering costs, and benefits both customers and
companies.
To accomplish the optimization tasks expressed in the study, an efficient stochastic optimization
strategy should be employed. To do so, an extensive study on the performance of well-known
stochastic optimization algorithms has also been conducted in this thesis, and based on their
performance, the best technique is chosen to be exploited for propelling the optimization
procedures.
The thesis objectives can be summarized as follows:
Determining an efficient stochastic optimization technique among other popular ones
Developing a new RBDO approach for reliability design of real-world applications with
small failure probabilities
Exploring the performance of the proposed RBDO approach on a practical case study (a
25-bar truss structure)
Constructing an RBDO framework for design and optimization of automotive aluminum
cross-car beam assemblies
Performing RBDO of an aluminum CCB to test the capability of the developed framework
5
1.4 Thesis Outline
The thesis is composed of five chapters. The topics covered in each of them can be briefly
expressed as:
Chapter 2: A comparative study on the performance of six most popular stochastic
optimization algorithms are performed based on a few known mathematical and structural
benchmarks. A concise introduction of each of the algorithms is stated, and the results of
their performance in all benchmarks are reported in details.
Chapter 3: A new RBDO approach is proposed in this chapter. Various terminologies in
the field of probability and reliability are defined. A brief introduction of subset simulation
and artificial neural networks are carried out. Eventually, the framework is presented and
tested on a 25-bar truss structure.
Chapter 4: A new design framework for automotive aluminum CCBs is submitted.
Different structural optimization techniques are reviewed and utilized to construct the
optimization architecture. The approach is examined on a CCB, currently manufactured in
industry; and finally, as the second case study for RBDO proposed in Chapter 3, the CCB
is designed for reliability performance.
Chapter 5: Contributions and achievements of the study are delivered in this chapter, and
a few suggestions are made for future research on the aforesaid matters.
6
CHAPTER 2: Evaluation of Metaheuristic
Optimization Algorithms
2.1 Overview
In this chapter, a comparative study on the performance of six more popular metaheuristics,
particle swarm optimization (PSO), genetic algorithm (GA), evolutionary strategies (ESs), firefly
algorithm (FA), harmony search (HS), and simulated annealing (SA) is accomplished based on
some well-known mathematical and structural benchmarks. The results of this chapter are
exploited later in the optimization of the aforementioned aluminum CCB.
2.2 Introduction
Optimization algorithms can be categorized into two different classes: Deterministic methods and
metaheuristic methods. Deterministic methods use analytical properties of the problem to obtain
the global or local optimal solution. Almost all deterministic methods need gradients of objective
or constraint functions to find the optimum solution. In case of noisy or discontinuous functions
with discrete or mixed discrete-continuous design variables, using deterministic methods and
calculating the gradients become considerably time consuming, and in many problems obtaining
desire results is impossible. In order to deal with these situations, metaheuristic, non-deterministic
or gradient-free methods could be employed. These algorithms are gradient-free and mostly bio-
inspired or nature-inspired which try to find the optimum solution by mimicking a natural
phenomenon.
Metaheuristic algorithms as scientific methods to solve optimization problems first were
developed by Ingo Rechenberg and Hans-Paul Schwefel in 1963 at the Technical University of
Berlin, by introducing evolutionary strategies [13]. Then in 1960s and 1970s genetic algorithm
7
was proposed by John Henry Holland at the University of Michigan [14]. The great advancements
in metaheuristic approaches happened in the 1980s and 1990s. In 1983 simulated annealing was
developed by Kirkpatrick, Gelatt, and Vecchi who were inspired by cooling process of molten
metals through annealing [15]. Macro Dorigo developed ant colony optimization method in his
PhD dissertation in 1992 [16]. This algorithm uses swarm intelligence of social ants using
pheromone as chemical messenger. The other important algorithm exploiting swarm intelligence
is particle swarm optimization developed by Kennedy and Eberhart in 1995 [17]. The technique,
first, was proposed as a stylized representation of the movements of organisms in bird flocks or
fish schools. Then, it was simplified and utilized as an optimization method. Other metaheuristics
developed recently are harmony search [18], cuckoo search [19], hunting search [20], firefly
algorithm [21], bat-inspired algorithm [22], big bang-big crush [23], charged system search [24],
bacterial foraging algorithm [25], honey bee algorithm [26], artificial bee colony [27], and eagle
strategy [28].
2.2 Metaheuristic Optimization Algorithms
2.2.1 Particle Swarm Optimization (PSO)
Inspired by the social behavior of flocks of birds, bees, and fish, in 1995, Kennedy and Eberhart
proposed an optimization algorithm called particle swarm optimization [17]. The technique was
founded by the resemblance between seeking an optimum and searching the best food source by
creatures in nature to avoid predators. In this method, each individual uses its own memory and
the information obtained by other individuals (swarm) to find the best food source. In PSO current
position of each particle is updated by velocity vector:
𝑥𝑘+1𝑖 = 𝑥𝑘
𝑖 + 𝑣𝑘+1𝑖 ∆𝑡 (2-1)
Where 𝑥𝑘+1𝑖 and 𝑣𝑘+1
𝑖 are the position of particle 𝑖 at iteration 𝑘 + 1 and its corresponding velocity
vector, respectively. The velocity vector is updated by
𝑣𝑘+1𝑖 = 𝑤𝑣𝑘
𝑖 + 𝑐1𝑟1
(𝑝𝑘𝑖 −𝑥𝑘
𝑖 )
∆𝑡+ 𝑐2𝑟2
(𝑝𝑘𝑔
−𝑥𝑘𝑔
)
∆𝑡 (2-2)
In this equation, 𝑤 is the inertia weight, 𝑟1 and 𝑟2 are two random numbers between 0 and 1, 𝑐1,
𝑐2, 𝑝𝑘𝑖 and 𝑝𝑘
𝑔 are respectively the cognitive parameter, social parameter, best position of particle
8
𝑖, and the global best position in the swarm up to iteration 𝑘. The inertia weight plays a key role
in convergence behavior of the algorithm. The larger the inertia, the more distributed search is
done in the design space. Also, 𝑐1 and 𝑐2 well define the confidence of the particle in itself and the
swarm. Perez and Behdinan have shown that if the following conditions are met, the convergence
of the algorithm is guaranteed [29]:
0 < 𝑐1 + 𝑐2 < 4 (2-3)
𝑐1+𝑐2
2− 1 < 𝑤 < 1 (2-4)
The inertia can be updated at each iteration or be constant during the run, while the former has
shown to yield a faster convergence rate to the optimum solution. It can be updated by
𝑤𝑘+1 = 𝛼𝑤𝑘 (2-5)
where 𝛼 is a constant between 0 and 1 (e.g. 0.975).
2.2.2 Genetic Algorithm (GA)
Genetic algorithms, introduced by John Holland in the mid 1960s, are the best known type of
evolutionary algorithms [14]. They were inspired by biological evolutions in nature. Genetic
algorithms are based on three essential operations: Selection, Crossover, and Mutation [30].
In the selection phase, parents are selected from the mating pool to breed the new generation (off-
springs). Those having higher fitness scores are more probable to be selected.
Crossover has the key role in GAs as optimization algorithms. In the crossover process the next
generation or off-springs are produced from the current population or parents. The new off-springs
share many characteristics of the mating parents.
Mutation is an operator which guarantees the diversity of the new population. In this process, a
number of new individuals are produced by altering one or more genes from the initial population.
This process is a key step in the procedure since it is likely that by selection and crossover some
useful genetic information is not transferred to the next generation and is missing in the rest of the
optimization process.
9
Although GAs have several advantages over other optimization algorithms, they are
computationally expensive approaches compared to the deterministic methods, and in case of
complex problems with a large number of design variables, may not lead to an acceptable solution
in a reasonable time frame.
2.2.3 Evolutionary Strategy (ES)
The other group of evolutionary algorithms is evolutionary strategies which were first proposed
by Ingo Rechenberg [31], Hans-Paul Schwefel [32], and their co-workers in 1960s and then
developed further in 1970s. The main difference between ES and GA is that individuals in GA
need to be coded as binary integers, yet in ES they are handled as real numbers. Furthermore, in
GA the selection is performed considering the fitness scores of parents, whereas in ES parents are
chosen randomly to generate off-springs. Similar to GAs, ESs have three principal steps:
Recombination, Mutation, and Selection [33].
2.2.4 Firefly Algorithm (FA)
Firefly algorithm is one of the most recent optimization algorithms, proposed by Yang in 2009
who was inspired by flashing characteristics of fireflies [21]. FA is based on three idealized rules:
All fireflies are unisexual, so all fireflies have the chance to be attracted to all other fireflies.
Attractiveness is proportional to firefly brightness, and both of them decrease as the distance
between two fireflies increases. The less bright firefly is attracted to the brighter one, and
moves toward it, and the brightest firefly moves randomly in the space.
The brightness of a firefly is determined by the landscape of the cost function.
The attractiveness 𝛽 of a firefly is defined as
𝛽(𝑟) = 𝛽0 𝑒𝑥𝑝(−𝛾𝑟2) (2-6)
where 𝛽0 is the attractiveness at 𝑟 = 0, and 𝛾 is the light absorption coefficient which can be taken
as a constant value. In this regard, the movement of firefly 𝑖 (less bright) toward another one 𝑗
(brighter) at iteration 𝑡 is calculated by
10
𝛥𝑥𝑖 = 𝛽0 𝑒𝑥𝑝(−𝛾𝑟2) (𝑥𝑗𝑡 − 𝑥𝑖
𝑡) + 𝛼휀𝑖, 𝑥𝑖𝑡+1 = 𝑥𝑖
𝑡 + 𝛥𝑥𝑖 (2-7)
where 𝛼 is the randomization parameter, and 휀𝑖 is a vector of random numbers derived from a
Gaussian distribution. For most problems 𝛼 can be taken 0.01, and 𝛾 varies between 0.01 and 100.
Also, it has been found that a population size of 10 to 25 individuals is sufficient to handle the
majority of optimization problems.
2.2.5 Harmony Search (HS)
Unlike previous metaheuristics which have biological basis, harmony search optimization
algorithm is derived from a musical phenomenon trying to find a perfect state of the harmony. HS
was proposed by Zong Woo Geem in 2001 [18]. He assumed that an optimization problem can be
simulated as a musical improvisation seeking for a musically pleasant harmony (objective) by
adjusting the pitch (design variables) of a musical instrument. The basic HS has four steps:
Step 1. Initializing a harmony memory (HM): An initial population like other metaheuristics.
Step 2. Improvising a new harmony memory from (HM): The objective value of the initial
population are evaluated and sorted in an ascending order (for a minimization problem).
Then, a new harmony whose members (design variables) are chosen form the whole space
or the harmony memory is improvised. The probability that a design variable in the harmony
memory transfers to the new harmony is controlled by a parameter called harmony memory
considering rate (HMCR), 0 ≤ HMCR ≤ 1. This parameter is normally between 0.7 to 0.95.
Design variables which are from the harmony memory can be pitch-adjusted to produce the
new population. The parameter controlling the probability of a design variable to be pitch-
adjusted or not is called pitch adjustment rate (PAR), 0 ≤ PAR ≤ 1.
Step 3. Updating the harmony memory: The objective value of the newly generated harmony
is calculated, and if the value is better than the worst harmony, the new candidate replaces
the current one in the memory, and the worst one is excluded from the memory.
Step 4. Going back the step 2, if the termination criteria are not met.
11
The main difference between HS and evolutionary algorithms is that, in former, new individuals
are chosen from the entire design space, while in the latter they are produced from the existing
population.
2.2.6 Simulated Annealing (SA)
Kirkpatrick, Gelatt, and Vecchi in 1983, relying on the resemblance between annealing process of
a molten metal and search for the optimum solution in a general system, proposed an optimization
algorithm called simulated annealing [15]. In this algorithm, temperature is the control parameter.
Like the annealing process in which a molten metal is cooled until acquires the minimum level of
energy, a system is optimized until reaches its lowest value. In SA, at each iteration, the objective
function at current design point and a number of neighbouring points are evaluated. Afterwards, a
probability parameter decides whether the current solution to be replaced by the new solution or
not. These probabilities ultimately lead the problem to obtain its minimum value. The principal
steps of basic SA are as follows:
Step 1. Initializing a design point: The algorithm starts by randomly selecting an individual.
Step 2. Setting a cooling schedule: First, a starting and a final acceptance probability (𝑃𝑠 and
𝑃𝑓), and the number of cooling cycles (𝑁𝑐) are chosen. Then, an appropriate cooling
schedule should be set. The parameters of cooling schedule are:
𝑇𝑠 = −1
𝑙𝑛(𝑃𝑠), 𝑇𝑓 = −
1
𝑙𝑛(𝑃𝑓), 𝜂 = [
𝑙𝑛(𝑃𝑠)
𝑙𝑛(𝑃𝑓)]
1
𝑁𝑐−1
(2-8)
where 𝑇𝑠, 𝑇𝑓, and 𝜂 are the starting temperature, final temperature, and cooling factor,
respectively.
Step3. Generating a neighbouring design point: a new individual is generated by randomly
choosing a design variable and giving a small perturbation to that, while keeping the rest of
the design variables fixed.
Step 4. Evaluating the objective function at the new point: Objective function of the new
candidate is calculated and compared to the previous one. If the objective has a lower value,
12
this point is accepted as a new solution. Otherwise, one of them should be selected. The
probability of accepting a poor solution is determined by:
𝑃 = 𝑒𝑥𝑝(−𝛥𝐸
𝐾𝑇) (2-9)
where 𝛥𝐸 is the difference between objective values of the new and the old design point, 𝑇
denotes the current temperature, and 𝐾 corresponds to the Boltzmann parameter.
Step 5. Repeating steps 3 and 4: Steps 3 and 4 are repeated while all design variables are
selected and perturbed to generate new design points.
Step 6. Updating the temperature: The temperature should be updated according to the
following equation:
𝑇𝑘+1 = 𝜂𝑇𝑘 (2-10)
Step 7. Terminating the process: Steps 3 through 6 are repeated until the temperature goes
equal or below the final temperature.
13
2.3 A Comparative Study on Metaheuristic Optimization
Algorithms
Choosing the most appropriate metaheuristic optimization algorithm for a specific problem has
been always a challenging issue. In essence, there are several parameters involved in the
formulation of optimization algorithms which have significant effects on the performance of these
methods. Metaheuristics are also problem dependent, and there is not a universal algorithm which
can be implemented for all type of optimization problems, and guarantees to yield the best possible
solution.
In recent years, a few comparative studies are carried out on the performance of metaheuristic
optimization algorithms [34], [35], [36]–[41], yet just limited types of problems are investigated
in these papers, and their results are not applicable on many other cases.
In this study, to cover a broader range of applications, optimization problems are divided into two
categories: Mathematical optimization and Structural optimization. In the former, design variables
are continuous, and functions have the closed-form clear relations which could be linear single
variable to highly nonlinear multi variable functions. In the latter, on the other hand, design
variables are discrete, continuous or mixed discrete-continuous, and functions are resulted from
physical rules applied on the structures.
In the following sections, the performance of the six most popular metaheuristics, ES, FA, GA,
HS, PSO, and SA are evaluated both in mathematical and structural application by a few well-
known benchmarks widely used in the literature [42]–[44]. Since metaheuristics use stochastic
search techniques to find the optimal solution of a problem, in each run of the algorithm the
obtained optimum solution could be different. Therefore, for each benchmark all algorithms are
executed 30 times, so that they have sufficient opportunity to produce their best possible
performance. All algorithms are coded in MATLAB and compared based on the best, worst, mean,
and standard deviation of their results.
14
2.3.1 Mathematical Benchmarks
As mentioned before, all the variables in this group of problems are continuous. The termination
criteria were the number of generations, which were the same for all algorithms in each problem.
The benchmark functions are mentioned in greater details in the appendix A. The summary of their
main properties are as follows:
Table 2- 1 Summary of the main properties of test functions
Test Fcn n 𝒇(𝒙) type LI NE NI
b01 13 quadratic 9 0 0
b02 20 nonlinear 1 0 1
b03 5 quadratic 0 0 6
b04 2 cubic 0 0 2
b05 10 quadratic 3 0 5
b06 2 nonlinear 0 0 2
b07 7 polynomial 0 0 4
b08 8 linear 3 0 3
b09 2 quadratic 0 1 1
b10 6 quadratic 2 0 0
where n is the number of design variables, LI denoted the number of linear inequalities, NE
corresponds to the number of nonlinear equalities, and NI is the number of nonlinear inequalities.
Table 2-2 contains the optimization results of the benchmarks with six different approaches.
15
Table 2- 2 Performance results of six algorithms
Test
Fcn
Statistical
Results
Non-deterministic Optimization Approach
ES FA GA HS PSO SA
b01
(min)
Best -15.000 -12.614 -15.000 -14.785 -15.000 -14.398
Worst -14.972 -6.502 -12.943 -11.186 -15.000 -9.233
Mean -14.993 -9.467 -14.706 -13.771 -15.000 -12.214
Std. Dev. 0.01398 1.72630 0.77753 0.80139 0 1.44629
No. of
Evaluations 100,000 100,000 100,000 100,000 100,000 100,000
b02
(max)
Best 0.78626 0.73571 0.72524 0.69292 0.78199 0.76806
Worst 0.64847 0.34127 0.49006 0.47526 0.63763 0.58934
Mean 0.74406 0.56637 0.60395 0.56049 0.71057 0.70113
Std. Dev. 0.03322 0.11885 0.05681 0.04921 0.03564 0.04814
No. of
Evaluations 150,000 150,000 150,000 150,000 150,000 150,000
b03
(min)
Best -30665.5387 -30665.5307 -30413.8912 -30370.5159 -30665.5387 -30665.4689
Worst -30476.6488 -30665.3568 -29042.6729 -29673.3875 -30665.5387 -30663.6678
Mean -30636.1692 -30665.5050 -29817.1075 -30042.3544 -30665.5387 -30664.8474
Std. Dev. 45.35082 0.03019 330.41727 208.44806 0 0.38866
No. of
Evaluations 100,000 100,000 100,000 100,000 100,000 100,000
b04
(min)
Best -6961.812 -6961.655 -6961.780 -6844.310 -6961.814 -6956.386
Worst -6894.829 -6959.029 -6960.805 -1515.307 -6842.438 -6881.125
Mean -6958.437 -6960.624 -6961.533 -3880.321 -6955.764 -6920.784
Std. Dev. 12.18861 0.66766 0.20767 1716.55154 22.80596 19.22084
No. of
Evaluations 60,000 60,000 60,000 60,000 60,000 60,000
b05
(min)
Best 24.531 24.402 26.258 34.700 24.356 25.896
Worst 268.515 27.407 51.790 264.693 29.871 28.899
Mean 39.908 25.227 35.068 68.987 25.594 26.565
Std. Dev. 45.02757 0.72034 5.94700 44.89739 1.24417 0.62592
No. of
Evaluations 100,000 100,000 100,000 100,000 100,000 100,000
16
Test
Fcn
Statistical
Results
Non-deterministic Optimization Approach
ES FA GA HS PSO SA
b06
(max)
ES FA GA HS PSO SA
Worst 0.09582504 -6.48E-08 0.02580484 0.02432997 0.09582504 0.09579405
Mean 0.09582504 0.02413717 0.06155956 0.07373537 0.09582504 0.09581538
Std. Dev. 0 0.02690272 0.03612776 0.03219843 0 8.4713E-06
No. of
Evaluations 60,000 60,000 60,000 60,000 60,000 60,000
b07
(min)
Best 680.641 680.647 681.164 683.788 680.641 680.931
Worst 683.996 681.245 705.074 751.858 680.877 681.791
Mean 680.915 680.849 686.911 705.629 680.722 681.214
Std. Dev. 0.65009 0.18507 6.99483 19.47802 0.05806 0.18511
No. of
Evaluations 100,000 100,000 100,000 100,000 100,000 100,000
b08
(min)
Best 7195.536 7195.591 9926.272 10797.951 7537.046 7577.050
Worst 8218.339 8140.376 25472.602 17645.428 17240.510 20338.087
Mean 7544.393 7452.466 18182.085 13032.135 10603.351 10707.342
Std. Dev. 290.291698 217.34881 4539.04015 2361.70914 2853.95757 3093.36032
No. of
Evaluations 100,000 100,000 100,000 100,000 100,000 100,000
b09
(min)
Best 1.39346 1.39347 1.39355 1.39347 1.39346 1.39347
Worst 1.39346 1.39347 1.39400 1.39349 1.39346 1.39354
Mean 1.39346 1.39347 1.39379 1.39348 1.39346 1.39349
Std. Dev. 0 0 1.6386E-04 8.10E-06 0 2.455E-05
No. of
Evaluations 60,000 60,000 60,000 60,000 60,000 60,000
b10
(min)
Best -213.000 -212.546 -212.191 -204.245 -213.000 -212.999
Worst -213.000 -208.871 -200.55 -179.337 -213.000 -210.881
Mean -213.000 -211.194 -207.792 -193.3811 -213.000 -212.366
Std. Dev. 0 1.0682 3.88765 8.52819 0 0.75684
No. of
Evaluations 100,000 100,000 100,000 100,000 100,000 100,000
The results in the boldfaces show the best produced solution of each criterion. As can be seen from
the table, there is not any algorithm which always surpasses other methods. One algorithm
produces the best result among others, while the other one have the best mean or standard
deviation. Therefore, it is impossible to choose one method as the one which has the best
17
performance between all other in all desired metrics. The summary of the performance of all
approaches based on the best results obtained in the benchmarks are shown in the table below. In
cases of equal best results, the mean values are considered. The scores are in descending order, in
that 1 is for the best performance.
Table 2- 3 Performance summary of all approaches
Test Fcn ES FA GA HS PSO SA
b01 2 6 3 4 1 5
b02 1 4 5 6 2 3
b03 2 3 5 6 1 4
b04 2 4 3 6 1 5
b05 3 2 5 6 1 4
b06 1 3 4 5 1 6
b07 2 3 5 6 1 4
b08 1 2 5 6 3 4
b09 1 3 6 4 1 5
b10 1 4 5 6 1 3
Total 16 34 46 55 13 37
This table verifies that in these test problems, PSO and ES excel all others, and HS has the poorest
performance among them.
2.3.2 Structural Benchmarks
All engineering problems are included in this class. Real-world problems are those whose variables
could be continuous, discrete or mixed continuous-discrete, and their objective functions could be
noisy or discontinuous. The majority of studies on metaheuristic algorithms exploit structural
optimization benchmarks to determine the performance and effectiveness of the proposed
approaches. In the following section, four popular non-truss design optimization benchmarks are
selected, and their performance are evaluated and compared.
18
2.3.2.1 Welded Beam Design
The welded beam shown in Figure 2-1 should be designed for minimum cost of welding labor and
material [42]. The beam is made of 1010 steel and is welded to a rigid support.
Figure 2- 1 The welded beam (from Ref. [42])
The force 𝑃 is applied on the free tip and the weld is subjected to constraints on shear stress,
bending stress, buckling, and geometry. The formulation of the problem is as follows:
Minimize 𝑓(𝑥) = 𝑓(ℎ, 𝑙, 𝑡, 𝑏) = 1.10471 ℎ2𝑙 + 0.04811 𝑡𝑏 (𝐿 + 𝑙) (2-11)
subject to
𝑐1(𝑥) = 𝜏(𝑥) − 𝜏𝑑 ≤ 0 (2-12 a-e)
𝑐2(𝑥) = 𝜎(𝑥) − 𝜎𝑑 ≤ 0
𝑐3(𝑥) = ℎ − 𝑏 ≤ 0
𝑐4(𝑥) = 𝑃 − 𝑃𝑐(𝑥) ≤ 0
𝑐5(𝑥) = 𝛿(𝑥) − 0.25 ≤ 0
in which,
𝜏(𝑥) = √(𝜏′(𝑥))2
+ (𝜏′′(𝑥))2
+2𝜏′(𝑥)𝜏′′(𝑥)𝑙
2𝑅
(2-13 a-h)
19
𝜏′(𝑥) =𝑃
√2ℎ𝑙, 𝜏′′(𝑥) =
𝑀𝑅
𝐽, 𝑀 = 𝑃 (𝐿 +
𝑙
2),
𝑅 = √𝑙2
4+ (
ℎ + 𝑡
2)
2
, 𝐽 = 2 {ℎ𝑙
√2[
𝑙2
12+ (
ℎ + 𝑙
2)
2
]}
𝜎(𝑥) =6𝑃𝐿
𝑡2𝑏, 𝛿(𝑥) =
4𝑃𝐿3
𝐸𝑡3𝑏, 𝑃𝑐(𝑥) =
4.013√𝐸𝐺𝑡2𝑏6
36
𝐿2(1 −
𝑡
2𝐿√
𝐸
4𝐺)
The design variables are bounded as: 0.1235 ≤ ℎ ≤ 5.005, 0.0975 ≤ 𝑙, 𝑡 ≤ 10, and 0.1 ≤ 𝑏 ≤
5. Also, the values of needed parameters are: 𝜏𝑑 = 13600 psi, 𝜎𝑑 = 30000 psi, 𝛿𝑑 =
0.25 in, 𝐸 = 30 × 106 psi, 𝐺 = 12 × 106 psi, 𝑃 = 6000 lb, and 𝐿 = 14 in. The values of ℎ
and 𝑙 are integer multiples of 0.0065 in, and 𝑡 and 𝑏 are continuous. Results are shown in Table 2-
4.
Table 2- 4 Optimization results for the welded beam design
Design Variables Non-deterministic Optimization Approach
ES FA GA HS PSO SA
ℎ 0.2600 0.2600 0.2665 0.2600 0.2600 0.2600
𝑙 4.9725 4.9660 4.7840 5.0050 4.9920 4.9920
𝑡 9.9157 9.9235 9.9100 9.9180 9.9180 9.8996
𝑏 0.2600 0.2600 0.2665 0.2601 0.2600 0.2602
Best 2.7245 2.7251 2.7620 2.7324 2.7289 2.7290
Worst 3.0373 3.1223 2.9368 3.0520 3.0369 3.0521
Mean 2.8443 2.8691 2.8439 2.8670 2.8456 2.8327
Std. Dev. 0.1023917 0.1090161 0.0501244 0.1011327 0.1023708 0.0787761
No. of Evaluations 100,000 100,000 100,000 100,000 100,000 100,000
20
2.3.2.2 Pressure Vessel Design
The pressure vessel shown in Figure 2-2 should be designed for the minimum total cost of forming,
welding, and material. This problem was introduced first by Sandgren in 1988 [45].
Figure 2- 2 The pressure vessel (from Ref. [42])
The vessel is capped at both ends with hemispherical heads. Variables 𝑇𝑠 and 𝑇ℎ are integer
multipliers of 0.0625, and the other two variables are continuous. The problem is expressed as:
Minimize 𝑓(𝑥) = 𝑓(𝑇𝑠, 𝑇ℎ, 𝑅, 𝐿) = 0.6224 𝑇𝑠𝑅𝐿 + 1.7781 𝑇ℎ𝑅2 + 3.1661 𝑇𝑠2𝐿 + 19.84 𝑇ℎ
2𝑅
(2-14)
subject to
𝑐1(𝑥) = 0.0193 𝑅 − 𝑇𝑠 ≤ 0 (2-15 a-d)
𝑐2(𝑥) = 0.00954 𝑅 − 𝑇ℎ ≤ 0
𝑐3(𝑥) = 1296000 − 𝜋𝑅2𝐿 −4
3𝜋𝑅3 ≤ 0
𝑐4(𝑥) = 𝐿 − 240 ≤ 0
where 0.0625 ≤ 𝑇𝑠, 𝑇ℎ ≤ 6.1875, 10 ≤ 𝑅 ≤ 200, and 10 ≤ 𝐿 ≤ 240. Results are shown in Table
2-5.
21
Table 2- 5 Optimization results for the pressure vessel design
Design
Variables
Non-deterministic Optimization Approach
ES FA GA HS PSO SA
𝑇𝑠 0.8750 1.0000 1.2500 1.1250 0.8750 0.7500
𝑇ℎ 0.4375 0.5000 0.6250 0.5625 0.4375 0.3750
𝑅 45.33678 50.29449 63.79859 57.81365 44.88550 38.73205
𝐿 140.25393 96.02993 16.5183 47.46965 145.37071 224.16519
Best 5574.02797 5808.43757 5919.37736 5817.78416 5610.29533 5560.51469
Worst 6029.15120 6115.25447 18154.88548 6961.24167 5937.66299 6063.88094
Mean 5809.04460 5920.21012 8863.72811 6231.80774 5791.44022 5771.51097
Std. Dev. 109.47069 89.18604 3077.20471 383.80744 84.77829 96.43163
No. of
Evaluations 100,000 100,000 100,000 100,000 100,000 100,000
2.3.2.3 Compression Spring Design
The aim of this optimization problem, proposed by Arora in 1989 [46], is to design a compression
spring to obtain the minimum spring volume and subsequently minimum mass.
Figure 2- 3 The compression spring (from Ref. [42])
The spring shown in Figure 2-3 is sought to be designed to handle shear stress, deflection, and
geometry constraints. The problem formulation is stated as:
Minimize 𝑓(𝑥) = 𝑑(𝐷, 𝑑, 𝑛) =𝜋2𝐷𝑑2(𝑛 + 2)
4 (2-16)
22
subject to
𝑐1(𝑥) =8𝐶𝑓𝑃𝑚𝑎𝑥𝐷
𝜋𝑑3− 𝑆 ≤ 0
(2-17 a-h)
𝑐2(𝑥) =𝑃𝑚𝑎𝑥
𝐾+ 1.05(𝑛 + 2)𝑑 − 𝐿𝑓𝑟𝑒𝑒 ≤ 0
𝑐3(𝑥) = 𝑑𝑚𝑖𝑛 − 𝑑 ≤ 0
𝑐4(𝑥) = (𝑑 + 𝐷) − 𝐷𝑚𝑎𝑥 ≤ 0
𝑐5(𝑥) = 3 −𝐷
𝑑≤ 0
𝑐6(𝑥) = 𝛿𝑝 − 𝛿𝑝𝑚 ≤ 0
𝑐7(𝑥) = 𝛿𝑤 −𝑃𝑚𝑎𝑥 − 𝑃𝑙𝑜𝑎𝑑
𝐾≤ 0
𝑐8(𝑥) =𝑃𝑚𝑎𝑥 − 𝑃𝑙𝑜𝑎𝑑
𝐾+ 𝛿𝑝 + 1.05(𝑛 + 2)𝑑 − 𝐿𝑓𝑟𝑒𝑒 ≤ 0
where
𝐶𝑓 =4
𝐷𝑑
− 1
4𝐷𝑑
− 4+
0.615
𝐷𝑑
, 𝐾 =𝐺𝑑4
8𝑛𝐷3, 𝛿𝑝 =
𝑃𝑙𝑜𝑎𝑑
𝐾 (2-18 a-c)
Table 2- 6 Allowable wire diameters for compression spring design
𝒅: Wire Diameters (in)
0.0090 0.0095 0.0104 0.0118 0.0128 0.0132 0.0140
0.0150 0.0162 0.0173 0.0180 0.0200 0.0230 0.0250
0.0280 0.0320 0.0350 0.0410 0.0470 0.0540 0.0630
0.0720 0.0800 0.0920 0.1050 0.1200 0.1350 0.1480
0.1620 0.1770 0.1920 0.2070 0.2250 0.2440 0.2630
0.2830 0.3070 0.3310 0.3620 0.3940 0.4375 0.5000
23
𝐷 is a continuous, and 𝑛 is an integer number. 𝑑 can take on the values listed in Table 2-6. Also,
the value of parameters in the formula are: 𝑃𝑚𝑎𝑥 = 1000 𝑙𝑏, 𝑆 = 189 × 103 𝑝𝑠𝑖, 𝐸 = 30 ×
106 𝑝𝑠𝑖, 𝐺 = 11.5 × 106 𝑝𝑠𝑖, 𝐿𝑓𝑟𝑒𝑒 = 14 𝑖𝑛, 𝑑𝑚𝑖𝑛 = 0.2 𝑖𝑛, 𝐷𝑚𝑎𝑥 = 3.0 𝑖𝑛, 𝑃𝑙𝑜𝑎𝑑 = 300 𝑙𝑏,
𝛿𝑝𝑚 = 6.0 𝑖𝑛, and 𝛿𝑤 = 1.25 𝑖𝑛. Table 2-7 demonstrates the optimization results.
Table 2- 7 Optimization results for compression spring design
Design Variables Non-deterministic Optimization Approach
ES FA GA HS PSO SA
𝐷 1.223042 1.223042 1.223042 1.224496 1.224496 1.223042
𝑑 0.283 0.283 0.283 0.283 0.283 0.283
𝑛 9 9 9 9 9 9
Best 2.658561 2.658561 2.658561 2.661718 2.661818 2.658561
Worst 3.465398 2.658572 3.465398 3.361833 3.537483 2.659216
Mean 2.833878 2.658565 2.739245 2.938948 2.990561 2.658787
Std. Dev. 0.218972 8E-06 0.255144 0.197258 0.219509 0.000176
No. of Evaluations 100,000 100,000 100,000 100,000 100,000 100,000
2.3.2.4 Stepped Cantilevered Beam Design
The cantilevered beam shown in Figure 2-4 is supposed to be designed for the minimum material
volume. This optimization problem was originally presented by Thanedar and Vanderplaats [47].
Figure 2- 4 The stepped cantilevered beam (from Ref. [42])
Design variables of fourth and fifth segment are continuous and their bounds are: 1 ≤ 𝑏4, 30 ≤
ℎ4, 𝑏5 ≤ 5, and ℎ5 ≤ 65. The other design variables are discrete values and are chosen from the
following values: 𝑏1: {1, 2, 3, 4, 5}, 𝑏2, 𝑏3: {2.4, 2.6, 2.8, 3.1}, ℎ1, ℎ2: {45, 50, 55, 60}, and
ℎ3: {30, 31, … , 65}. In this regard, 𝑃 is 50 kN, 𝜎𝑑 is 14 kN/cm2, 𝐸 is 2 × 104 kN/cm2, ∆𝑚𝑎𝑥 is
2.7 cm, and 𝑙𝑖 (𝑖 = 1, 2, … , 5) are 100 cm. The problem can be defined as:
24
Minimize 𝑓(𝑥) = 𝑓(𝑏1, ℎ1, 𝑏2, ℎ2, 𝑏3, ℎ3, 𝑏4, ℎ4, 𝑏5, ℎ5) =
100(𝑏1ℎ1 + 𝑏2ℎ2 + 𝑏3ℎ3 + 𝑏4ℎ4 + 𝑏5ℎ5) (2-19)
subject to
𝑐1(𝑥) =6𝑃𝑙5
𝑏5ℎ52 − 𝜎𝑑 ≤ 0, 𝑐2(𝑥) =
6𝑃(𝑙5 + 𝑙4)
𝑏4ℎ42 − 𝜎𝑑 ≤ 0 (2-20 a-k)
𝑐3(𝑥) =6𝑃(𝑙5 + 𝑙4 + 𝑙3)
𝑏3ℎ32 − 𝜎𝑑 ≤ 0
𝑐4(𝑥) =6𝑃(𝑙5 + 𝑙4 + 𝑙3 + 𝑙2)
𝑏2ℎ22 − 𝜎𝑑 ≤ 0
𝑐5(𝑥) =6𝑃(𝑙5 + 𝑙4 + 𝑙3 + 𝑙2 + 𝑙1)
𝑏1ℎ12 − 𝜎𝑑 ≤ 0
𝑐6(𝑥) =106𝑃
3𝐸(
1
𝐼5+
7
𝐼4+
19
𝐼3+
37
𝐼2+
61
𝐼1) − ∆𝑚𝑎𝑥≤ 0
𝑐7(𝑥) =ℎ5
𝑏5− 20 ≤ 0, 𝑐8(𝑥) =
ℎ4
𝑏4− 20 ≤ 0
𝑐9(𝑥) =ℎ3
𝑏3− 20 ≤ 0, 𝑐10(𝑥) =
ℎ2
𝑏2− 20 ≤ 0
𝑐11(𝑥) =ℎ1
𝑏1− 20 ≤ 0
Optimization results are presented in Table 2-8.
25
Table 2- 8 Optimization results for stepped cantilevered beam design
Design Variables Non-deterministic Optimization Approach
ES FA GA HS PSO SA
𝑏1 3 4 3 4 3 4
ℎ1 60 55 60 55 60 55
𝑏2 3.1 3.1 3.1 3.1 3.1 3.1
ℎ2 55 55 55 55 55 55
𝑏3 2.6 2.6 2.6 2.6 2.6 2.6
ℎ3 51 50 51 51 52 50
𝑏4 2.281 2.286 2.226 2.389 2.756 2.238
ℎ4 43.916 44.687 44.494 42.58 40.283 44.74
𝑏5 1.936 1.815 1.750 2.021 1.843 1.768
ℎ5 34.298 35.046 34.993 33.171 34.625 35.054
Best 64967.332 68618.095 64338.139 69183.621 66031.886 68260.359
Worst 80415.294 74175.496 71820.647 72854.528 73028.421 69148.702
Mean 69607.232 70363.548 66210.175 70467.069 68548.639 68427.021
Std. Dev. 3126.525 1475.184 2198.125 1383.008 1676.413 358.608
No. of Evaluations 150,000 150,000 150,000 150,000 150,000 150,000
Based on the best objective attained in each benchmark, optimization results are summarized in
Table 2-9. Algorithms are ranked based on their best solutions from 1 to 6; 1 as the best and 6 as
the worst and those with the same performance have the same ranks. Cases in which there are more
algorithms with the same best solutions are rated according to their mean values.
Table 2- 9 Performance summary of all approaches
Test Prob. PSO GA ES FA HS SA
Case I 3 6 1 2 5 4
Case II 3 6 2 4 5 1
Case III 6 3 4 1 5 2
Case IV 3 1 2 6 5 4
Total 15 16 9 13 20 11
26
This table shows that ES is superior to other methods. In all cases ES is among the first two best
algorithms except for the third case. In this case, ES, FA, GA, and SA all give the same best
solutions, and their difference is due to their mean, standard deviation, and worst values. Therefore,
they can be counted to have the same performance.
2.4 Closing Remarks
Based on the presented results, in each benchmark the algorithm producing the best solution is
different, and in many cases the algorithm with the best answer does not necessarily produce the
best mean, worst, and standard deviation values. Furthermore, As Wolpert and Macready stated in
“No free Lunch Theorem” that there is no universally superior optimization algorithm [48], we
cannot strongly conclude that ES is the best method to optimize any problem. Yet for this thesis,
this approach will be utilized in all optimization procedures.
27
CHAPTER 3: Reliability-Based Design Optimization
of Practical Structure Using Subset Simulation and
Artificial Neural Networks
3.1 Overview
As mentioned in the introduction, the literature lacks a robust RBDO strategy that can be
implemented to design large-scale engineering structures with nonlinear performance functions
within a reasonable execution time. In this chapter, a methodology for designing the practical
structures with small failure probability criteria is proposed by leveraging artificial neural networks
(ANNs) and subset simulation (SS). In the end, to investigate the effectiveness of the proposed
approach, it is tested on a 25-bar truss structure.
3.2 Introduction
Proposing new variance reduction techniques and RBDO algorithms which alleviate the reliability
design of engineering systems has been of special interests in recent years. Importance sampling
[49], Gibbs sampling [50], and Metropolis-Hasting algorithm [51] are all a few techniques which
highly reduce the sampling effort that seems to be the main obstacle in MCS. Another barrier to
efficient reliability analysis, by the use of standard MCS, is to generate samples from rare failure
events. In these conditions, a novel sampling approach called subset simulation has been proposed
[11]. Despite the existence of all these variance reduction techniques, designing a complex
structure satisfying reliability expectations remains an intensive computational task and other
remedies should be sought to pave the RBDO path of practical applications. To further diminish
the number of function evaluations, surrogate models could be implemented to approximate the
value of performance functions at each design point which incredibly improves the cost efficiency
28
of the process. Utilizing surrogate models in structural reliability has expanded the boundary of
RBDO from purely academic applications to more industrial purposes in the recent years.
Polynomial response surfaces [52], [53], support vector machines [54], [55], artificial neural
networks (ANN) [56], and kriging [57] are among the most renowned surrogates in the literature.
In this study, ANNs are chosen due to their efficiency in handling nonlinear and non-smooth
performance functions. By training proper number of input-output pairs of samples extracted from
actual analysis, a multi-layer network is made which can be exploited for appraising deterministic
and reliability constraints. For complex systems where actual simulation time is a hassle, this
shortcut model extensively saves the computational costs. From recent progresses in training
algorithms, the application of neural networks in reliability-based design of industrial components
is hastened [58]–[61]. However, the required CPU time for more sophisticated systems is still a
barrier and other treatments are needed to be deliberated in order for further speeding up the
implementation of RBDO in real-world practices.
In the present study, a methodology for designing the practical structures with small failure
probability criteria is proposed by leveraging ANN and SS. First, by generating a number of input-
output pairs a network is constructed that can be then applied to approximate the deterministic
constraints. Afterwards, by the use of SS, an adequate number of probabilistic samples are
produced to create another ANN to estimate the reliability performance functions. The two ANNs
created in the process strikingly reduce the computation time. By taking advantage of an
optimization algorithm, the structure will be optimized and designed satisfying probabilistic
constraints.
29
3.3 Problem Definition
In mathematical terms, an RBDO problem can be formulated as follows:
min𝑋
𝐶(𝑋, 𝑍) (3-1 a-c)
Subject to 𝑃𝐹(𝑋, 𝑍) ≤ �̃�𝐹
𝑔𝑖(𝑋, 𝑍) ≤ 0, 𝑖 = 1, … , 𝑘
where 𝑋 = [𝑥1, 𝑥2, … , 𝑥𝑛] and 𝑍 = [𝑧1, 𝑧2, … , 𝑧𝑞] denote the design vector and uncertainty variable
vector, respectively. Also, 𝐶(𝑋, 𝑍) is the system fitness function, such as structure weight or
fabrication cost; 𝑃𝐹(𝑋, 𝑍) is the limit state function representing the probability of failure of the
structure which should not violate the threshold failure probability, �̃�𝐹; and 𝑔𝑖(𝑋, 𝑍), 𝑖 = 1, … , 𝑘,
corresponds to deterministic functions or performance functions which separates the design space
into a safe,𝑔𝑖(𝑋, 𝑍) ≤ 0, and failure, 𝑔𝑖(𝑋, 𝑍) ≥ 0, regions. The random vector, 𝑍, embraces the
uncertainties in the material properties and external loading conditions of the system. The other
source of uncertainty imposed on the system is laid within the design vector 𝑋 which might be due
to the manufacturing tolerances inherent in real-world cases.
To determine the probability of failure, an 𝑛-fold integral over the failure region should be
calculated by
𝑃𝑓 = ∫ ∫ ⋯ ∫ 𝑓(𝑋, 𝑍)𝑑𝑥1 𝑑𝑥2 … 𝑑𝑥𝑛 𝑑𝑧1 … 𝑑𝑧𝑞
𝑔(𝑋,𝑍)≥0
(3-2)
in which 𝑓(𝑋, 𝑍) is the joint probability density function (PDF) of random variables. Without loss
of generality, in engineering problems, random variables can be treated as independent variables,
and so the joint PDF can be stated as
𝑓(𝑋, 𝑍) = ∏ 𝑓𝑖(𝑥𝑖) ∏ 𝑓𝑗(𝑧𝑗)
𝑞
𝑗=1
𝑛
𝑖=1
, (3-3)
30
where 𝑓𝑖(𝑥𝑖) and 𝑓𝑗(𝑧𝑗) are one-dimensional PDF of each design and uncertain variables,
respectively.
Practically, deciding on whether a system with several performance criteria fails at a design point
relies on the scheme that the constraint functions are related. In a series system, failure of any of
the components results in the failure of the whole system. In mathematical expression, the ultimate
failure probability is
𝑃𝐹(𝑋, 𝑍) = max(𝑃𝐹)𝑖 , 𝑖 = 1,2, … , 𝑘. (3-4)
In a parallel arrangement, on the contrary, a system fails when all components fail. In other words:
𝑃𝐹(𝑋, 𝑍) = min(𝑃𝐹)𝑖 , 𝑖 = 1,2, … , 𝑘. (3-5)
3.4 Subset Simulation Method
Calculating the failure probability relies on the evaluation of the multi-fold integral expressed in
Equation 3-2. In the high-dimensional design space and complicated failure region geometry, the
computation of the integral becomes an intensive task. In these conditions, sampling methods such
as Monte Carlo simulation (MCS) has been shown to be a reliable method which is insensitive to
the type and dimension of the problem. The probability of failure can be attained by standard MCS
as
𝑃𝐹 ≈1
𝑁∑ 𝐼𝐹𝑖
(𝑋, 𝑍)
𝑁
𝑖=1
, (3-6)
where 𝐼𝐹(𝑋, 𝑍) is an indicator function defined as
𝐼𝐹(𝑋, 𝑍) = {1 if 𝑋, 𝑍 ∈ 0 if 𝑋, 𝑍 ∈
failure region
safe region (3-7)
As the number of samples increases to infinity, 𝑃𝐹 converges to the exact probability of failure
[62]. The main shortcoming of standard MCS is its incapability in calculating the rare failure
events. The coefficient of variation of 𝑃𝐹 calculated by direct MCS can be determined by
31
𝛿𝑃𝐹= √
1 − 𝑃𝐹
𝑁. 𝑃𝐹, (3-8)
so for a 𝑃𝐹 = 10−4 with a 𝛿𝑃𝐹 of 10−2, by approximately 108 samples from the design space
should be generated which forces a massive computational costs on the process. Although by
employing some sampling techniques, such as importance sampling, the sample size reduces to
5%-20% of that of standard MCS, MCS still remains an inappropriate choice in high-dimensional
spaces [63]. In order to overcome the aforementioned drawbacks, subset simulation (SS) has been
introduced which considerably diminishes the heavy computational hassle of brutal MCS in
estimating small failure probabilities [11]. Subset simulation is based on the concept that a rare
failure event can be broken down into a sequence of larger intermediate failure events which are
more probable to occur. Thus, a small probability is expressed as a product of greater conditional
ones needing substantially fewer samples. The samples for each subset are generated using Markov
chain MCS along with Metropolis-Hastings algorithm. It has been showed that for the failure
probability of 10−6, the required sampling size of SS is 1030 times smaller than that of standard
MCS [11].
3.4.1 Subset Simulation Algorithm
If 𝐹 denotes the targeted failure region, then a decreasing sequence of 𝑚 intermediate failure events
can be defined as:
𝐹1 ⊃ 𝐹2 ⊃ ⋯ ⊃ 𝐹𝑚 = 𝐹, (3-9)
so that 𝐹 = ⋂ 𝐹𝑖𝑚𝑖=1 , and each 𝐹𝑖 is corresponding to the exceedance of the performance function
over a limit value:
𝐹𝑖 = {𝑋, 𝑍: 𝑔(𝑋, 𝑍) > 𝑔𝑖}, (3-10)
Accordingly, the sequence of intermediate limits is: 𝑔1 < 𝑔2 < ⋯ < 𝑔𝑚 = 𝑔. Therefore, the final
failure probability is computed by the multiplication of conditional probabilities of each subset:
32
𝑃𝐹 = 𝑃(𝐹1) ∏ 𝑃(𝐹𝑖+1|𝐹𝑖)
𝑚−1
𝑖=1
. (3-11)
The main idea of this approach is that intermediate limits are not specified in advance, and they
will be chosen in a way that 𝑃(𝐹1) and 𝑃(𝐹𝑖+1|𝐹𝑖) are sufficiently large so that their values can
be determined by standard MCS. For instance, to produce a 10−4 probability, three conditional
limits, 𝑔𝑖: 𝑖 = 1,2,3, can be selected to have 𝑃(𝐹1) and 𝑃(𝐹𝑖+1|𝐹𝑖): 𝑖 = 1,2,3, as 0.1 which are
more likely to happen than the original event. In Equation 3-11, the first probability, 𝑃(𝐹1) can be
estimated by direct MCS:
𝑃(𝐹1) ≈1
𝑁∑ 𝐼𝐹1
(𝑋𝑘, 𝑍𝑘)
𝑁
𝑘=1
, (3-12)
as it is for Equation 3-6, 𝐼𝐹1, here, is the indicator function of the first failure region which takes 1
when 𝑔(𝑋, 𝑍) > 𝑔1 and zero within the safe boundaries. To evaluate conditional probabilities,
𝑃(𝐹𝑖+1|𝐹𝑖), a sufficient number of samples should be generated from the conditional distribution
function defined as
𝑓(𝑋, 𝑍|𝐹𝑖) = 𝐼𝐹𝑖(𝑋, 𝑍)𝑓(𝑋, 𝑍) (3-13)
In other words, the samples for the (𝑖 + 1)th intermediate failure event should lie in the 𝑖th failure
region, 𝐹𝑖. The samples can be gathered using Markov chain Monte Carlo simulation based on the
modified Metropolis algorithm. The conditional probability at (𝑖 + 1)th subset, therefore, can be
achieved by
𝑃(𝐹𝑖+1|𝐹𝑖) ≈1
𝑁𝑖+1∑ 𝐼𝐹𝑖+1
(𝑋𝑘, 𝑍𝑘)
𝑁𝑖+1
𝑘=1
. (3-14)
Since, in the simulation, the value of intermediate probabilities are appointed prior to their
estimation, at each subset level, the limit value, 𝑔𝑖, is determined in a way that 𝑁𝑖 . 𝑃(𝐹𝑖|𝐹𝑖−1)
samples are located in the failure region where 𝑔(𝑋, 𝑍) > 𝑔𝑖. The process continues until 𝑔𝑖 ≥
𝑔𝑚 = 𝑔. A simple schematic of a three-state subset simulation is shown in Figure 3-1.
33
Figure 3- 1 A four-level subset simulation
3.4.2 Markov Chain Monte Carlo Simulation
Markov chain Monte Carlo simulation (MCMCS) is the process of simulating samples from an
arbitrary probability distribution function (PDF) using Markov chain mechanism. The mechanism
is constructed in a way that the samples are concentrated more in the most important regions [64].
The technique, first, invented by Metropolis et al. in 1953 to simulate a liquid in equilibrium with
its gas phase [65]. They realized that to study the thermodynamic equilibrium of this system the
exact dynamics simulation of it from the very beginning step of the process was not required.
(a) First subset level (b) Second subset level
(c) Third subset level
Ultimate
limit value
First
limit
value Second
limit
value
Third limit
value
Fourth or
ultimate
limit value
34
Instead, they could investigate the equilibrium state by just building a Markov chain having the
same equilibrium distribution. The algorithm they used to generate the Markov chain is called
Metropolis algorithm which was further completed later by Hastings in 1970 by the advent of
modern computers [51], and thereafter the algorithm has been called Metropolis-Hastings
algorithm.
3.4.2.1 Markov Chains
Let 𝑋(𝑡) denotes the value of a random variable at time 𝑡, and 𝒳 is the state space which
encompasses all possible values of 𝑋. In other words, 𝑋(𝑡)𝜖 (𝒳 = {𝑠1, 𝑠2, … , 𝑠𝑖}) where 𝑠𝑖
corresponds to the current state. The sequence of random variables, {𝑋(1), 𝑋(2), … , 𝑋(𝑡)}, comprises
a Markov chain if
𝑃(𝑋(𝑡) = 𝑠𝑖|𝑋(1) = 𝑠1, 𝑋(2) = 𝑠2, … , 𝑋(𝑡−1) = 𝑠𝑖−1) = 𝑃(𝑋(𝑡) = 𝑠𝑖|𝑋
(𝑡−1) = 𝑠𝑖−1), (3-15)
which means that the transition probabilities between different states in the state space depends
solely on the current state of the chain. Normally, a Markov chain is specified by its transition
probability, 𝑃(𝑋(𝑡)|𝑋(𝑡−1)), which is the probability that the process moves from state 𝑠𝑖−1 to 𝑠𝑖.
The chain is homogeneous if this probability remains constant during the entire process.
Suppose 𝑝𝑖−1(𝑡 − 1) = 𝑃(𝑋(𝑡−1) = 𝑠𝑖−1) denotes the probability that the chain is at state 𝑠𝑖−1 at
time 𝑡 − 1, and 𝑝(𝑡 − 1) is a vector which contains the values of the state space probabilities at
time 𝑡 − 1. Also, 𝐏 represents the probability transition matrix whose (𝑖, 𝑗)th element, 𝑃𝑖𝑗,
corresponds to the probability of moving from state 𝑖 to state 𝑗. Therefore, considering Equation
3-15, 𝑝(𝑡), the probability vector at time 𝑡 can be obtained as
𝑝(𝑡) = 𝑝(𝑡 − 1)𝐏, (3-16)
and since the probability at each time depends only on the previous state, 𝑝(𝑡) can be stated as
𝑝(𝑡) = 𝑝(1)𝐏𝑖, (3-17)
35
where 𝑝(1) is the probability vector of the initial state. The transition matrix or the chain is
irreducible if all states communicate with each other. That is, there is always a possibility to go
from any of the states to all other states in a finite number of steps 𝑛. In other words:
∀𝑖, 𝑗 ∈ ℝ, ∃𝑛 𝑃𝑖𝑗𝑛 > 0, (3-18)
Also, the chain is said to be aperiodic if the greatest common divisor of 𝐷𝑖, gcd(𝐷𝑖), is equal to 1,
where 𝐷𝑖 is the set consisting all integers 𝑛 ≥ 1 such that 𝑃𝑖𝑖𝑛 > 0. If gcd(𝐷𝑖) ≥ 2 then it is the
period of state 𝑖. A Markov chain converges to a stationary distribution provided that it is
irreducible and aperiodic. Put differently, after a sufficient number of iterations, the chain stabilizes
at an invariant distribution regardless of the probability of the initial state.
3.4.2.2 Modified Metropolis Algorithm
The principal step in utilizing MCS is to generate samples from a joint PDF. Metropolis algorithm
is the most common MCMCS method for obtaining samples from a complex probability
distribution. Suppose the goal is to draw samples from a multi-dimensional PDF, 𝑓(𝜃) = 𝑝(𝜃)/𝐾
where 𝐾, the normalizing constant, is not known or difficult to calculate. Let 𝑓𝑖(𝜃𝑖), 𝑖 = 1,2, … , 𝑛
be the one-dimensional PDF of variable 𝜃𝑖, so the sequence of samples from this distribution by
Metropolis algorithm can be drawn as follows:
1. Initiate the process from an arbitrary initial guess such as 𝜃1𝑖 , so that 𝑓𝑖(𝜃1
𝑖) > 0.
2. Given the current value of 𝜃𝑘𝑖 , generate a candidate sample 𝜃𝑘+1
𝑖 from a symmetry proposal
PDF, 𝑞(𝜃𝑘+1𝑖 |𝜃𝑘
𝑖 ), which is the probability of producing 𝜃𝑘+1𝑖 given 𝜃𝑘
𝑖 . The symmetry
property of the distribution implies that 𝑞(𝜃𝑘+1𝑖 |𝜃𝑘
𝑖 ) = 𝑞(𝜃𝑘𝑖 |𝜃𝑘+1
𝑖 ).
3. Calculate the ratio
𝛼 =𝑓𝑖(𝜃𝑘+1
𝑖 )
𝑓𝑖(𝜃𝑘𝑖 )
(3-19)
4. Accept the candidate 𝜃𝑘+1𝑖 as a new sample if 𝛼 > 1. If 𝛼 < 1 then accept 𝜃𝑘+1
𝑖 as the new
sample with the probability of 𝛼. Otherwise, reject the candidate, and take 𝜃𝑘𝑖 as the current
sample (i.e., 𝜃𝑘+1𝑖 = 𝜃𝑘
𝑖 ), and return to step 2.
36
In the context of probability assessment, since a design space could be multi-dimensional, the
modified version of Metropolis algorithm should be implemented. To that end, after the sample is
updated in each dimension, the following step is added to the original method:
5. Check the location of 𝛉𝑘+1 = {𝜃𝑘+11 , 𝜃𝑘+1
2 , … , 𝜃𝑘+1𝑛 }. If 𝛉𝑘+1 ∈ 𝐹𝑗 accept it as the next
sample. Otherwise, reject it and set 𝛉𝑘+1 = 𝛉𝑘. 𝐹𝑗 is the failure region of the previous
intermediate level.
The last step ensures that the samples lie in 𝐹𝑗, and so can be used to compute the subset failure
probabilities. The other matter in the algorithm is the choice of the proposal PDF. It has been
proved that the sampling effort is insensitive to the proposal PDF, and accordingly, the more
straightforward distribution will comply with the simulation requirements [61].
3.5 Artificial Neural Networks
An artificial neural network (ANN) is an information processing model inspired by biological
neural networks that can be used to approximate the response of an unknown complex function by
training a sufficient number of input-output pairs. An ANN performs as a black box containing a
web of interconnected neurons, similar to that in the human brain, which is capable of predicting
the output of a system in a fraction of the real computation time with an acceptable prescribed
error relative to the true output. Therefore, the implementation of ANNs extensively reduces the
required processing time of an intensive numerical analysis task which makes the technique a
superior tool to handle the large-scale real-world problems. Nowadays, by ever-accelerating pace
of computing technology, the application of ANNs is hastening in numerous areas, such as stock
market, medicine, and optimization [66]–[68].
The first step of creating an ANN is to determine its architecture. A network, as shown in Figure
3-2, is consisted of one input layer, one output layer, and multiple hidden layers, each of them
containing a group of neurons or processing elements. The principal parameters of an ANN
topology are the number of hidden layers, number of neurons in each layer, and connection patterns
across the layers.
37
Figure 3- 2 Fully connected ANN configuration
The number of processing elements in the hidden layers has an important impact on the network
efficiency. However, this value cannot be determined prior to the process, and is specified by trial
and error [69].
Generally, ANNs can be classified into two main categories based on the connection patterns
between the units: feed-forward networks and recurrent networks. In the former, the information
from input to output layers strictly flow in the forward direction, while in the latter, there are
feedback connections between units within the network. In this study, feed-forward neural network
trained by back-propagation algorithm is used. The basic idea of this approach is stated in the
following section. A more thorough study can be found in [70].
3.5.1 Feed-Forward Back-Propagation Network
Backward propagation of errors or in abbreviation back-propagation algorithm is a method to
monitor the learning process of a data set to construct a multi-layer neural network. The first and
last layers are input and output layers, respectively, and the middle layers are called hidden layers,
Input Layer
1st Hidden
2nd Hidden
Output Layer Input
layer
1st hidden
layer
2nd hidden
layer Output
layer
38
which can be more than one. In this method, by the use of an optimization method, such as gradient
descent, the connection weights between the neurons are modified, so the error between the
network output (observed) and the target output (real) is minimized. Since updating the weight
values is performed in a backward layer-by-layer manner from output to input, this algorithm is
named back-propagation. The creation of a feed-forward back-propagation network is carried out
in two phases: feed-forward phase and back-propagation of errors phase.
3.5.1.1 Feed-Forward Phase
Suppose 𝑥𝑞𝑖 , 𝑖 = 1,2, … , 𝑛 denote the inputs to neuron 𝑖 in layer 𝑞. In this phase, the value of the
output of each processing element, and the required derivatives are calculated. The output of
neuron 𝑗 in layer 𝑝 is expressed as
𝑦𝑝𝑗
= 𝐹 (∑ 𝑤𝑝𝑖𝑗
𝑥𝑞𝑖
𝑛𝑞
𝑖=1
+ 𝑏𝑝𝑗) (3-20)
where 𝑤𝑝𝑖𝑗
is the connecting weight of neuron 𝑗 in layer 𝑝 (target layer) and neuron 𝑖 in layer 𝑞
(source layer); 𝑏𝑝𝑗 is the bias factor; and 𝐹() is a differentiable function called activation function
which produces the output of a neuron. The most commonly used activation functions are binary
step function, sigmoid function, and bipolar sigmoid function [71]. Due to the ability of the
sigmoid function in handling both large and small inputs, in this study, it is selected as the
activation function which is defined as
𝐹(𝑥) =1
1 + 𝑒−𝑥. (3-21)
Also, the derivative of the function can be calculated by
𝑑𝐹 = 𝐹(1 − 𝐹). (3-22)
39
3.5.1.2 Back-Propagation of Errors Phase
In this stage, the weights are modified backwardly, from the output to input layers, to reduce the
discrepancy between the target and observed values. The error of each neuron in the output layer
are obtained by
𝐸𝑚𝑗
= 𝑇𝑚𝑗
− 𝑦𝑚𝑗
(3-23)
where 𝑇𝑚𝑗 and 𝑦𝑚
𝑗 are target and computed outputs of neuron 𝑗 in the output layer 𝑚, respectively.
Subsequently, the weight changes in this layer are calculated using the following equations:
𝛿𝑚𝑗
= 𝑑𝐹 (∑ 𝑤𝑚𝑖𝑗
𝑥𝑞𝑖
𝑛𝑞
𝑖=1
+ 𝑏𝑚𝑗
) . 𝐸𝑚𝑗
= (∑ 𝑤𝑚𝑖𝑗
𝑥𝑞𝑖
𝑛𝑞
𝑖=1
+ 𝑏𝑚𝑗
) . (1 − ∑ 𝑤𝑚𝑖𝑗
𝑥𝑞𝑖
𝑛𝑞
𝑖=1
+ 𝑏𝑚𝑗
) . 𝐸𝑚𝑗
(3-24 a-b)
∆𝑤𝑚𝑖𝑗
= 𝜂. 𝛿𝑚𝑗
. 𝑦𝑞𝑖
in which 𝑞 is the layer before the output layer, and 𝜂 corresponds to the learning rate coefficient
which normally takes a value between 0.01 and 0.9. One may also consider the influence of the
previous weight change in the current weight update by employing a momentum term 𝛼, which
eventually yields the following expression for the weight modifications at iteration 𝑡 + 1:
(∆𝑤𝑚𝑖𝑗
)𝑡+1
= 𝛼. (∆𝑤𝑚𝑖𝑗
)𝑡
+ 𝜂. 𝛿𝑚𝑗
. 𝑦𝑞𝑖 . (3-25)
The weight changes for the neurons in the hidden layers can be acquired by
𝛿𝑞𝑖 = 𝑑𝐹 (∑ 𝑤𝑞
𝑟𝑖 𝑥𝑘𝑟
𝑛𝑘
𝑟=1
+ 𝑏𝑞𝑖 ) . (∑ 𝛿𝑝
𝑗. 𝑤𝑝
𝑖𝑗
𝑛𝑝
𝑗=1
) (3-26 a-b)
(∆𝑤𝑞𝑟𝑖)
𝑡+1= 𝛼. (∆𝑤𝑞
𝑟𝑖)𝑡
+ 𝜂. 𝛿𝑞𝑖 . 𝑦𝑘
𝑟
40
where the layers 𝑝 and 𝑘 denote one layer after and before the desired hidden layer 𝑞, respectively.
Accordingly, the modified connection weight for the next iteration can be computed by
(𝑤)𝑡+1 = 𝑤𝑡 + (Δ𝑤)𝑡+1. (3-27)
To initiate the process, the connection weights are arbitrarily spread throughout the network, and
then modified until the favorable level of accuracy for the network output is achieved. Hence,
creating an ANN is an optimization problem in which the goal is to minimize the error of the
network responses relative to the real values. The error function mentioned in Equation 3-23 is the
simplest cost function defined for this optimization problem. The other common function, utilized
in this study, is the sum of squares error (SSE) expressed as
𝐸 =1
𝑁𝑖𝑁𝑜∑ ∑(𝑇𝑖𝑗 − 𝑦𝑖𝑗)
𝑁𝑜
𝑗=1
𝑁𝑖
𝑖=1
(3-28)
where 𝑁𝑖 is the total number of training pairs, and 𝑁𝑜 is the number of output elements.
41
3.6 Proposed Reliability-Based Design Optimization Framework
As aforementioned in the introduction, the proposed framework aims to optimize the real-world
structures for probabilistic constraints in a realistic time frame which is impossible by conventional
RBDO approaches. This strategy proceeds in three principal stages:
1. Constructing ANNs to determine the deterministic constraints: Due to the complexity
of the majority of engineering structures, the actual simulation of them is an intensive task.
Hence, considering the capabilities of ANNs, a network can be constructed as a
replacement for the real model to predict the deterministic constraints. The input-output
pairs for the network should be acquired by running the simulation and collecting the
essential data.
2. Creating ANNs to evaluate failure probability at each design candidate: The principal
obstacle against RBDO problems is the reliability assessment phase. A remedy to
disburden the procedure is the implementation of SS which significantly reduces the
sampling efforts. The prime rationale for composing the deterministic ANNs, in the first
stage, is that the reliability estimation, at each design point, even by using SS demands a
considerable number of samples whose execution time is a hefty barrier if the direct
analysis is performed. The deterministic network acts as a shortcut model which
significantly reduces the computational costs of assessing the reliability of the structure at
a design point. Hence, by leveraging SS and deterministic ANNs, training samples for
constructing probabilistic ANNs can be generated. These networks can then be utilized to
determine the probabilistic constraints.
3. Optimizing the structure considering both deterministic and probabilistic
constraints: In order to propel the optimization process, an optimization method, ES here,
is employed, and at each design candidate, both deterministic and probabilistic constraints
are checked to ensure the feasibility of the solution.
The application of subset simulation, deterministic ANNs, and probabilistic ANNs can
substantially decrease the computational time. The optimization architecture is demonstrated in
Figure 3-3.
42
Figure 3- 3 The proposed optimization framework
Generate Samples for
Deterministic Constraints
Construct the
Deterministic ANN
Structural
Simulation
Generate Samples for
Probabilistic Constraints
Construct the
Reliability ANN
Subset
Simulation
Optimization
Process
(ES)
Start
End
43
3.7 RBDO of a 25-bar Structure
In order to assay the performance of the proposed framework a 25-bar truss structure, as described
by Schmit and Fleury [72] and shown in Figure 3-4, is sought to be designed for the expected
reliability.
Figure 3- 4 The 25-bar truss structure (from Ref. [72])
The objective is to minimize the weight of the structure. Young’s modulus and the material density
of the links are 104 ksi and 0.1 lbm/in3, respectively. The design variables are cross-sectional
area of individual members, which are divided into 8 groups as depicted in Table 3-1. These
variables are discrete values selected from the set mentioned in Table 3-2. The structure loading
condition is also presented in Table 3-3.
44
Table 3- 1 The members of the 8 groups
Group Truss members
1 1
2 2-5
3 6-9
4 10, 11
5 12, 13
6 14-17
7 18-21
8 22-25
Table 3- 2 The discrete values of bar areas
Choice A
(𝒊𝒏𝟐) Choice
A
(𝒊𝒏𝟐) Choice
A
(𝒊𝒏𝟐) Choice
A
(𝒊𝒏𝟐) Choice
A
(𝒊𝒏𝟐)
# 1 0.1 # 7 0.7 # 13 1.3 # 19 1.9 # 25 2.5
# 2 0.2 # 8 0.8 # 14 1.4 # 20 2.0 # 26 2.6
# 3 0.3 # 9 0.9 # 15 1.5 # 21 2.1 # 27 2.8
# 4 0.4 # 10 1.0 # 16 1.6 # 22 2.2 # 28 3.0
# 5 0.5 # 11 1.1 # 17 1.7 # 23 2.3 # 29 3.2
# 6 0.6 # 12 1.2 # 18 1.8 # 24 2.4 # 30 3.4
Table 3- 3 The loading condition of the structure
Node 𝑷𝒙 (𝒍𝒃) 𝑷𝒚 (𝒍𝒃) 𝑷𝒛(𝒍𝒃)
1 1000 -10000 -10000
2 0 -10000 -10000
3 500 0 0
6 600 0 0
The deterministic constraints are imposed on the displacement of each node and the allowable
stress of each member. In that, the displacements should be smaller than 0.35 in along all
directions and the stress in each link, in both tension and compression, should not exceed 40 ksi.
The deterministic constraints are in a serial configuration in which the failure of any will lead to
the collapse of the whole structure. Due to uncertainties available in the system, the structure is
meant to satisfy the maximum failure probability of 10−4. The random variables are the external
loads, elastic moduli, and cross-section of members whose distribution type and parameters are
expressed in Table 3-4.
45
Table 3- 4 The characteristics of the random variables
Random Variable Distribution Mean Dispersion
P1, P2, P3 and P4 Extreme Value I P1, P2, P3 and P4 10%
Cross sections Uniform A 5%
Elastic Modulus Normal E 8%
Since the design variables are chosen from a set of discrete values, and no closed-form relation for
the constraints are available, a gradient-free optimization algorithm should be involved to carry
out the optimization task. In this study, ESs are taken as the optimization technique, due to its
ability in handling nonlinear structural problems [73] as reviewed in chapter 2.
In the presented case study, a (𝜇 + 𝜆)-ES with 𝜆 = 𝜇 = 25 and 500 iterations is utilized; to
construct the ANNs, both probabilistic or deterministic networks, 1000 proper training pairs, are
generated for each one; and for the probability calculations, a 4-intermediate-level SS with a
conditional failure probability equal to 0.1 is elected. The number of samples used at each level is
400 which makes the total of 1480 samples for evaluating the probability at each design point.
Gradient-free optimization algorithms, in essence, are stochastic search techniques, and each time
they are run, they do not necessarily produce the exact same best optimal result. Thus, to increase
the chance of finding the best optimum, the optimization process is executed 15 times.
The most time consuming part of a structural RBDO is its reliability assessment portion whose
evaluation time is a function of two factors: (i) the number of real structural simulations to gather
the required information to calculate the reliability of the structure or create desired ANNs; (ii) the
number of ANN calls to determine deterministic or probabilistic performance functions. By the
same token, the simulation time of a complete optimization process depends on the programming
skills and the computer specifications. Accordingly, the computation time may not be an
appropriate index to compare the efficiency of different approaches. In this study, therefore, the
number of real structural simulations and ANN calls, both deterministic and probabilistic, are
considered as the indicator.
All required codes were written in MATLAB, and run on a 32GB RAM-Core i7 CPU platform.
The optimization results produced by each approach are reported in Table 3-5.
46
Table 3- 5 The performance summary of optimization approaches
Design variables
Optimization approaches
DBO DBO-
DANN RBDO-MCSa RBDO-
MCSa&DANN
RBDO-
DANN&PANN
A1 (𝒊𝒏𝟐) 0.1 0.1 0.1 0.2 0.2
A2 (𝒊𝒏𝟐) 0.4 0.4 2.5 2.6 2.6
A3 (𝒊𝒏𝟐) 3.4 3.4 3.4 3.4 3.4
A4 (𝒊𝒏𝟐) 0.1 0.1 0.1 0.1 0.1
A5 (𝒊𝒏𝟐) 2.2 2.2 0.2 0.3 0.3
A6 (𝒊𝒏𝟐) 1 1 1.3 1.2 1.2
A7 (𝒊𝒏𝟐) 0.4 0.4 1.7 1.7 1.7
A8 (𝒊𝒏𝟐) 3.4 3.4 3.4 3.4 3.4
Optimum weight
(𝒍𝒃) 484.3278 484.3278 679.8821 680.1066 680.1066
Mean 492.3455 492.0089 680.0120 680.8126 680.9769
Standard
deviation 8.5490 10.8373 0.3429 1.0025 1.2598
Failure
probability 0.4386 0.4386 9.7780 × 10−5 7.7869 × 10−5 7.7869 × 10−5
No. of structural
simulations 12500 1000 1.25 × 109 1000 1000
No. of ANN calls 0 12500 0 1.25 × 109 1.4925 × 106
a 100,000 simulations for probability calculation at each point
In Table 3-5, DBO denotes the conventional deterministic-based optimization approach, in which
probabilistic constraints are ignored; DBO-DANN stands for the deterministic optimization where
the deterministic ANNs are used to estimate the deterministic constraints; RBDO-MCS
corresponds to the reliability-based design optimization by the use of direct MCS considering both
deterministic and probabilistic constraints; RBDO-MCS&DANN is the RBDO in which MCS is
utilized for the probability evaluations, and ANNs are exploited to determine the deterministic
47
constraints; and RBDO-SS&PANN, the proposed approach in this study, denotes the RBDO where
both deterministic and probabilistic constraints are calculated by leveraging SS and ANNs.
As can be seen from this table, by taking into account the reliability expectations, the optimum
weight has increased by approximately 40%. Also, the failure probability of the structure with the
DBO solution is 0.4386 which is notably larger than the target value. This well endorses the
significance of considering uncertainties in the design process of critical structures in order to
prevent unforeseen failures. Tables 3-6 to 3-9 show the stresses and displacements of the structure
for DBO and RBDO-SS&PANN optimum designs.
As mentioned before, the computation time is an important issue in optimization problems. In
order to rank the performance of different techniques implemented for this case study, one can
refer to the number of structural simulations and ANN calls presented in Table 3-5. Generally
speaking, the execution time of a single structural simulation is considerably greater than that of
one ANN call. As an instance, in this benchmark, a single simulation takes about 100 times longer
than an ANN call. For more sophisticated structures, the difference is more crucial. Consequently,
it is apparent from the table that DBO-DANN is superior to DBO in case of optimizing the
structure satisfying just deterministic constraints. The dominance of using ANN is even more
noticeable in the reliability approaches. Although RBDO-MCS has the best mean and standard
deviation among other RBDO techniques, the huge number of simulations required in this
approach, makes it the last choice for RBDO applications. Also, the number of deterministic and
probabilistic ANN calls strongly confirms the advantage of the proposed approach over RBDO-
SS&DANN.
Table 3- 6 Truss stresses in deterministic optimality
Truss
No.
Stress
(𝒌𝒔𝒊)
Truss
No.
Stress
(𝒌𝒔𝒊)
Truss
No.
Stress
(𝒌𝒔𝒊)
Truss
No.
Stress
(𝒌𝒔𝒊)
Truss
No.
Stress
(𝒌𝒔𝒊)
#1 0.2279 #6 2.3249 #11 -0.8168 #16 1.8036 #21 -4.4569
#2 0.8675 #7 -5.4560 #12 1.6782 #17 -4.2215 #22 -5.6910
#3 3.0145 #8 2.5316 #13 -4.0657 #18 2.0352 #23 2.9290
#4 -5.3149 #9 -5.2446 #14 2.0937 #19 1.6472 #24 2.4411
#5 -3.1190 #10 -0.7246 #15 -3.9242 #20 -3.8608 #25 -6.2073
48
Table 3- 7 Nodal displacements in deterministic optimality
Node
No.
Displacement (𝒊𝒏)
𝒙-direction
Node
No.
Displacement (𝒊𝒏)
𝒚-direction
Node
No.
Displacement (𝒊𝒏)
𝒛-direction
#1 0.0332 #1 -0.3498 #1 -0.0475
#2 0.0350 #2 -0.3474 #2 -0.0510
#3 -9.3863e-4 #3 0.0083 #3 0.0579
#4 0.0116 #4 0.0078 #4 0.0557
#5 -0.0081 #5 0.01040 #5 -0.1243
#6 0.0224 #6 0.0137 #6 -0.1240
Table 3- 8 Truss stresses in reliability optimality
Truss
No.
Stress
(𝒌𝒔𝒊)
Truss
No.
Stress
(𝒌𝒔𝒊)
Truss
No.
Stress
(𝒌𝒔𝒊)
Truss
No.
Stress
(𝒌𝒔𝒊)
Truss
No.
Stress
(𝒌𝒔𝒊)
#1 4.3301 #6 1.4278 #11 -0.7296 #16 1.4944 #21 -3.6326
#2 1.5618 #7 -4.2056 #12 -0.0733 #17 -3.5220 #22 -4.7433
#3 1.8970 #8 1.6375 #13 -3.7115 #18 1.3173 #23 2.6208
#4 -2.8114 #9 -3.9967 #14 1.7511 #19 1.2285 #24 2.1307
#5 -2.4776 #10 -0.79366 #15 -3.2662 #20 -3.4898 #25 -5.2514
Table 3- 9 Nodal displacements in reliability optimality
Node
No.
Displacement (𝒊𝒏)
𝒙-direction
Node
No.
Displacement (𝒊𝒏)
𝒚-direction
Node
No.
Displacement (𝒊𝒏)
𝒛-direction
#1 -0.0012 #1 -0.2780 #1 -0.0395
#2 0.0313 #2 -0.2775 #2 -0.0468
#3 0.0032 #3 0.0054 #3 0.0481
#4 0.0026 #4 0.0055 #4 0.0431
#5 -0.0105 #5 0.0110 #5 -0.1070
#6 0.0174 #6 0.0113 #6 -0.1024
49
3.8 Closing Remarks
Uncertainty in the design conditions of practical structures is an inevitable fact, and so proper
precautions should be taken into account in the design process of a system to prevent its failure in
unexpected operating situations. Due to the complexity of real-world structures and mainly the
nonlinearity of their behaviors, reliability assessment applies a costly computational burden on the
design stage. Therefore, developing new RBDO approaches paves the path of designing safer and
more reliable structures within the feasible timeframe. The present study proposes a new RBDO
framework for practical structures by taking advantage of artificial neural networks and subset
simulation technique. In order to diminish the computation time of real structural simulations,
deterministic and probabilistic ANNs are exploited to determine the constraints and check the
feasibility of the solution in the optimization procedure. Deterministic ANNs are constructed using
input-output training pairs generated by direct simulation of the structure; the inputs are the
random variables and outputs are the response of the structure corresponding to those inputs. To
create the probabilistic ANNs, another batch of training pairs is required which contains the failure
probability of the structure at different values of random variables. Using SS in the evaluation of
small probabilities is another technique, exploited in the proposed approach, which further lightens
the computational cost of an RBDO procedure.
The proposed RBDO framework was applied on a case study to prove its efficiency. In both
deterministic-based and probabilistic-based optimizations, using ANNs remarkably reduced the
computation time of the procedures. The use of subset simulation, deterministic, and probabilistic
artificial neural networks in developing our approach, RBDO-SS&PANN, performed the RBDO
of this case study by approximately 78500 and 785 times faster than the conventional RBDO-MCS
and RBDO-MCS&DANN, respectively. These differences are more immense in more
sophisticated structural applications.
The results of this chapter are used for reliability design of automotive aluminum cross-car beam
assemblies presented in the following chapter.
50
CHAPTER 4: Reliability-Based Design Optimization
of Aluminum Cross-car Beam Assemblies
4.1 Overview
Nowadays, moving toward more lightweight designs is the key goal of all major automotive
industries, and they are always looking for more mass saving replacements. In this chapter, a new
methodology for the design and optimization of cross-car beam (CCB) assemblies is proposed to
obtain a more lightweight aluminum design as a substitution for the steel counterpart considering
targeted performances. In the end, to assay the RBDO framework developed in the thesis, this
CCB is designed for reliability performance.
4.2 Introduction
To redesign and optimize a CCB, one strategy is to modify the existing design, such as varying
the thicknesses or altering the design of existing parts in the assembly. This approach does not
necessarily yield the best possible design and reduces chances of finding a probable fundamentally
novel substitute. In some cases, also, this approach may lead to infeasible industrial or oversized
solutions.
The other strategy, proposed in this study, to handle the modifications, while not being restricted
to only minor variations of the existing model, is to start the design process from a block of
aluminum fitting in the permissible design space and apply multi-step topology optimization in
order to acquire the conceptual design of the CCB. Thereafter, by implementing shape and size
optimization concurrently, the final detailed design of the assembly is obtained. This new method
does not get trapped in a design loop, and the results are the best probable solution if the procedure
is implemented accurately.
51
In this context, shape optimization addresses the size and location of beads to increase the torsional
and bending rigidities of different segments. Size optimization also takes care of the cross-
sectional dimensions and the thickness of components. The implementation of topology, shape,
and size optimization techniques for the conceptual and detailed designs of industrial components
has been performed in many studies. Shin et al. [74] applied various structural optimization
methods on the inner panel of an automotive door to minimize its weight and increase the rigidity.
Topology optimization determined the number of parts and welding lines in the inner panel, and
size optimization found the optimal thicknesses. Pedersen and Nielsen [75] applied shape and size
optimization simultaneously to design 3D practical trusses. Lee et al. [76] modified an aluminum
control arm, under-floor panel, and engineering plastic hood for the lowest possible mass,
maximum rigidity, plastic strain, and residual deformations. Sekulski [77] employed topology and
size optimizations on a vehicle-passenger catamaran structure by using genetic algorithms.
In this thesis, structural optimization techniques are utilized to seek the best-in-class design of
CCBs. Topology optimization is implemented to find the conceptual physical boundaries of CCBs,
and size and shape optimizations are exploited to polish the model obtained in the topology stage.
The development of this chapter is as follows: Firstly, a brief introduction of different structural
optimization categories is presented. Then, the proposed methodology for designing CCBs is
mentioned, followed by a case study where the application of the methodology is assayed on a
CCB currently manufactured in Van-Rob Kirchhoff Inc, and finally, the CCB is designed by taking
into account the probabilistic constraints.
4.3 Structural Optimization
Structural optimization based on material distribution methods can be categorized into three
classes: size optimization, topology optimization, and shape optimization.
4.3.1 Size Optimization
Size optimization deals with geometrical parameters of a structure to obtain the optimum value of
the objective function. Therefore, design variables are plate thicknesses, geometrical lengths,
widths, and cross-sectional dimensions [78]. In this context, the topology of the object remains
unvaried during the whole process, and hence, the topology and boundaries of the structure are
52
determined prior to the process. Figure 4-1 depicts an example of size optimization applied to a
truss structure whose design variables are bar cross-sections.
Figure 4- 1 Size optimization of a structure: a) Initial design b) Optimum design
4.3.2 Topology Optimization
Topology optimization is the process of finding the best material layout of an object within the
design space satisfying the required performance. In the design domain, connectivity of different
sections of the object, as well as, the shape, number and location of holes is found, so that the
objective function is maximized or minimized, Figure 4-2.
Figure 4- 2 Topology optimization of a structure: a) Initial domain b) Optimum design
The mathematical formulation of topology optimization can be expressed as [79]
(a) (b)
(a) (b)
53
min𝜌
𝑓(𝜌)
subject to
(4-1 a-c)
∫ 𝜌 𝑑𝑥
𝛺
≤ 𝑉∗
𝜌(𝑥) = 0 𝑜𝑟 1, ∀𝑥 ∈ 𝛺
in which, 𝛺 is the design domain, 𝑉∗ is the allowable volume of the object, and 𝑓 is the objective
function such as the compliance or natural frequencies. In this manner, topology optimization tries
to find a subdomain, 𝛺𝑠, within the design domain, 𝛺, that minimizes the objective function. The
density function, 𝜌, takes the value 1 in 𝛺𝑠 and zero elsewhere. During the past decades, topology
optimization has received much attention, and various topology optimization techniques have been
developed. Among them, evolutionary structural optimization (ESO) [80], homogenization [81],
solid isotropic material with penalization (SIMP) [82]–[84], and level-set methods [85], [86] can
be mentioned.
Evolutionary structural optimization is based on gradual elimination of unnecessary material from
the design domain to achieve the optimality. In this concept, a fixed model is meshed with standard
finite elements, and the contribution of each element to the response function, a behavior of the
structure under study, such as its stiffness, is assessed by a proper criterion. Following the
procedure, elements with lesser contribution are removed from the design domain. This method
exploits evolutionary strategy as the optimization algorithm which makes it computationally an
expensive process.
Homogenization is the other method for handling topology problems. In homogenization method,
the material is considered as a medium with micro-scale voids which is supposed to be optimized
according to the prescribed design criteria. Since the geometrical properties of voids play the role
of design variables in this method, the complex topology problem turns into a simpler sizing
problem. However, this approach needs multi-scale modeling of the structure, which is a time-
consuming process, and may produce infinitesimal pores in the material, questioning the
manufacturability of the final design [7].
54
To overcome the drawbacks associated with homogenization method, other strategies such as
SIMP have been proposed. SIMP discretizes the domain into elements with constant material
properties. The correlation between the material properties (e.g. Young’s modulus or conductivity)
and design variables, the density of the cells, is expressed by an explicit relation, such as power
law, to steer the process from having intermediate densities to 0-1 pattern. The mathematical
definition of this method is expressed as
𝐸𝑖𝑗𝑘𝑙(𝑥) = (𝜌(𝑥))𝑝
𝐸𝑖𝑗𝑘𝑙0 (𝑥), 𝑝 > 1 (4-2 a-b)
∫ 𝜌 𝑑𝑥
𝛺
≤ 𝑉∗, 0 ≤ 𝜌(𝑥) ≤ 1, 𝑥 ∈ 𝛺
where 𝐸𝑖𝑗𝑘𝑙0 and 𝐸𝑖𝑗𝑘𝑙 are the original and modified stiffness tensor for the given isotropic
material, respectively. Having a value greater than unity for 𝑝 leads the intermediate densities to
tend to 0 or 1. It is shown that for cases with constraints on the volume being active, values greater
than 3 for 𝑝 yield a fairly 0-1 distribution pattern [83].
4.3.3 Shape Optimization
Shape optimization is utilized to find the optimum shape of structures, so design requirements are
satisfied and certain fitness functions are minimized or maximized. Shape optimization is different
from topology optimization, for in the former, structure topology will be preserved. In this type
of optimization, the shape of existing boundaries is altered to maximize or minimize the objective
function, and new boundaries are not allowed to be created or defined. As an instance, adding a
hole to a metal sheet is unacceptable since it makes a new boundary in the geometry [87]. Figure
4- 3 is a simple demonstration of shape optimization.
55
Figure 4- 3 Shape (topography) optimization of a structure: a) Initial design b) Optimum design
A sub-category of shape optimization is topography optimization, where shapes are perturbed
perpendicular to surface grids. This results in bead patterns on the component surfaces, and
subsequently yields more bending and torsional rigidities that improve structural performance. To
that end, the perturbed shape is achieved by adding a perturbation vector to the location vector of
design variables forming the surface. In the mathematical expression, it reads as
𝑋 = 𝑎1𝑋1 + 𝑎2𝑋2 + ⋯ + 𝑎𝑛𝑋𝑛 (4-3)
where 𝑋𝑖 (𝑖 = 1, . . 𝑛) are vectors by which the surface shape is defined, and 𝑎𝑖 (𝑖 = 1, … , 𝑛) are
their contribution coefficients in the shape. Each 𝑋𝑖 is a linear combination of original vector in
the initial shape and the perturbation vector.
(a) (b)
56
4.4 Methodology
The main components of a CCB assembly are driver-side and passenger-side beams which bear
most of the loads applied to the structure. The attachment of the assembly to vehicle’s body and
the modules mounted on it (e.g. the steering column, air conditioning system, and dashboard) is
carried out through the other components of the CCB (Figure 4-4).
Figure 4- 4 A cross-car beam assembly and its main components
In order to optimize or modify a CCB to meet the new design requirements, two strategies can be
utilized: one is to alter the existing components, particularly more crucial parts, and the other is to
ignore the available design and commence the procedure as if there is no such a solution presented
complying with the expectations. In the first approach the focus is mainly on the central beams,
end brackets, vertical braces and other prominent components in the driver side or passenger side.
Accordingly, fundamental changes in the assembly, such as altering the topology of critical
components or repositioning the parts on the central beams are less likely to take place. Therefore,
major concentration will be restricted to thicknesses, cross-sectional dimensions, or minor
modifications on existing parts. In order to obtain the best-in-class design, all possibilities should
be investigated which builds the basis of the proposed approach; Integration of topology, shape
and size optimization to deliver CCB assemblies compatible with design criteria. Hence, first, the
Driver-Side End Bracket
Driver-Side Vertical Brace Passenger-Side Vertical Brace
Passenger-Side End Bracket
Main or Central Beams
57
CCB designed conceptually by leveraging topology optimization, and thereafter by use of shape
and size optimizations, final detailed design is achieved.
4.4.1 Conceptual Design Stage
In this stage, a 3D finite element (FE) model of the CCB is built and meshed by solid elements.
This FE model is a solid aluminum geometry which contains the permissible CCB design space.
By defining the design criteria (e.g. NVH performance, stiffness) and objective function (e.g.
structure weight, fabrication cost) the conceptual process will be initiated. Since CCB assembly is
a complicated structure, the final conceptual design is achievable in multiple steps which makes it
an iterative process. Further, raw results of topology optimization are not realistic and to make
them manufacturable they should be smoothed. This task has to be repeated at each step until the
desired final conceptual design is accomplished.
4.4.2 Detailed Design Stage
The majority of CCB components are made by stamping or extrusion of metal sheets, and so the
FE model of the CCB in the conceptual design stage should be converted from a solid meshed
format into a one with shell elements. From the conceptual design step, the general geometry and
position of main parts, such as the central beams, end brackets, steering column supports, and
vertical braces are figured out. In this stage, optimal cross-sectional dimensions, thicknesses, and
bead patterns are explored. The proposed optimization framework is demonstrated in Figure 4-5.
58
Figure 4- 5 The proposed optimization framework
Creating a Solid Model
Fitting in the Design Space
Start
Topology
Optimization
(Conceptual Design)
Design Criteria
and
Manufacturability
Creating the Model
with 2D Elements
Shape Optimization
(Bead Patterns)
Size Optimization
(Thicknesses and
Cross-Sections)
Design Criteria and
Manufacturability Finalizing the Design
NO
YES
Detailed Design
NO YES
NO
59
4.5 Case Study
The case study selected to test the proposed optimization framework is redesigning an aluminum
CCB currently manufactured by Van-Rob Kirchhoff Inc., a tier-one automotive supplier in
Canada. The targeted CCB is sought to meet the NVH requirements. NVH criteria guarantee that
natural frequencies of the structure are greater than those of principal vibration sources. This
prevents resonance in the structure, and subsequently diminishes the excessive noise inside and
outside of the vehicle [88]. For CCBs, NVH performance is essentially concentrated on the
analysis of natural frequencies of the assembly, ensuring less oscillation of the instrument panel
and particularly the steering wheel in the driver’s hands. Prevailing excitement sources for a CCB
are rotary elements of automotive front section, such as the engine, air conditioner, and electrical
motor. For this study, the first and second natural frequencies should be, respectively, over 40 Hz
and 44 Hz in both conceptual and final design stage. Another constraint is on the manufacturability
of the design, as most of the components are created by stamping or extrusion, the designed parts
should also be able to be constructed by these processes. To deliver the structure for reliability
performance, a failure probability of 10−3 is assumed. The objective in all stages is to achieve the
lowest possible weight. Finite element simulations are conducted in Altair Hypermesh software
using Optistruct 12.0 as the solver.
4.5.1 Conceptual Design
The solid meshed finite element model of the CCB design domain is pictured in Figure 4-6. Green
region is corresponding to the designable section, and red elements, which are fixed throughout
the whole process, represent the attachment spots of the assembly to the vehicle’s body and
different components installed on it. These linkages are modeled as rigid elements. Hence, green
section, the allowable CCB design space, is subjected to topology optimization.
60
Figure 4- 6 Design space of the CCB meshed by solid elements
The topology optimization technique utilized in this study is SIMP method due to its applicability
to a wide range of industrial purposes. Penalty factor is assigned as 3 which has been shown that
produces a decent 0-1 distribution pattern [83]. The design space is made of aluminum with
properties presented in Table 4-1. The whole structure including the steering section and CCB are
meshed by roughly one million solid elements, 80 percent of which belong to the design space,
whose mass is 99.21 kg.
Table 4- 1 Mechanical and physical properties of design aluminum
Al Alloy Yield Stress
(𝑴𝑷𝒂)
Young’s Modulus
(𝑮𝑷𝒂)
Density (𝒌𝒈/𝒎𝟑)
Poisson’s
Ratio
AA 6082 T4 170 71 2700 0.33
Figure 4-7 depicts the outcome of the first topology optimization. In order to interpret topology
results, a threshold value should be introduced, 0.3 here, so only elements with densities greater
than or equal to this number are preserved, and the rest are of less importance and eliminated. Also,
since the main beams, in the driver-side or passenger-side, are made by extrusion, topology
optimization on these sections is driven by manufacturability control.
Attachment Spots
Designable Region
61
Figure 4- 7 Topology of the CCB after the first optimization
In Figure 4-7, red color corresponds to elements with densities equal to 1, and the elements with
densities smaller than 0.3 are omitted. As can be seen, results are remarkably rough and need to
be smoothened. An indispensable fact is, as long as the general cross-sectional geometry of the
main beam is not devised, finding an acceptable conceptual design is basically impossible.
Topology optimization result of the central beam, Figure 4-8, suggests that this part can be realized
by a hollow circular cross-section beam. Topology results declare that to minimize the weight, the
internal elements of the main beam can be removed, whereas none of the constraints are violated.
Using a thin-walled tube to construct the main beam, which could be counted as the most important
outcome of the topology stage, completely matches conventional designs of CCBs.
Figure 4- 8 Topology results of the main beams
In order to make the model more realistic, it should be prepared for another topology optimization.
Figure 4-9 represents the input model of the second step optimization; main beams are considered
62
as hollow thin-walled beams and are excluded from the designable area. Also, some redundant
elements are removed from the design domain.
Figure 4- 9 Input model of the second topology optimization
Figure 4-10 contains the optimization result, in which red elements are those with the density as
1, and since the central beams are non-designable they have appeared as red.
Figure 4- 10 Second topology optimization results
This process can be continued for multiple steps. However, finding a design which can be created
by stamping or extrusion just by topology optimization is practically improbable. Therefore, at
some point, designers have to convert the model to the one which is manufacturable and also
satisfies all design requirements. Thus, for the next step, the input is created by noting these facts:
Designable Region
Non-designable Region
63
1. Main beams are circular hollow tubes.
2. Components should be able to be produced only by stamping or extrusion.
3. Attachment spots of the CCB to the other components of the automotive are unvarying
throughout the process.
Figure 4-11 shows an 18.30 kg-input CCB for the next optimization round.
Figure 4- 11 Input model for the third phase of topology study
Based on the optimization results, the finalized conceptual design of the CCB satisfying NVH
performance is delivered. A principal consequence of this step is that the passenger-side main
beam, which has a great contribution to the structure weight, can have a smaller diameter than that
of the driver-side beam. Figure 4-12 illustrates optimization results of the last step along with the
modified components and finalized conceptual design.
64
Figure 4- 12 Finalized conceptual design of the CCB
65
4.5.2 Detailed Design
In the previous stage, a 7.53 kg-CCB was conceptually designed from a design space filled with
solid aluminum elements. The first and second natural frequencies of the assembly were obtained
as 40.05 Hz and 45.77 Hz, respectively. Although none of the constraints were violated and the
weight was reduced by approximately 92 percent, the best design is yet to be sought. The first step
for commencing the detailed design procedure is to reconstruct the conceptual design using shell
instead of solid elements. Subsequently, shape and size optimizations are applicable on the
assembly to further reduce the structure weight. Figure 4-13 depicts the CCB made by 2D
elements.
Figure 4- 13 Conceptual design modeled by shell meshes
As stated before, the central beams are the key components in the assembly on which the design
of other members is dependent. Therefore, diameters of these tubes are two main design variables
which will be specified by size optimization. The other design variables involved in the process
are the part thicknesses, and bead patterns on them along with element densities for topology
optimization on the two end brackets. The initial diameter of driver-side and passenger-side beams
are 55 mm and 45 mm, respectively, and the initial thickness of all components is 3.5 mm. For
further clarification, following figure demonstrates the assembly with the name of different
components on it.
66
Figure 4- 14 The CCB and its components
If 𝑆𝑖 (𝑖 = 1,2, … ,10) stands for size design variables (excluding thicknesses), 𝐵𝑖 (𝑖 = 1,2)
represents bead design variables applied to two end brackets, and 𝑇𝑖 (𝑖 = 1,2, … ,11) is
corresponding to thicknesses, Table 4-2 presents allowable ranges of shape and size design
variables.
Table 4- 2 Range of shape and size optimization design variables
Design Variable Lower Bound Upper Bound Perturbation Vector (𝒎𝒎)
𝑆𝑖 (𝑖 = 1,2, … ,10) -1.00 1.00 10.00
𝐵𝑖 (𝑖 = 1,2) 0 1.00 5.00
𝑇𝑖 (𝑖 = 1,2, … ,11) (mm) 2.00 8.00 ---
As depicted in Figure 4-15, the potential locations of beads are identified. Also, the diameter of
driver-side tube and that of the passenger-side tube changed from 55 mm and 45 mm to 64.58 mm
and 52.84 mm, respectively.
Passenger-Side
End Bracket
Driver-Side Beam Passenger-Side Beam
Driver-Side
End Bracket
Steering Support 3
Dashboard
Support 1
Steering
Support 2
Steering Support 1
Dashboard
Support 2
Passenger-Side
Vertical Brace
Driver-Side
Vertical Brace
67
Figure 4- 15 Potential locations of the beads
The mass of the optimized model decreased from 7.50 kg to approximately 6.00 kg. The optimal
values of design variables can be found in Table 4-3.
Table 4- 3 Optimal value of design variables
Design Variable Optimal Value Design Variable Optimal Value
𝑆1 1.00 𝑇1 (𝑚𝑚) 4.98
𝑆2 0.95 𝑇2 (𝑚𝑚) 2.00
𝑆3 1.00 𝑇3 (𝑚𝑚) 2.37
𝑆4 1.00 𝑇4 (𝑚𝑚) 3.08
𝑆5 1.00 𝑇5 (𝑚𝑚) 2.00
𝑆6 1.00 𝑇6 (𝑚𝑚) 4.16
𝑆7 1.00 𝑇7 (𝑚𝑚) 2.00
𝑆8 1.00 𝑇8 (𝑚𝑚) 3.68
𝑆9 1.00 𝑇9 (𝑚𝑚) 5.00
𝑆10 1.00 𝑇10 (𝑚𝑚) 5.00
𝐵1 1.00 𝑇11 (𝑚𝑚) 3.26
𝐵2 1.00
Although the goal, designing a CCB meeting all requirements, is achieved so far, the cross-car
beam configuration attained in this stage is manufacturing-wise infeasible. Based on the presented
results in Table 4-3 and topology optimization, the new CCB for final optimization process is
schemed (Figure 4-16).
68
Figure 4- 16 Input CCB model for the last step optimization
Since the model is altered extensively relative to the optimum model in the previous step, its weight
is no longer the optimal value. The input CCB has a mass of 8.17 kg, driver-side and passenger-
side tubes have 64 mm and 54 mm diameters, and thicknesses are all those obtained earlier. In
order to have a more weight-efficient assembly, size optimization (to find optimal thicknesses)
along with shape optimization (to find the bead patterns on applicable parts) processes are applied
simultaneously to the assembly to consummate the model. Since in the manufacturing processes,
the components are made of stamping of standard metal sheets, except the main beams, the final
thicknesses have to be selected from available standard metal sheet thicknesses. The allowable
values are reported in Table 4-4.
Table 4- 4 The discrete values of thicknesses
Choice Thickness (𝒊𝒏) Choice Thickness (𝒊𝒏)
#1 0.0808 #6 0.1443
#2 0.0907 #7 0.1620
#3 0.1019 #8 0.1819
#4 0.1144 #9 0.2043
#5 0.1285 #10 0.2294
The final size optimization was carried out by ES as the optimization algorithm, and due to its
metaheuristic nature, the process was repeated for 15 times. The best produced CCB solution has
10.9952 lb or 4.9918 kg mass and its first and second natural frequencies are 40.7037 Hz and
44.6790 Hz, respectively. The optimum thicknesses of all components are presented in Table 4-5,
and Figure 4-17 shows the final design of the assembly. Figure 4-18, also, includes the
convergence history of a few runs executed for this problem.
69
Table 4- 5 Optimal value of thicknesses
Component Optimum Thickness (𝒊𝒏)
Driver-Side Beam 0.1144
Passenger-Side Beam 0.0808
Driver-Side End Bracket 0.0808
Passenger-Side End Bracket 0.0808
Driver-Side Vertical Brace 0.1443
Passenger-Side Vertical Brace 0.0808
Steering Support 1 0.0808
Steering Support 2 0.2294
Steering Support 3 0.0907
Dashboard Support 1 0.0808
Dashboard Support 2 0.0808
Braces Cross Link 0.1443
Final Weight (𝒍𝒃) 10.9952
70
Figure 4- 17 Ultimate Optimum Design of the CCB
71
Figure 4- 18 Convergence history of the final step optimization
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
0 20 40 60 80 100 120 140 160 180 200
Ob
ject
ive
Fu
nct
ion
(W
eigh
t) (
kg)
Generations
Run 2
Run 4
Run 6
Run 8
Run 10
Run 12
Run 14
72
4.6 Reliability–Based Design of the CCB
In Chapter 3, a new RBDO approach was proposed to design and optimize practical structures for
reliability performance. In this section, to investigate the applicability of this framework in more
sophisticated cases, the developed strategy is applied to the assembly.
In a real manufacturing process, uncertainties in component thicknesses and material properties
are so likely to happen. As discussed before, in the design stage of life-critical structures such as
airplanes, the effects of tolerances is more crucial, and to prevent any probable disaster, they must
be studied with special thoughtfulness. Furthermore, the higher the level of structural reliability is,
the more expensive the design process will be. Therefore, overestimating the required reliability
can exert unnecessary costs on the process. For our problem, as a benchmark, a failure probability
of 10−3 is considered even though in reality for such a structure a lower probability may suffice.
The random variables and their distribution are presented in Table 4-6.
Table 4- 6 The characteristics of the random variables
Random Variable Distribution Mean Dispersion
Thicknesses Uniform t 6%
E Normal 71000 10%
Similar to deterministic-based optimization, thicknesses are discrete values selected from Table 4-
4. To construct the deterministic ANNs, 500 samples are generated from the design space, and
then, these networks are exploited to gather 800 training pairs for creating the probabilistic
networks. For reliability assessment using SS, 600 samples are utilized at each intermediate level
which makes the total number of samples to be 1680 for the expected failure probability. A (𝜇 +
𝜆)-ES with 𝜆 = 𝜇 = 25 and 500 iterations is utilized to propel the optimization process. Table 4-
7 contains the results of deterministic-based and reliability-based optimizations.
73
Table 4- 7 The results of optimization approaches
Design variables
(𝒊𝒏)
Optimization approaches
DBO-
DANN
RBDO-
DANN&PANN
Driver-Side Beam 0.1144 0.1285
Passenger-Side Beam 0.0808 0.0808
Driver-Side End Bracket 0.0808 0.0808
Passenger-Side End Bracket 0.0808 0.0808
Driver-Side Vertical Brace 0.1443 0.1144
Passenger-Side Vertical Brace 0.0808 0.0808
Steering Support 1 0.0808 0.0808
Steering Support 2 0.2294 0.2294
Steering Support 3 0.0907 0.0907
Dashboard Support 1 0.0808 0.0808
Dashboard Support 2 0.0808 0.0808
Braces Cross Link 0.1443 0.2294
Optimum weight (𝒍𝒃) 10.9952 11.1819
Mean 11.0179 11.2484
Standard deviation 0.0243 0.0478
Failure probability 0.4792 1.1440 × 10−5
Unlike 25-bar truss case study in Chapter 3, reliability design of the current problem does not lead
to a considerable weight difference compare to the DBO solution. The optimum weight of the two
approaches differs by approximately 0.1867 lb or 84.7618 kg, and the first and second natural
frequencies are 40.5745 Hz and 44.7403 Hz, respectively. As it comes from the table, although the
weights in two different strategies have slightly changed, the probability of failure have modified
remarkably. This proves, in some cases, how minor changes in the design can make strikingly a
safe structure. The optimization convergence history of RBDO-DANN&PANN strategy is
depicted in Figure 4-19.
74
Figure 4- 19 Convergence history of the RBDO optimization
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
0 25 50 75 100 125 150 175 200 225 250
Ob
ject
ive
Fu
nct
ion
(W
eigh
t) (
kg)
Generations
Run 2
Run 4
Run 6
Run 8
Run 10
Run 12
Run 14
75
4.7 Closing Remarks
In this chapter, an optimization framework was introduced and examined to design aluminum
cross-car beam assemblies for both conceptual and detailed design stages. Despite the fact that the
proposed framework originally aims to design aluminum CCBs to replace their steel counterparts,
it can be also employed for optimizing other components in automotives. For the case study, a
design domain of a CCB meshed with solid aluminum elements was chosen. The whole design
process contained two principal stages: conceptual and detailed designs. In the former, by use of a
multiple-step topology optimization process, a 7.53 kg-CCB was delivered conceptually. In this
stage, an idea of the general configuration of the assembly was achieved. Then, by applying shape
and size optimizations on the assembly, the modeled was finalized in the detailed design stage.
The optimal mass of the assembly was obtained as 4.9918 kg, and its first and second natural
frequencies were 40.7037 Hz and 44.6790 Hz, respectively. Based on the information from the
sponsor of the project, Van-Rob Kirchhoff Inc., this design is by approximately 2.00 kg lighter
than its existing aluminum counterpart, which entails a huge saving, financially and materially, in
a large-scale production.
The case study was conducted by considering only the NVH performance to demonstrate the
efficacy of the approach; other requirements, such as fatigue life, can be also taken into account
for other applications. Further, crashworthiness was not deliberated, for it is associated with high
computational costs, and there are no such safety regulations particularly for cross-car beam in
different impact conditions. Hence, crashworthiness performance is achievable by modeling the
entire car which requires a huge evaluation time. One way of treating this complication is to replace
non-designable sections of a complex structure by superelements containing all necessary
information for investigating designable sections. The same procedure can also be applied to CCBs
which hopefully will be addressed in future studies.
In this chapter, also, reliability-based design of the CCB was carried out by the developed strategy
presented in Chapter 3. The procedure led to an optimum RBDO solution whose weight differs
negligibly from that of the DBO design. This is a quite noteworthy outcome of this benchmark
that reliability design of a structure does not necessarily make a great change in the optimum
weight, and subsequently fabrication costs relative to the deterministic-based solution, and
sometimes by a minor variation in the design, a far more reliable structure is achievable.
76
CHAPTER 5: Conclusion
5.1 Thesis Summary
A comparative study on six most popular metaheuristic optimization algorithms, along with
developing a robust reliability-based design optimization framework for practical structures, and
proposing a novel strategy for the design and optimization of automotive aluminum cross-car beam
assemblies was performed in this thesis.
Particle swarm optimization, genetic algorithm, evolutionary strategy, firefly algorithm, harmony
search, and simulated annealing optimization algorithms were coded in MATLAB, and examined
on 10 mathematical and 4 structural benchmarks. Comparing the results well proves that
evolutionary strategy surpasses other methods in both types of problems. Therefore, this algorithm
was utilized in other chapters to handle the optimization task. This part of the thesis, with a few
minor alterations, was presented in The Canadian Society for Mechanical Engineering conference
held in Toronto in 2014 [73].
One of the main achievements of this study is the introduction of a new powerful RBDO approach
which can be exploited to design real-world structures with small failure probabilities. In this
technique, artificial neural networks are employed to replace deterministic and probabilistic
performance functions. This will remarkably diminish the computation time of reliability
assessment of the structure at design points. To generate probabilistic training pairs, subset
simulation was used which greatly reduced the required sampling effort. To verify the excellence
of the proposed approach over existing ones, a 25-bar truss structure was designed for reliability.
The results show that the proposed strategy, RBDO-SS&PANN, performs the process 78500 and
785 faster than conventional RBDO-MCS and RBDO-MCS&PANN, respectively. For more
complex engineering systems, this achievement will substantially drop the design procedure time
77
and subsequently the project costs. This portion of the research is ready to be submitted to a
renowned scientific journal in this field.
Developing a new framework for designing automotive aluminum CCBs is the second principal
contribution of this work. The use of this strategy leads to the best possible design for the desired
component. Although the application was demonstrated in CCBs, it can be implemented for other
automotive components as well. To test the proposed framework, an aluminum CCB was sought
to be designed for NVH performance. The design procedure was initiated from a block of
aluminum fitting in the permissible design space. Then, various structural optimization techniques
were applied to it to deliver the most functional aluminum CCB for the demanded NVH
performance. In the end, as the second case study for the proposed RBDO framework, the CCB
was designed for reliability constraints. Comparing the optimum weights of two solutions,
deterministic and probabilistic constraints, shows a negligible difference which entails that
considering failure probabilities in design process does not necessarily end up to a heavier design.
This part of the research will be soon submitted to SAE International Journal of Materials &
Manufacturing.
5.2 Future Work
Despite the successful implementation of the proposed RBDO architecture in this thesis, there is
still ample room for progressing in this field. There are numerous parameters involved in the
definition of ANNs which greatly influence their performance in different applications. Therefore,
to improve the speed of an RBDO process, one can investigate other learning algorithms and
network topologies.
In this study, a single-hidden layer ANN with the feed-forward back propagation training
algorithm is utilized to replace complex performance functions. The time consuming part of
constructing an ANN is the sampling task since it demands direct simulation of the structure.
Hence, generating a sufficient number of samples could avoid unnecessary computation effort on
this portion of the process. The literature seems to still demands research to unravel the problem
of finding an optimum ANN architecture for specific purposes.
Another interesting direction in this area is the employment of other surrogate meta-models instead
of ANNs for RBDO procedure. As pointed out in Chapter 3, support vectors machines, response
78
surfaces, and Kriging are all techniques which can be utilized to hasten obtaining the response of
sophisticated systems. Valuable studies have been carried out on them, yet much attention can still
be allotted in engineering applications.
In Chapter 4, where a new design framework for aluminum CCBs is developed, a case study is
accomplished in which the fitness function is the assembly weight, and constraints are on first and
second natural frequencies. To draw the case study toward a more realistic project, one can
consider other parameters in the fitness function, such as fabrication costs which comprise material
and labor costs. Furthermore, by modeling the entire vehicle in which the CCB is implemented,
the assembly can be studied for crashworthiness and safety regulations.
79
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APPENDICES
Appendix A. Mathematical Benchmarks of Chapter 2
Followings are the test problems for mathematical optimization benchmarks. Test problems 1
through 8 are from Runarsson (2005) [43] and test problems 9 and 10 are from Parsopoulos and
Vrhatis (2002) [44].
(1) b01
Minimize 𝑓(𝑥) = 5 ∑ 𝑥𝑖
4
𝑖=1
− 5 ∑ 𝑥𝑖2
4
𝑖=1
− ∑ 𝑥𝑖
13
𝑖=5
subject to
𝑐1(𝑥) = 2𝑥1 + 2𝑥2 + 𝑥10 + 𝑥11 − 10 ≤ 0,
𝑐2(𝑥) = 2𝑥1 + 2𝑥3 + 𝑥10 + 𝑥12 − 10 ≤ 0,
𝑐3(𝑥) = 2𝑥2 + 2𝑥3 + 𝑥11 + 𝑥12 − 10 ≤ 0,
𝑐4(𝑥) = −8𝑥1 + 𝑥10 ≤ 0,
𝑐5(𝑥) = −8𝑥2 + 𝑥11 ≤ 0,
𝑐6(𝑥) = −8𝑥3 + 𝑥12 ≤ 0,
𝑐7(𝑥) = −2𝑥4 − 𝑥5 + 𝑥10 ≤ 0,
𝑐8(𝑥) = −2𝑥6 − 𝑥7 + 𝑥11 ≤ 0,
89
𝑐9(𝑥) = −2𝑥8(𝑥) − 𝑥9 + 𝑥12 ≤ 0,
in which 0 ≤ 𝑥𝑖 ≤ 1 (𝑖 = 1, 2, … , 9), 0 ≤ 𝑥𝑖 ≤ 100 (𝑖 = 10, 11, 12), and 0 ≤ 𝑥13 ≤ 1. The
global optimum is at 𝑥∗ = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1) where 𝑓(𝑥∗) = −15, and constraints
𝑐1, 𝑐2, 𝑐3, 𝑐4, 𝑐5, and 𝑐6 are active.
(2) b02
Maximize 𝑓(𝑥) = ||∑ cos4(𝑥𝑖)
𝑛𝑖=1 − 2 ∏ cos2(𝑥𝑖)𝑛
𝑖=1
√∑ 𝑖𝑥𝑖2𝑛
𝑖=1
||
subject to
𝑐1(𝑥) = 0.75 − ∏ 𝑥𝑖
𝑛
𝑖=1
≤ 0,
𝑐2(𝑥) = ∑ 𝑥𝑖
𝑛
𝑖=1
− 7.5𝑛 ≤ 0,
in which 𝑛 = 20 and 0 ≤ 𝑥𝑖 ≤ 10 (𝑖 = 1, 2, … , 𝑛). The best known solution is 𝑓(𝑥∗) =
0.803619, where constraint 𝑐1 is close to being active (𝑐1 = −10−8).
(3) b03
Minimize 𝑓(𝑥) = 5.3578547𝑥32 + 0.8356891𝑥1𝑥5 + 37.293239𝑥1 − 40792.141
subject to
𝑐1(𝑥) = 85.334407 + 0.0056858𝑥2𝑥5 + 0.0006262𝑥1𝑥4 − 0.0022053𝑥3𝑥5 − 92 ≤ 0,
𝑐2(𝑥) = −85.334407 − 0.0056858𝑥2𝑥5 − 0.0006262𝑥1𝑥4 + 0.0022053𝑥3𝑥5 ≤ 0,
𝑐3(𝑥) = 80.51249 + 0.007131𝑥2𝑥5 + 0.0029955𝑥1𝑥2 + 0.002183𝑥32 − 110 ≤ 0,
𝑐4(𝑥) = −80.51249 − 0.007131𝑥2𝑥5 − 0.0029955𝑥1𝑥2 − 0.002183𝑥32 + 90 ≤ 0,
𝑐5(𝑥) = 9.300961 + 0.0047026𝑥3𝑥5 + 0.0012547𝑥1𝑥3 + 0.0019085𝑥3𝑥4 − 25 ≤ 0,
90
𝑐6(𝑥) = −9.300961 − 0.0047026𝑥3𝑥5 − 0.0012547𝑥1𝑥3 − 0.0019085𝑥3𝑥4 + 20 ≤ 0,
in which 78 ≤ 𝑥1 ≤ 102, 33 ≤ 𝑥2 ≤ 45, 27 ≤ 𝑥𝑖 ≤ 45 (𝑖 = 3, 4, 5). The optimum solution is at
𝑥∗ = (78.33, 29.995056025682, 45, 36.77581205788), where 𝑓(𝑥∗) = −30665.539, and 𝑐1
and 𝑐6 are active constraints.
(4) b04
Minimize 𝑓(𝑥) = (𝑥1 − 10)3 + (𝑥2 − 20)3
subject to
𝑐1(𝑥) = −(𝑥1 − 5)2 − (𝑥2 − 5)2 + 100 ≤ 0,
𝑐2(𝑥) = (𝑥1 − 6)2 + (𝑥2 − 5)2 − 82.81 ≤ 0,
in which 13 ≤ 𝑥1 ≤ 100, 0 ≤ 𝑥2 ≤ 100. The optimum solution is at 𝑥∗ = (14.095, 0.84296),
where 𝑓(𝑥∗) = −6961.81388, and both constraints are active.
(5) b05
Minimize 𝑓(𝑥)
= 𝑥12 + 𝑥2
2 + 𝑥1𝑥2 − 14𝑥1 − 16𝑥2 + (𝑥3 − 10)2 + 4(𝑥4 − 5)2 + (𝑥5 − 3)2
+ 2(𝑥6 − 1)2 + 5𝑥72 + 7(𝑥8 − 11)2 + 2(𝑥9 − 10)2 + (𝑥10 − 7)2 + 45
subject to
𝑐1(𝑥) = −105 + 4𝑥1 + 5𝑥2 − 3𝑥7 + 9𝑥8 ≤ 0,
𝑐2(𝑥) = 10𝑥1 − 8𝑥2 − 17𝑥7 + 2𝑥8 ≤ 0,
𝑐3(𝑥) = −8𝑥1 + 2𝑥2 + 5𝑥9 − 2𝑥10 − 12 ≤ 0,
𝑐4(𝑥) = 3(𝑥1 − 2)2 + 4(𝑥2 − 3)2 − 7𝑥4 − 120 ≤ 0,
𝑐5(𝑥) = 5𝑥12 + 8𝑥2 + (𝑥3 − 6)2 − 2𝑥4 − 40 ≤ 0,
𝑐6(𝑥) = 𝑥12 + 2(𝑥2 − 2)2 − 2𝑥1𝑥2 + 14𝑥5 − 6𝑥6 ≤ 0,
91
𝑐7(𝑥) = 0.5(𝑥1 − 8)2 + 2(𝑥2 − 2)2 + 3𝑥52 − 6𝑥6 − 30 ≤ 0,
𝑐8(𝑥) = −3𝑥1 + 6𝑥2 + 12(𝑥9 − 8)2 − 7𝑥10 ≤ 0,
in which −10 ≤ 𝑥𝑖 ≤ 10 (𝑖 = 1, 2, … , 10). The global optimum is at 𝑥∗ =
(2.171996, 2.363683, 8.773926, 5.095984, 0.9906548, 1.430574, 1.321644, 9.828726,
8.280092, 8.375927), where 𝑓(𝑥∗) = 24.3062091, and 𝑐1, 𝑐2, 𝑐3, 𝑐4, 𝑐5 and 𝑐6 are active.
(6) b06
Maximize 𝑓(𝑥) =sin3(2𝜋𝑥1) sin(2𝜋𝑥2)
𝑥13(𝑥1 + 𝑥2)
subject to
𝑐1(𝑥) = 𝑥12 − 𝑥2 + 1 ≤ 0,
𝑐2(𝑥) = 1 − 𝑥1 + (𝑥2 − 4)2 ≤ 0,
in which 0 ≤ 𝑥1, 𝑥2 ≤ 10. The global optimum is at 𝑥∗ = (1.2279713, 4.2453733), where
𝑓(𝑥∗) = 0.095825, and none of the constraints is active.
(7) b07
Minimize 𝑓(𝑥)
= (𝑥1 − 10)2 + 5(𝑥2 − 12)2 + 𝑥34 + 3(𝑥4 − 11)2 + 10𝑥5
6 + 7𝑥62 + 𝑥7
4 − 4𝑥6𝑥7
− 10𝑥6 − 8𝑥7
subject to
𝑐1(𝑥) = −127 + 2𝑥12 + 3𝑥2
4 + 𝑥3 + 4𝑥42 + 5𝑥5 ≤ 0,
𝑐2(𝑥) = −282 + 7𝑥1 + 3𝑥2 + 10𝑥32 + 𝑥4 − 𝑥5 ≤ 0,
𝑐3(𝑥) = −196 + 23𝑥1 + 𝑥22 + 6𝑥6
2 − 8𝑥7 ≤ 0,
𝑐4(𝑥) = 4𝑥12 + 𝑥2
2 − 3𝑥1𝑥2 + 2𝑥32 + 5𝑥6 − 11𝑥7 ≤ 0
92
in which −10 ≤ 𝑥𝑖 ≤ 10 (𝑖 = 1, 2, … , 7). The global optimum is at 𝑥∗ = (2.330499,
1.951372, −0.4775414, 4.365726, −0.6244870, 1.038131, 1.594227), where 𝑓(𝑥∗) =
680.6300573, and 𝑐1 and 𝑐4 are active.
(8) b08
Minimize 𝑓(𝑥) = 𝑥1 + 𝑥2 + 𝑥3
subject to
𝑐1(𝑥) = −1 + 0.0025(𝑥4 + 𝑥6) ≤ 0,
𝑐2(𝑥) = −1 + 0.0025(𝑥5 + 𝑥7 − 𝑥4) ≤ 0,
𝑐3(𝑥) = −1 + 0.01(𝑥8 − 𝑥5) ≤ 0,
𝑐4(𝑥) = −𝑥1𝑥6 + 833.33252𝑥4 + 100𝑥1 − 83333.333 ≤ 0,
𝑐5(𝑥) = −𝑥2𝑥7 + 1250𝑥5 + 𝑥2𝑥4 − 1250𝑥4 ≤ 0,
𝑐6(𝑥) = −𝑥3𝑥8 + 1250000 + 𝑥3𝑥5 − 2500𝑥5 ≤ 0,
in which 100 ≤ 𝑥1 ≤ 10000, 1000 ≤ 𝑥2, 𝑥3 ≤ 10000, and 10 ≤ 𝑥𝑖 ≤ 1000 (𝑖 = 4, 5, … , 8).
The global optimum is at 𝑥∗ = (579.3167, 1359.943, 5110.071, 182.0174, 295.5985,
217.9799, 286.4162, 395.5979), where 𝑓(𝑥∗) = 7049.3307, and 𝑐1, 𝑐2 and 𝑐3 are active.
(9) b09
Minimize 𝑓(𝑥) = (𝑥1 − 2)2 + (𝑥2 − 1)2
subject to
�̂�1(𝑥) = 𝑥1 − 2𝑥2 + 1 = 0,
𝑐1(𝑥) =𝑥1
2
4+ 𝑥2
2 − 1 ≤ 0,
in which design variables are not bounded. The global optimum is at 𝑥∗ =
(0.8228757, 0.911437828), where 𝑓(𝑥∗) = 1.393464981, and all constraints are active.
93
(10) b10
Minimize 𝑓(𝑥) = −10.5𝑥1 − 7.5𝑥2 − 3.5𝑥3 − 2.5𝑥4 − 1.5𝑥5 − 10𝑥6 − 0.5 ∑ 𝑥𝑖2
5
𝑖=1
subject to
𝑐1(𝑥) = 6𝑥1 + 3𝑥2 + 3𝑥3 + 2𝑥4 + 𝑥5 − 6.5 ≤ 0,
𝑐2(𝑥) = 10𝑥1 + 10𝑥3 + 𝑥6 − 20 ≤ 0,
in which 0 ≤ 𝑥𝑖 ≤ 1 (𝑖 = 1, 2, … , 5), and 0 ≤ 𝑥6. The global optimum is at 𝑥∗ = (0, 1, 0, 1, 1, 20),
where 𝑓(𝑥∗) = −213.0, and 𝑐2 is active.