Design and Optimization of Aluminum Cross-Car Beam ... · Design and Optimization of Aluminum...

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.

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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

viii

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

ix

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

2

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


b04 2 cubic 0 0 2


b06 2 nonlinear 0 0 2

b07 7 polynomial 0 0 4

b08 8 linear 3 0 3



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


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


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


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


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


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


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


A


A


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.


𝒚-direction

Node

No.


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.


𝒙-direction

Node

No.


𝒚-direction

Node

No.


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



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



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.

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