1
Introduction
In a continuously evolving world, the field of civil engineering plays an increasingly
fundamental role in a nation’s development. The need to create sustainable and
resilient infrastructure, capable of minimizing environmental degradation and pre-
serving natural resources for future generations, has become an absolute priority. In
fact, Long et al. [1] affirm that the impact of global emissions and climate change
is significantly influenced by urban centers worldwide, encompassing only 2 % of
the Earth’s surface yet contributing to nearly 80 % of carbon emissions from human
activities. The urban setting directly shapes our living standards, social welfare, and
health. Therefore, the attributes and excellence of our infrastructures play a pivotal
role in favoring urban sustainability and the overall health of our surroundings,
considering that they constitutes at least 50 % of our national wealth [1]. In this
context, parametric designs and optimization algorithms emerge as essential tools
for guiding the decision-making process in civil engineering, enabling the planning of
efficient and environmentally responsible solutions. Hence, extensive research and
practical applications of new materials, construction methods, and design techniques
are necessary [2].
This thesis focuses on modern optimization algorithms and parametric designs to
evaluate the environmental impact of tied-arch bridges (a common infrastructure
type worldwide). The focus is directed towards the arch’s shape and geometries
with the aim of reducing steel requirements, consequently lowering costs and carbon
dioxide emissions. To achieve this objective, a real tied-arch bridge was analyzed to
validate the algorithm. This first chapter serves as an introduction to the topic.
The second chapter begins with an introduction to the fundamental concepts of the
optimization process, guiding the reader to the subsequent discussion on structural
optimization. It further delves into the properties of tied-arch bridges and influence
lines. Finally, the chapter addresses the analyses conducted on them in accordance
with the Eurocodes, followed by the definition of the Global Warming Potential
(GWP) [3], [4].
In the third chapter, an extensive exploration of the algorithm is given. This includes
an in-depth look at the modeling of the optimization problem, the process of sizing
and modeling the arch, the development of a finite element model (FEM), and the
management of constraints. This chapter is the most comprehensive, and great care
and attention were given to its content.
1
1. Introduction
The algorithm was assessed by utilizing it to optimize an existing tied-arch bridge
by comparing it to five alternative design scenarios. The methodology and results of
these case studies are detailed in the fourth chapter.
The fifth and final chapter of the thesis concludes by examining the degree to
which the intended goals have been achieved. Additionally, it offers suggestions into
potential areas for future research within this field.
2
2
Background
Numerical optimization methods serve as powerful tools that allow to uncover
solutions that maximize efficiency, minimize material usage, or achieve desired
outcomes and this can be exploited very well in civil engineering problems [2], [5]–
[7]. Most traditional optimization algorithms are deterministic [5]. To address the
intricaciesofnumerousreal-worldoptimizationchallenges, manyturntometaheuristic
algorithms, which draw inspiration from stochastic natural phenomena through the
integration of randomization [8]–[10]. However, due to the limited availability of
information and instrumentation, this thesis will address a manual optimization
approach [11].
The chapter begins by presenting the fundamental theory of numerical optimization,
followed by the problem of carbon dioxide emissions in steel production industry.
Then, the optimization in structural design are discussed, and influence lines are
introduced. We focus on handling resistance and buckling analyses for tied-arch
bridges, concluding with the definition of the Global Warming Potential [3], [4].
2.1 The Optimization Process
The conventional process of optimization involves modifying the input of a function
or procedure with the aim of either diminishing or augmenting the resulting output
value [2], [5], [6]. The function under consideration for optimization is commonly
referred to as the objective function and an array of input values constitutes a solution.
Additionally, constraints are typically imposed, delimiting the permissible range of
input values. A solution that satisfies constraints is called a feasible solution, and the
collection of allowable input values defines the search space. Consequently, the search
space delineates the domain encompassing all feasible solutions. Mathematically, the
formulation of an optimization problem
1
is as follows:
minimize f
i
(x), i = 1, 2,...,M (2.1)
subject to the constraints:
1
This can be a minimization but also a maximization.
3
2. Background
g
j
(x)≤ 0, j = 1, 2,...,J
h
k
(x) = 0, k = 1, 2,...,K
(2.2)
where f
i
(x), g
j
(x) and h
k
(x) are functions of the design vector:
x = (x
1
,x
2
,...,x
D
), x∈R
D
(2.3)
The components x
i
of the design vector are called design variables, the functions
f
i
(x) are the objective functions, and the inequalities g
j
(x) and equalities h
k
(x) are
the constraints.
An optimization problem can have an arbitrary number of objective functions. The
optimization process is classified as single-objective optimization if there is only one
objective function, and as multi-objective optimization if there are more than one
objective function. Therefore, a single-objective optimization has the aim to find
a single optimal solution, while multi-objective optimization seeks to find a set of
solutions instead. These solutions are known as the Pareto front [2], [5], where no
solution is better than another in all objectives [12].
2.1.1 Objective Function
The objective function is a numerical representation of the process that we are
seeking optimal input values to [2], [5], [6]. Thus, the output of the objective function
is the quantity to be optimized. For example, if the purpose is to minimize the
environmental impact of a bridge construction by reducing the amount of its CO
2
(the carbon dioxide) equivalent, the output of the objective function should be the
CO
2
equivalent emitted. The input to the function are the design variables. All
other parameters necessary to determine the CO
2
equivalent emitted is provided as
constants (or functions of the design variables) within the objective function.
2.1.2 Search Space
The number of design variables determines the dimensionality of the optimization
problem, where n variables correspond to an n-dimensional problem [2], [5]. A
simple problem with only two design variables can be visualized in a comprehensible
manner with a surface plot, usually called landscape. In such a plot, each pair of
x and y coordinates represents a point in the search space, and the corresponding
z-coordinate represents the value of the objective function at that particular point.
Typically, objective functions have several local optima, but only one global optimum.
What is referred to as optimum depends on whether we are seeking a minimum or a
maximum of the function. By convention, it is assumed that the objective function
should be minimized. If the objective function should be maximized instead, we can
simply invert the sign of the function and minimize.
4
2. Background
2.1.3 Constraints
Constraints refer to the limitations, conditions, or requirements that guide the process
of improving an algorithm [2], [5], [6]. These constraints can encompass various
aspects, such as time, accuracy and compatibility. In this way, constraints set the
boundaries and expectations for the optimization process, ensuring that the algorithm
aligns with specific criteria or limitations. For this reason, optimization problems
can be classified according to the constraints: a problem without constraints is an
unconstrained optimization problem and a problem with constraints is a constrained
optimization problem [5]. Whilemanyoptimizationalgorithmsaremosteffectivewhen
dealing with unconstrained variables, real-world problems often involve constraints.
The basic idea of constraint-handling techniques is to transform a constrained
optimization problem into an unconstrained one [13]. An example of this is the
penalty method [14]. Constraints can fall into two primary categories: linear and
non-linear [15]–[17]. Linear constraints are relatively easy to manage since they
delineate infeasible regions and narrow down the search space directly. They can
be evaluated without the need of computationally intensive operations. In contrast,
non-linear constraints make the optimization problem more challenging to solve. In
this thesis we focus on penalty method [14] to handle non-linear constraints.
The penalty method is a widely efficient approach thanks to its fundamental principle
which involves assigning a penalty value to a solution that fails to satisfy the
constraints. This penalty value is then added to the objective function, forming the
penalized objective function [5], which can be formulated as:
Π(x,µ ,ν) =f(x) +P (x,µ ,ν) (2.4)
wheref(x) is the objective function and P (x,µ ,ν) is the penalty term. The penalty
term has several popular definitions, one of them can be defined as:
P (x,µ ,ν) =
J
j=1
µ j
max(0,g
j
(x))
2
+
K
k=1
ν
k
|h
k
(x)|, (2.5)
where µ j
> 0 and ν
k
> 0 are the penalty factors. Another popular definition of the
penalty term is:
P (x,µ ,ν) =
J
j=1
µ j
H
j
[g
j
(x)] g
2
j
(x) +
K
k=1
ν
k
H
k
[h
k
(x)] h
2
k
(x), (2.6)
where µ j
> 0 and ν
k
≫ 1. The factors H
j
[g
j
(x)] and H
k
[h
k
(x)] fulfill the conditions:
H
j
[g
j
(x)] =
0 if g
j
(x)≤ 0 ,
1 otherwise ,
H
k
[h
k
(x)] =
0 if h
k
(x) = 0 ,
1 otherwise .
5
2. Background
To simplify the implementation, it is also possible to consider µ =µ j
for all j and
ν =ν
k
for all k. In this case, the penalized objective function of the Equation 2.4
can be written as:
Π(x,µ ,ν) =f(x) +µ J
j=1
H
j
[g
j
(x)] g
2
j
(x) +ν
K
k=1
H
k
[h
k
(x)] h
2
k
(x). (2.7)
In Equation 2.7, it is clear that the penalty term is directly proportional to the
magnitude of the constraint violations. This relationship is evident because significant
violations of the constraints should result in a substantial penalty within the objective
function. This penalty serves to guide the algorithm away from regions in the search
space where feasibility is compromised. In a manual approach, however, it alerts us
whether the path we are following will actually lead to optimization or not.
2.2 Optimization in Structural Design
A structure to be optimized can be defined through a collection of quantities, which
can be categorized into two different type of values [2], [5], [6]:
• Preassigned parameters;
• Variables;
Preassigned parameters maintain a constant value throughout the optimization
process. Their optimal values are not fixed, but are necessary for defining the
computational structure. Conversely, variables are changeable quantities that can be
further divided into two categories [5], [6], [18], [19]:
• Independent variables: these are the focus of optimization algorithms, and their
optimal values are adjusted or varied to enhance the performance or efficiency
of a given structure;
• Dependent variables: these variables are influenced by the choices made for the
independent variables and represent the desired optimal outcomes.
In particular, independent variables are also known as design variables and they can
be adjusted or modified within certain predefined bounds or constraints to find the
best solution to our problem [20]. Because of that, they are the main focus of our
study. Dependent variables, on the other hand, encompass all quantities that are
neither independent variables nor preassigned parameters.
The decision to treat a quantity in a structure design as a design variable, a dependent
variable or preassigned parameter requires thoughtful consideration, guided by several
key inquiries [5]:
• How does a variation in the quantity impact the objective?
• Are there challenging-to-quantify constraints, such as aesthetics?
• Is the parameter value open for the designer’s choice, or is it predetermined?
6
Ricevi informazioni sui nostri servizi, sulle offerte e non perdere news e consigli su università e lavoro.
