predicting the probable propagation – Global Homework Experts

Analysis of crack trajectories using layout optimization techniques
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The purpose of this research was to assess a new approach to
predicting the probable propagation of fractures in structural
elements. This becomes more important in a broad range of

problems when structures approach their Ultimate Limit State.
Steel supports and joints may develop cracks over time.
structural designers might use a trustworthy (in terms of
computational labor) approach for forecasting tier propagation
in concrete pillars or walls. Also, the directions of the primary
stresses in a solid subjected to the same loads and supported in
the same manner coincide with the locations of the tension and
compression bars. In addition, fractures should start out
perpendicular to the direction of the highest tension loads. By
doing a Layout Optimisation (LO) study, we may determine the
starting direction of the crack and then place a “hole” in the
design space facing in that direction. Next, we’ll re-run the study
with this “hole” in place and see how the maximum tension
shifts as a result. To test if the LO method reliably forecasts the
crack’s growth direction, we may conduct the analysis many
times. If this works, we may have discovered a promising new
way for predicting cracks.

Abstract ​2
Contents ​3
Chapter 1: Introduction ​4
Chapter 2: Literature Review ​6
Chapter 3: Methodology ​18
Chapter 4: Results and Discussion ​31
Chapter 5: Conclusions and Recommendations ​34
References ​38
Chapter 1: Introduction
Recently, there has been an unbelievable amount of progress in
the approaches of structural optimization. When one is trying to
optimize a structure, one of the goals that they should have is to
decrease the mass of the structure while maintaining its stiffness
and satisfying other design criteria within a particular range of
possibilities. Utilizing optimization methods may result in
improvements to the object’s size, form, and topology (
). While size and form optimization are used to simply
maximize the cross-sectional features and the design domain
boundaries, respectively, topology optimization is used to

determine the number of cavities in continuous structures, as
well as their size and location within those structures. Since
concrete has a relatively low tensile strength owing to
weathering, creep, and aging effects, cracking is to be
anticipated in components subjected to service loads. The
stiffness, energy absorption, capacity, and ductility of reinforced
concrete structures are all negatively impacted by cracking. Loss
of a building’s strength and rigidity may cause severe cracks and
even collapse. Numerous research and investigations have been
conducted to ascertain the impact of crack location and depth on
the static and dynamic behavior of concrete structural element.
The growing need for low-cost, light-weight, high-performance
structures has led to the categorization of structural optimization
techniques into the three broad categories of size, shape, and
topology. By reducing or maximizing an objective function
while adhering to design restrictions, topology optimization
determines the best possible material arrangement within a
specified design domain (
Réthoré, 2008). Topology optimization
has been widely used in several engineering domains,
particularly the automotive and aerospace sectors, since it offers
the greatest practical and prospective design space compared to
other structural optimization approaches. Several topology
optimizations models, for instance, have been presented during
the last several decades. Shear pivot stiffness in pantographic
metamaterials may be predicted with startling accuracy by using
topology optimization techniques.
Although some of its fundamental principles were developed
about a century ago, layout or topology optimization is one of
the most recent and rapidly increasing topics of structural design
because it concerns itself with the selection of the ideal
configuration for structural systems. This means that it is one of
the most important aspects of modern structural design. In spite
of the fact that it presents the most challenging design problem
from an analytical and computational standpoint, it offers the

highest reward. It is essential to make use of the conservative
calculations that are included into the standard design standards
in order to eliminate the possibility of failure mechanisms.
However, more research is required to investigate the
mechanical behavior of the structure if non-standard concrete
types that are not specified in the design guidelines are used, or
if non-static stress or temperature stress is applied (
). Within the framework of the circular economy, a new
generation of environmentally friendly concretes that make use
of by-products of industrial processes is now under
development. Quite a few studies have been carried out in order
to assess their performance and possible uses in the building
industry. On the other hand, there is a scarcity of research that
investigates the mechanical behavior of these organisms at the
structural or macro level. During the process of Ultimate Limit
State Design, the resistance to shear failure, which is also known
as shear capacity, is an essential property that must be verified in
SRC beams. In regions where the risk of earthquakes is high,
structural monitoring may benefit from the use of nondestructive methods like acoustic emissions. When the shear
strength of a material is surpassed, a critical diagonal fracture
will occur. As the load continues to increase, the diagonal
fracture will continue to expand until it is no longer adequate to
induce failure. The load at which a diagonal crack emerges for
the very first time is referred to as the diagonal crack load. The
beginning of diagonal cracking must be located before one can
conduct an analysis of the behavior of structural concrete
elements when subjected to shear pressure. On the one hand, the
occurrence of the diagonal fracture is a significant factor in
determining the overall shear strength of the material (
). However, the reserve shear strength factor is something
that can only be determined with the assistance of the diagonal
shear crack load. The reserve shear strength factor assesses the
resistance to diagonal shear cracking in comparison to the

ultimate shear force. When determining the reserve shear
strength factor, it is very necessary to take into account the
impact that concrete mixes have on shear resistance, such as the
interlocking of the aggregate. The buffer zone that exists in each
material between the point at which the first diagonal fracture
begins and the point at which shear failure occurs is an
important consideration. The reserve shear strength parameter
was estimated in subsequent studies while examining the
behavior of SRC beams with stirrups in the case of concrete
with recycled materials and self-compacting concrete, even
though the initial work was performed on high-strength concrete
deep beams without stirrups (
Jung, 2014). In other words, the
reserve shear strength parameter was estimated while examining
the behavior of SRC beams with stirrups in the case of concrete
with recycled materials and self-compacting concrete (SCC).
Chapter 2: Literature Review
The study of the design, construction, operation, and
maintenance of a wide variety of buildings and systems, such as
houses, bridges, and roads, is referred to as “civil engineering,”
and the phrase “civil engineering” refers to the area of study that
encompasses this topic. The architecture, engineering, and
construction (AEC) sector is frequently criticized for its high
labor intensity, poor efficiency, and significant environmental
repercussions (
Bontempi, 2001). This is despite the fact that this
sector is extremely important to the economy. The construction
industry is responsible for around nine percent of the world’s
gross domestic product (GDP). According to the findings of yet
another study, in the year 2017, the construction industry was
responsible for approximately 20% of China’s total energy
consumption, approximately 23% of the country’s total power
consumption, and approximately 30% of the country’s total CO2
emissions (
Bolander, 2000). Each of these factors had a
significant impact on the environment.

As a direct consequence of this, there has been a surge in the
number of initiatives that aim to improve the track record of the
civil engineering business in terms of its effects on society, the
economy, and the environment. Since the introduction and
development of computer systems for structural design and
analysis in the 20th century, optimization strategies based on
mathematical programming techniques have been developed and
utilized in the field of civil engineering in the recent decades
Uta, 2009).
The procedure of getting the very best outcome that can be
accomplished is referred to as “optimization,” and the word
“optimization” is used to characterize this process. When it
comes to projects involving civil engineering, optimization may
be carried out throughout any step of the process, beginning
with planning and continuing through design, construction,
operation, and maintenance. When it comes to making anything
as efficient as possible, one of the most common approaches is
called structural optimization (
Bolourian, 2020). In the context
of this study, the term “structural optimization” refers to an
optimization strategy that pays no attention to the inherent
characteristics of the chosen materials. Instead, the focus is
placed on identifying the optimal configuration of structures or
structural components in order to achieve a predetermined set of
objectives within a specific set of constraints. The structures
made by civil engineers rely heavily on the materials they
utilize, since such materials are crucial to the functioning of the
finished product. The building industry and civil engineering
infrastructure both make substantial use of composite site
materials that are based on concrete. These materials include
plain concrete, reinforced concrete, pre-stressed concrete, and
many more (
Kaddah, 2019). However, due to the computational
challenges involved in taking material distribution into
consideration, structural optimization in civil engineering is
often only researched for buildings comprised of a single

material type. This is because of the nature of the field. The
process of optimizing a structure may be segmented into four
separate subfields, which are as follows:
Size optimization: also known as sizing optimization, which
treats the cross
sectional areas of structures or structural
members as the design variables;
Shape optimization: also known as configuration optimization,
which treats the nodal coordinates of structures as the design
Topology optimization: focuses on how nodes or joints are
connected and supported, aiming to delete unnecessary
structural members to achieve the optimal design;
objective optimization: simultaneously considers two or
more of the above op
timization objectives for better
optimization results; an optimization involving size, shape, and
topology at the same time is also known as layout optimization.
In the beginning stages of structural optimization research
within the domain of civil engineering, the primary focus is
placed on mathematical theories and programming methods that
are founded on fundamental structures as standards. Particularly,
topology optimization has seen increased use as computational
and construction techniques have advanced, allowing for the
optimization of larger and more complicated civil engineering
structures. This has led to an increase in the number of
companies that offer topology optimization services. The Qatar
National Convention Center (QNCC) in Doha, which is one of
the largest civil engineering structures created by generative
tools based on topology optimization, utilized topology
optimization during the design process in order to minimize
structural compliance (
Zhou, 2016). The topology optimization
was based on iterative 3D Extended Evolutionary Structural
Optimization (EESO) algorithms. The Shenzhen CITIC
Financial Centre, which can be found in Shenzhen, China, is a
further illustration of the use of structural optimization to a

large-scale civil engineering building project. Thanks to
topology optimization-assisted design, the exo-skeleton truss
arrangement was redesigned to be more efficient in its use of
materials while preserving the structure’s overall stiffness.
One of the key objectives of structural optimization is to bring
down the total cost of the construction. When it comes to
construction projects, it is almost always desirable to meet, or
even exceed, the criteria of structural performance at a lower
cost (
Robertson, 2007). According to a number of studies,
cutting down on the amount of weight a building has may
drastically cut down on construction expenses. In recent years,
thanks to a growing focus on environmental issues and
sustainable development, the substantial quantity of CO2
emissions produced by the civil engineering sector has made it
an additional important goal of structural optimization to
minimize the environmental consequences of the sector’s
activities. In addition, a number of research publications on
structural optimization center their attention on improving
certain structural performances, such as mechanical behavior,
aerodynamic performance, and dynamic seismic performance, in
order to adapt structures to a variety of environments.
Many different optimization solutions have been proposed and
put into action in an effort to accomplish the aforementioned
aims. As a result of its applicability to combinatorial
optimization problems, metaheuristic approaches have quickly
risen to prominence as one of the most sought-after optimization
strategies in the field of civil engineering structural optimization
research. This is due to the fact that combinatorial optimization
problems can be solved using these approaches. On the other
hand, these metaheuristic techniques come with a number of
downsides as well, such as their complexity and their inability
for high-dimensional scenarios. As a direct consequence of this,
there has been a rise in the amount of research conducted with
the objective of improving the effectiveness of optimization

strategies (Kumar, 2019). This research can take the form of
either refining and expanding upon existing metaheuristic
approaches or developing entirely new optimization strategies.
For example, Mortazavi developed an extra fuzzy decision
mechanism that increased the effectiveness of the interactive
search algorithm (ISA) in the process of optimizing the
structural size and topology. The fuzzy tuned interactive search
algorithm, often known as FTISA, is a hybrid strategy that
enhances solution accuracy while simultaneously reducing the
amount of computation time required. For the purpose of truss
size optimization, Degertekin suggested using not one, but two
improved versions of the harmony search technique known as
the efficient harmony search algorithm and the self-adaptive
harmony search algorithm. Experimental data gathered from a
variety of contexts demonstrates that the new approaches are
superior than the traditional harmony search algorithm in terms
of the amount of money spent on computation, the rate at which
convergence is achieved, and the results of optimization
Koyama, 2012). In addition, the transformable triangular mesh
(TTM) method offers an explicit topology optimization strategy
for structural topology optimization. This approach beats other
state-of-the-art algorithms in terms of its capacity to produce the
best feasible solution.
Researchers in the field of structural optimization have
demonstrated its potential and displayed its successes in the
aforementioned works, which aim to improve the productivity
and sustainability of civil engineering. These researchers have
shown that structural optimization has a lot of potential.
However, despite the fact that several research and survey
reports were published in this sector, not one of them was able to
offer a thorough review of the scientific achievements made on
structural optimization (
Mohan, 2018). As a result, the purpose
of this paper is to conduct a comprehensive review of the
literature on structural optimization in the field of civil gndissertation.cndc
engineering. This review will include an examination of the
optimization objectives and their temporal and spatial trends, an
examination of the optimization processes, which consist of four
main steps, as well as discussions of research limitations and
recommendations for future works.
A significant amount of study has been conducted on diagonal
cracking in SRC beams that do not have stirrups. The section
height, denoted by h, the concrete compression strength, denoted
by fc, and the shear span to effective depth ratio, denoted by a/d,
are the primary parameters that influence it. The effect that the
ratio of shear span to shear depth has on the resistance of
materials to diagonal shear cracking has been the subject of
discussion in a number of studies. Cracking along the diagonal
in high-strength SRC beams and the influence of early-age
shrinkage on the beams (
Talischi, 2010). The measurement of
the beams’ strength when subjected to flexural loading is a rather
basic problem; however, the analysis when shear loads are
applied is much more complicated. There is a dearth of
analytical explanations about the distribution of shear stress in
the cross sections of composite materials. A certain amount of
simplification of the problem and the experimental correlations
is needed, and because of this, it is necessary to include certain
assumptions. This is similar to the situation with the design code
equations. A number of expressions that may be utilized to
predict the beginning of diagonal cracking can be found in both
ACI 318 and Eurocode 2, and both of these building codes were
written in the United States (
Negri, 2017). It is essential to
highlight the fact that these codes are used in the design of
genuine projects due to the straightforward and allencompassing nature of their formulas.
The design of industrial facilities, the placement of equipment
and machinery, and even the organization of furniture in
professional settings are some of the most fundamental aspects
that have a direct influence on productivity and, by extension, 0th articce ,
the success of a firm. Experts in manufacturing and logistics
need to put in an incredible amount of effort in order to maintain
their competitive edge (
Holmberg, 2014). The layout is an
essential part of production in this effort to make improvements,
and it has a significant bearing on costs and the reduction of
travel lengths, both of which contribute to greater operational
efficiency. In this context, “betterment” refers to the process of
making something better. The use of the material that involves
“staying put” is the most effective usage of it. Ineffective layout
design and material management are responsible for between
twenty and fifty percent of all the costs associated with
production. Finding an efficient answer to the problem of
department placement is projected to result in a cut of between
10 and 30 percent in the costs of managing and running the
business, in addition to an increase in the effectiveness of the
production process.
Errors in the layout design can cause supply interruptions, which
can then lead to internal and external customer dissatisfaction,
production delays, which can then cause confusing and
unnecessary queues and stocks, and high costs related to
inefficiency in creating synergy between the physical
arrangement set.
Manufacturers are rethinking their production methods in order
to make room for newly developed manufacturing technologies
or improvements to existing products. Because of this, the
planning stage must take a significant amount of time before the
final design modifications can be executed. A layout that is welldesigned is supposed to help businesses save money by
optimizing their use of space, streamlining the movement of
workers and machines within the building, streamlining the
management of day-to-day operations like task assignment and
supply management, reducing the amount of downtime that
occurs between processes, and streamlining the overall operation
Bird, 2018). All of these things should help streamline the
The literature review in this part also has nothing to do with the theme of the article, which I emphasized many
times before, so these should be irrelevant. You should look for the results of applying Lo method to analyze
the fracture trajectory through the references I sent you. It can be paraphrase by literature review in these
references or summarized by the results of these articles.

overall operation.
In light of these positive aspects, the purpose of this article is to
investigate the procedures and sources that are engaged in layout
management in an attempt to get an understanding of the
environment in which it operates and the rules that govern it.
The purpose of this investigation is to carry out a comprehensive
literature review of the procedures that have been effectively
applied and investigated by manufacturing companies over the
course of the last ten years (
Rezaie, 2020). The findings are
extremely restricted to instances of uneven size and single row
layout; however, it is anticipated that the practical application of
these characteristics will become even more widespread in the
future, eventually encompassing a wider variety of scenarios.
This is despite the fact that the findings only cover a small
portion of the possible circumstances.
It is standard procedure to terminate a shear loading test as soon
as a diagonal fracture becomes visible. Once the test has been
terminated, the stress level at which the fracture first formed
may be recorded. During a three-point flexural test, this may be
accomplished by inserting linear crack gauges at the midspan in
a direction that is perpendicular to the principal stress axes. The
fact that these gauge types are so cumbersome, on the other
hand, makes it potentially difficult to include more experimental
devices. An alternative experimental strategy might be of
interest if the goal is to avoid interrupting the shear cracking
load prediction test in order to do a visual examination.
The well-known ‘ground structure’ based truss layout
optimization method recently saw the addition of a true
modeling of distributed self-weight as one of its optimization
criteria. The addition of equally stressed catenaries into the
ground structure helps to reduce the number of non-conservative
errors that are caused by forgetting about the bending effects
that occur inside members that are carrying their own weight
Wang, 2019). It is possible that the self-weight of a structure
will play a substantial role in the process of carrying the
imposed pressures; nevertheless, this may result in solutions that
need components to be arranged in intricate, overlapping
patterns. A redesigned formulation for layout optimization is
offered here in order to tackle this challenge. This formulation
allows for favorable unstressed masses like counterweights to be
included in the final design. The equation takes into account the
costs of abutments and anchorages, in addition to the frictional
support system (
Gaynor, 2013). We demonstrate that the method
that was presented is efficient by using both benchmark
examples and the conceptual design of a simplified long-span
bridge construction. This design takes into consideration both
the ground anchored and self-anchored choices for the building
of the bridge.
At the moment, a significant amount of effort has been put into
attempting to anticipate the form development of a fatigue
fracture with a traditional numerical analytic approach. On the
other hand, not a lot of work has been put into the majorization
of propagation spots and increment for the sake of simulation
accuracy. In this article, a fresh approach to the error
computation is given with the purpose of quantifying the
simulation findings and the experimental data. On the basis of
this information, the impact of the distribution and amount of
propagation sites on the outcomes of the simulation of crack
propagation has been analyzed and optimized (
Fontanesi, 2013).
In addition, in order to get a greater crack development
increment by using the calculus approach, a form of equivalent
stress intensity factor amplitude expression is produced. This
expression is based on the premise that the stress intensity factor
increases exponentially with fracture depth. The simulation
results are compared with already known expressions of the
stress factor, which allows the validity to be checked.
Back in the late 1970s, a pioneering effort was made to optimize
computer layouts, sometimes known as “topologies.” However,

despite enormous advances in the amount of computer power
that is accessible and the implementation of more effective
optimization algorithms, even in the present day, the old “ground
structure” technique can only be used to solve issues that are
relatively small in scale. This is due to the need that the latter
must, in general, include every possible member bringing
together the nodes in an issue. We provide a solution approach
that is not only straightforward but also successful, and it is
capable of addressing issues involving enormous numbers of
prospective members (such as more than one hundred million)
Prasad, 2003). Despite the fact that the approach is based on the
linear programming methodology known as “column
generation,” it is presented as an iterative “member addition”
method since layout optimization-specific heuristics are used in
the process. It is necessary for the first iteration of the procedure
to employ a ground structure that has a limited amount of
connection. After that, members are added in successive
iterations as needed until the (provably optimum) solution is
Limit State:
FORM is a powerful software application for engineers who
want to design strong and light components and structures. It is
particularly suitable for exploiting the design freedoms
associated with additive manufacturing (AM). The software
automatically identifies minimum-weight truss designs for
specified material stress or deflection limits.
Limit State: FORM provides a number of advantages over other
It can rapidly identify high fidelity optimized lattice design
solutions. It produces parametric geometry output, rather than a
mesh. Engineers can move quickly from defining the design
domain, to an optimized form, and back again, in one fluid
workflow. It is easy to refine the optimized design. Optimized
forms are highly efficient and light weight – in fact they are very

close to theoretical mathematical optima (Leonel, 2010).
There is no need to specify a target volume fraction – the
software will automatically determine the optimum component
geometry for any given set of input parameters. It includes a
range of analysis tools, and models can be easily exported in
standard formats for analysis in FEA tools. The software is built
on ANSYS SpaceClaim, a powerful direct modelling CAD
package, which offers a rich suite of editing tools. Models can
be analyzed in ANSYS Mechanical with a one-click process.
The tool has been proven in projects in the aerospace,
automotive, space and construction engineering sectors
VelascoHogan, 2018). Using the conservative formulas of
standard design codes is a safeguard against failure mechanisms.
However, if a non-conventional concrete type that is not
specified in the design codes is used or non-static or temperature
loading is applied, then the mechanical behavior requires further
investigation. Currently, a new generation of green concretes
based on the use of industrial by-products is under development
in the framework of the circular economy. Several studies have
been carried out, in order to assess their behavior and
applicability to building. However, research on their mechanical
behavior at the structural or macro level is scarce. A key
property that must be checked during the Ultimate Limit State
Design of Steel Reinforced
Concrete (SRC) beams is resistance to shear failure or shear
capacity. For instance, when the risk of earthquakes is high, it
would be of interest to implement structural monitoring by
means of non-destructive methods such as acoustic emissions.
The first sign of shear failure is a critical diagonal crack, the
width of which increases until the applied load reaches the
ultimate shear strength. The diagonal crack load is referred to as
the load at which the first diagonal crack occurs (
Portioli, 2016).
Determination of the onset of diagonal cracking in structural

concrete elements is essential to analyze their behavior under
shear loading. On the one hand, the diagonal crack plays a
decisive role in the ultimate shear strength. On the other hand,
the diagonal shear crack load is necessary to determine the
reserve shear strength factor. The reserve shear strength factor is
defined as the ratio of the ultimate shear load to the diagonal
shear cracking resistance (
Farahani, 2017). When analyzing the
reserve shear strength factor, the effects of concrete mixtures on
shear resistance, such as aggregate interlock, must be taken into
account. The safety margin of each material between the first
diagonal crack and the ultimate shear failure is a key parameter.
Although the pioneering study of high-strength concrete deep
beams without stirrups, the reserve shear strength parameter was
computed later in other studies when analyzing the behavior of
SRC beams with stirrups in the case of concrete with recycled
aggregates and self-compacting concrete (SCC).
Identification of cracks
All kinds of engineering structures inevitably have cracks, holes
and other defects due to the influence of many factors, such as
construction quality, applied load and temperature change
(Figure 1 shows some common defects in typical concrete
structures). As a representative type of defect, the existence and
evolution of cracks will lead to the decrease of structural bearing
capacity, the degradation of serviceability and durability, and
even lead to more serious consequences such as collapse,
endangering the safety of life and property. Therefore, it is of
great significance to detect and identify cracks in various
structures quickly and accurately (
Garcin, 2015).
Crack identification in structures is a typical inverse analysis
The literature review in this part also has
nothing to do with the theme of the article,
which I emphasized many times before, so
these should be irrelevant. You should look
for the results of applying Lo method to
analyze the fracture trajectory through the
references I sent you. It can be rewritten by
literature review in these references or
summarized by the results of these articles.

problem to quantify crack geometries such as the location, size
and type of cracks. At present, the most frequently adopted
crack identification methods include manual detection
technology, ultrasonic detection technology and image
recognition technology, but these methods have some
limitations. For example, manual detection technology is
laborious and inefficient; ultrasonic detection technology will
lead to inaccurate measurement under high temperature, and the
image recognition method is easily affected by the environment
Dung, 2019). In recent years, the rapid development of
numerical technologies and intelligent optimization (IO)
schemes provides a new way for crack detection and
identification. In the collaborative numerical modeling-IO
approaches, the numerical methods are developed to perform
forward crack simulation, upon which the required data is
obtained and then used in the IO algorithm to further identify the
crack configuration. In the following, we will give a brief review
on the research status of representative numerical methods and
IO schemes for crack modeling and identification.
Chapter 3: Methodology
Depending on a number of circumstances, including the local
soil and material conditions and availability, as well as other
site-specific considerations, a number of different types of
foundations may be suitable for use with ground-anchored
systems. These factors include: A gravity-type foundation is
supported by a massive anchoring block that is positioned in
such a way that the amount of frictional force that exists
between its base and the ground is sufficient to sustain the
horizontal stress. It is possible that the mass, in this
configuration, may be made up of locally accessible and
inexpensive materials like sand, which would represent a major
cost reduction. The use of direct anchoring into the bedrock is
one alternative design that, depending on the conditions of the

site, may be acceptable.
The use of counterweights has allowed for the creation of
innovative geometries for bridges with reduced spans. For
example, the ingenious Alamillo Bridge designed by Calatrava
does away with the need for back stays by hanging a cable net
from an incline pylon. However, many experts in the subject
have pointed out that the form that arises as a result of this
situation is very wasteful from a structural standpoint. In the
current body of optimization literature, there are very few
resources that engineers may consult in order to develop
efficient gravity-balanced structural forms. This is true whether
the engineers choose to make use of distributed self-weight
components or explicit lumped masses.
The literature that already exists on optimization seldom takes
into consideration scenarios that use support types other than the
usual fixed pin or pin/roller supports. frictional support types
where the cost grows with the amount of the reaction force are
examined, and different solutions based on these types are
given. The cost of the frictional support types increases with the
amount of the reaction force. These, on the other hand, need
prior acquaintance with the fundamental structure of the optimal
solution. In addition, self-weight, which may be employed in
combination with an anchoring block to bear horizontal forces,
was not taken into consideration. This was a significant
omission from the analysis.
The costs of unstressed material, such as those found in
anchoring or abutment structures, are included into the
distributed self-weight method. This allows for the costs of
unstressed material to be taken into consideration in conjunction
with the impacts of the shifting boundary conditions. Because of
this, it will be feasible to study the prospect of cost reductions in
the self-anchored and ground-anchored cases, in addition to the
more realistic circumstances in which friction will offer the sole
form of restriction. It will be feasible to analyze different
When you modify this part, you can refer to the article written by my tutor and other articles. You can analyze the
optimization processing of the structure by using layopt software or limitstate: form, and then use this method to analyze
the crack trajectory and apply similar data processing methods to modify it.

situations by making use of materials that are not strained,
which will make it possible to include counterweights into the
structural solution.
Layout optimization
The classical ground structure-based truss layout optimization
procedure is shown diagrammatically below. When distributed
self-weight is included, each straight-line connection between
nodes is replaced by a pair of equal strength (i.e., equally
stressed) catenary elements, one to carry compressive forces and
the other tensile forces. However, the resulting problem
formulation differs from the standard formulation only in the
composition of the coefficient matrices such that linear
programming can still be used to obtain solutions; thus, for a
problem comprising
n nodes and m potential elements the
formulation can be written as:
For this part, we refer to some literatures to analyze the crack with layopt or limitstate: form.
After analyzing the structural optimization with these two software, we can analyze the crack
trajectory. This is probably done through tension and compression truss (I guess so).
So, if we can identify that direction quickly and efficiently from a Layout Optimisation (LO)
analysis(*) we can then position a “hole” in the design space, which represents the initial
direction of the crack. Then we can re-run the analysis with this “hole” and see how the
direction of the maximum tension changes. This way, by running the analysis several times, we
can see if the LO approach accurately predicts the direction of the crack as it grows. If it does,
then we may have a very useful new crack prediction method.

Layout optimization stages: a problem
b design domain discretized with grid of
c form of ground structure for a problem without
self-weight—employing straight truss members connecting
each pair of nodes;
d resulting optimal solution; e ground
structure for a problem with distributed self-weight—
employing two equally stressed catenaries connecting each
pair of nodes;
f resulting optimal solution, comprising
tensile members sagging downwards and compressive
members arching upwards due to distributed self-weight.
This part is just like what you wrote, but it should be more specific. More examples
should be added, that is, the optimal solution can be generated by setting different
specifications and numbers of examples and data analysis. This part needs a large
number of numbers and pictures to describe, and then the fracture trajectory can be
analyzed in combination with pressure and tension.

Member with distributed self-weight: a geometry; b end
force in the case of a single equally stressed catenary
member. Dashed lines correspond to corresponding
member without self-weight.
It should be noted that when self-weight is considered, members
that would, in the non-self-weight formulation, have overlapped
and been superfluous should now be explicitly included in the
model. Such members are included in the ground structure
shown, which shows curved elements spanning across two or
three nodal divisions (e.g., along the top and bottom edges of the
domain). It is evident that, although the end nodes of members
may lie on the same straight line, the elements themselves are
not coincident, and thus more than one element may exist in the
optimal solution.
Finding a crack trajectory
A crack’s path has been mapped out over the course of the last
three decades thanks to the application of the variational
principle, which is a central tenet of the majority of the
competing crack theories. Considerations of both energy and
force may provide starting criteria for fracture propagation. A.
A. Griffith proposed the concept of an energy fracture criterion
in the year 1920, and G. R. Irwin produced the force criterion in
The whole part is to use lo to analyze and predict the crack trajectory, that is,
Lo itself is to find the optimal solution of the structure. However, it is necessary
to find and predict the crack trajectory from this method. If feasible, this is a
new method to predict the crack trajectory. Modify it after understanding my
remarks above.

the year 1957, concurrently establishing that the two criteria are
equivalent. It has been shown that the limiting equilibrium state
of a cracked continuous elastic body may be identified by
making use of the Irwin force criteria for crack extension in
conjunction with the corresponding Griffith energy criterion.
[Citation needed] This formulation may be commonly
recognized, but it is just one of several that may be used to
calculate the limiting equilibrium state of a broken body. Other
formulations include: The models created by Leonov and
Panasyuk in 1959, Dugdale in 1960, Wells in 1961, Novozhilov
in 1969, and McClintock in 1970 are among the most wellknown examples of this type of model (1958). Finding the
limiting load and the correlated crack propagation direction is a
variational problem that is typically reduced to finding the
extrema of a function that has multiple variables [9]. The crack
propagation problem is formulated on a discrete level in this
paper, as opposed to the variational approach that is typically
taken, which is more common. The numerical computation is
then carried out using a cutting-edge technique known as the
Cell Approach, which is a revolutionary numerical method for
solving field equations (CM). The absence of the requirement to
establish a model for the treatment of the area right before the
crack’s leading edge is one of the advantages of adopting a
discrete formulation rather than a variational one. Other
advantages include:
As an example, take the scenario in which the plastic
deformation is confined to a certain area. When examining crack
issues for an elastic-perfectly plastic body using the energy
equilibrium criteria, a solution is frequently supplied
immediately in front of the crack edge. This occurs just before
the crack edge. The amount of distance that this area is able to
cover is on the order of elastic displacements. In addition, if the
elastic-plastic issue is changed into a purely elastic one by
reducing the size of the plastic zone in front of the edge, the

problem will be solved. This is feasible due to the fact that the
action of the plastically deforming material is replaced in the
linearized formulation by the action of applied forces along the
face of an extra cut, which results in a narrower plastic zone. As
a consequence of this, it is brought to the attention of the reader
that the plastic non-linear effects region in the model that is
being considered varies with the external load and represents a
plastically deforming material in which the state of stress and
strain must be determined from the solution of an elastic-plastic
problem. This information is brought to the reader’s attention as
a result of the fact that the model is being considered. In
contrast, the discrete formulation does not make any
assumptions about the size and geometry of the plastic zone.
Instead, the computation is performed immediately, without first
translating the issue into an equivalent elastic one.
Crack extension criterion
Numerous criteria, including as the maximum normal stress
criterion, the maximal strain criterion, the minimal strain energy
density fracture criterion, the maximal strain energy release rate
criterion, and the damage law criteria, may be used to calculate
the limiting load. The current work investigates the fracture
extension condition in the Mohr-Coulomb plane. When
calculating the limiting load, the condition of tangency between
Mohr’s circle and the Leon limit surface is used (Fig. 2).
My understanding is to analyze the following situations by selecting the initial structure with
different number of components.
1. Crack trajectory analysis when the same load is applied to the same position of these different
2. Analysis of crack trajectory when different loads are applied at the same position of these
different structures.
3. Crack trajectory analysis when the same load is applied at different positions of these different
4. Crack trajectory analysis when different loads are applied at different positions of these
different structures.
5. Analysis of crack trajectory when different loads are applied at different positions of the same
6. Analysis of crack trajectory when the same load is applied at different positions of the same
7. Analysis of crack trajectory when different loads are applied at the same position of the same

In the Mohr-Coulomb plane, the Leon criterion may be
expressed as follows: where c represents the cohesion, f
represents the compressive strength, tb represents the tensile
strength, n represents the shear stress, and sn represents the
normal stress on the orientation of the external normal n.
Because the CM domain discretization is of such significant
relevance to the physical world, it is possible to calculate the
Mohr circle for the tip neighborhood. The CM divides the
domain into two parts by making use of two different cell
complexes. The first cell complex is a simplicial complex,
which means that each cell in that complex only contains a
single node from the second cell complex. In order to construct
the two meshes that are necessary for the two-dimensional
domains, the study in question makes use of a Delaunay and
Voronoi mesh generator. The first mesh, known as the Delaunay
mesh, is produced when the domain is segmented into triangles.
The perimeter of the triangle does not include any extra
locations or places (Fig. 3). The Voronoi mesh is the outcome
that you get when you take the polygons whose centers are the
circumcenters of the main mesh and put them together (Fig. 3).
Every point that is geographically close to a Voronoi site is in a
position where it is physically closer to that site than it is to any
other Voronoi site. Every main vertex has a dual polygon that
corresponds to it, and this dual polygon is required to follow the

conservation rule. We were able to find the Mohr’s circle that
defines the tip neighborhood by inserting a hexagonal piece near
the apex, which served as a marker (Ferretti in press, Fig. 3). In
order for the mesh generator to function, the hexagonal element
must first be sliced into two Delaunay triangles with equal sides.
Next, a quasi-regular tip Voronoi cell must be constructed from
these triangles (the cell filled in gray in Fig. 3). Because of this,
we were able to calculate the relationship between the
orientations of the Voronoi cells at the vertices of the tip and the
stress field at the tip itself. The tension points provide an exact
description of the Mohr’s circle in the Mohr-Coulomb plane with
each rotation of the hexagonal element around the tip.
As a consequence of this, we are able to identify the direction of
propagation by following the path of the line that starts at the
tangent and terminates at Mohr’s pole (Fig. 2).
Topology optimization formulation considering material and
construction costs
In order to begin the process of topology optimization, the first
step is to formally formulate an optimization problem statement
for the design issue. In this investigation, we use a method to
topology optimization that is predicated on attaining the lowest
possible compliance, which is analogous to achieving the
highest possible stiffness. As a result of the fact that this
formulation reduces the amount of external effort, or,
After the above modification and
analysis, it can be changed into the
causes of cracks and how to solve
the problems of cracks, such as the
treatment methods when cracks
appear in different structures.
This part can be written but the number of words should be reduced, because the focus of this paper is not here
but the analysis part I mentioned above.

equivalently, the amount of internal strain energy, it may be used
in the construction of structures in which forces are directed
along the load path that is the most rigid. The topology
optimization process is then used to mesh the design domain; in
the hybrid truss-continuum technique, the truss ground structure
serves as a stand-in for the steel, while the continuum finite
element discretization models the concrete (Figure 2).
Calculating the volume percentage c of the continuum elements,
where c = 1 indicates compression-carrying concrete and c = 0
indicates non-load-carrying concrete, and the cross-sectional
area t of the truss components is the job of the optimizer.
Therefore, the design of the steel reinforcement (the ties) is
represented by truss elements with non-zero cross-sectional
areas, and the path that the compression load will take is
represented by continuum elements with a value of c equal to
one. (the support beams)
Let’s describe the equilibrium condition as Ku F, where K is the
global stiffness matrix that is determined by t and c in the
design. In addition, let’s express the external work as F uT,
where F are the applied nodal loads and u are the nodal
displacements. This will give us the equilibrium condition.

Therefore, the optimization problem for the lowest possible
compliance (the maximum possible stiffness) is as follows:
where the second constraint is the total cost TC, which is
composed of the material cost M and the construction cost C;
the third set of constraints is the bounds on the design variables
for the continuum elements in the domain (denoted as c), with
cmin chosen as a small positive number to preserve the positive
definiteness of the global stiffness matrix; and the fourth set of
constraints is the bounds on the design variables for the truss
elements in the domain (denoted as t). We make use of the
formula in order to calculate the cost of the materials within the
parameters of the total cost restriction.
where denotes the length of the steel bars and indicates the
volume of the element in continuous concrete, and e c and e t are
the unit costs of the concrete and steel, respectively. Constraints
on material volume use such as this one are rather typical in the
minimum compliance conundrum. Asadpoure et al. 23 recently
introduced the construction cost to represent the manufacturing
cost in discrete structures, and the formula that they used may be
found as follows:
Any truss element that has a cross-sectional area that is more
than zero is regarded as a needed element, and the function H,
also known as the Heaviside step function, is the symbol used to
symbolize this concept. Since the continuum elements are what
represent the concrete domain, it is essential to notice that this

function contains just the truss (steel) components. This is
because the concrete domain is the one that is being represented.
The term e f is shorthand for the amount of money that must be
spent to construct element e. This element in truss constructions
accounts for the labor expenditures associated with inserting a
member (including the time required for the use of a crane) and
establishing two connections (one at each end). Although the
amount of the element construction cost will ultimately be
determined by the local market and the building procedures
used, the purpose of this article is to explain how this cost term
may be leveraged to effect the buildability of rebar timescales.
Because H is discrete, it has to be regularized before optimizers
that are dependent on gradients may utilize it. Following is an
explanation of the regularization function that should be used for
projection approaches in continuum topology optimization. This
function was first proposed by Guest et al. 24.
According to Guest et al. (25), the value of the regularization
parameter dictates how well the step function is approximated.
This work has the value set to 10. If a steel truss component
achieves a non-zero cross-sectional area, this equation gives a
magnitude of one, which puts the element’s unit cost onto the
total cost function. This is because the equation yields a
magnitude of one. When e e c s and 0 e f are both satisfied, it is
clear that this total cost limitation is equivalent to the
fundamental volume constraint that Yang et al. 20 utilized. In
order to enable a comparison between the costs of materials and
construction and to highlight the influence of the latter on
optimizing the ideal position of reinforcing steel, the fixed and
equal to unit values of e c and e s are employed. These values
may be found in the equations. Therefore, if you change the
value of e and f, the ratio of the cost of constructing to the cost
of materials will change. We want to emphasize that the values
of these components may be determined by utilizing actual

market pricing for concrete, steel, and the cost of labor in the
area for placing steel bars.
As a result of this, the Heaviside Projection Method, also known
as HPM, is applied (Guest et al. 24, Guest 26) in order to get
around the numerical instability of checkerboards and the mesh
dependence of continuum components. The adjoint method is
used to compute sensitivities (see Gaynor et al. 19 for related
equations for the hybrid topology optimization), and the
gradient-based optimizer, the Method of Moving Asymptotes
(MMA) (Svanberg 27), is utilized because it is efficient for
structural optimization. Both of these methods can be found in
Svanberg. The algorithmic details for this technique is presented
in detail in Guest et al. 25, which can be found here. Last but not
least, we want to point out that the hybrid truss-continuum
method is based on a bilinear, stress-dependent mechanics
model in which the truss components (steel) carry only tension
and the continuum elements (concrete) carry only compression.
This model assumes that the truss components carry only
tension and the continuum elements carry only compression.
Concrete is assumed to have a Young’s modulus of 24.9 GPa
(3600 ksi) in compression and 2.0 GPa (290 ksi) in tension in
the numerical examples presented in this work. On the other
hand, steel is assumed to have a modulus of 200 GPa (29000
ksi) in tension but no modulus at all in compression. This
assumption was made for the sake of simplicity.
Chapter 4: Results and Discussion
The first numerical example is the classic simply supported
beam problem with a load in the middle of the beam, as shown
in Fig. 3a. This problem is a simple example of a beam issue.
Figure 3b exhibits a typical STM, whereas Figure 3c shows a
topology-optimized solution that just takes into consideration
the cost of the materials and ignores the cost of the building. We
have allowed a fine structural topology to closely approach this
and highlight the difference when assessing constructability
Finally, the remaining parts will be summarized, analyzed and
rewritten according to the above modifications

because it is common knowledge that minimal strain energy
topologies will mirror the primary stress trajectories. This was
done in order to take advantage of the fact that this is the case.
Naturally, simpler topologies might be achieved by making
adjustments to the fundamental ground structure (see Gaynor et
al. 19 for a discussion on this). The solution may be found in
Figure 3d, which depicts the situation where the ratio of unit
labor expenses to unit material costs is very large. This design
not only has a lower bar count, but it is also more
straightforward, and as a result, it would have a cheaper overall
construction cost. Because these bars have a considerably larger
cross-sectional area, the total cost of the materials is much
greater than what is depicted in Fig. 3c. The solution shown in
Figure 3e was reached as a result of placing a higher value on
inclined rebar in contrast to horizontal and vertical rebar. Due to
the fact that this was the situation, the algorithm came to the
conclusion that it would not be appropriate to make use of
inclined rebar, despite the fact that this specific design example
would benefit from the latter’s greater structural efficiency. Keep
in mind that two inclined bars were included in the final design
because the price was not high enough to negate the structural
efficiency of the inclined bars and the accompanying material
cost savings that resulted from the usage of the inclined bars.

A standard example of another type is a deep beam that has
openings, as can be seen in Figure 4a. Figure 4b illustrates a
typical STM that has been constructed using only horizontal and
vertical steel ties. Figure 4c illustrates an answer that has been
topology-optimized while taking into account only the material
cost. It has been noted that it is made up of a significant amount
of steel rebar, which lessens the utility of the proposed STM. If
you take a look at Fig. 4d, you’ll notice that the STM becomes a
great deal easier to understand once you take into account the
total cost of materials and labor. If the construction cost of these
steel rebar is increased, as shown in Figure 4e, the resulting steel
rebar will have a shallower angle of inclination. This will
produce a different STM. These findings are preliminary, and
they should be interpreted as such; however, they do
demonstrate the potential of including labor cost in STM
optimization. [Citation needed] A significant challenge is the
estimation of labor costs, which are significantly impacted by
the economies of individual regions. In this work, we simply
express these costs as a ratio to the material costs in order to

demonstrate the concept and investigate the tradeoffs that exist
between the costs of material and constriction.
Chapter 5: Conclusions and Recommendations
In this research, we describe an automated technique for
constructing optimum strut-and-tie models in reinforced
concrete structures using a performance-based evolutionary
topology optimization approach. To demonstrate the utility of
the suggested optimization approach, we have offered five
examples that span many kinds of reinforced concrete
components. Strut-and-tie models derived using the current

optimization approach have been demonstrated to be consistent
with both analytical solutions and experimental findings.
Optimal strut-and-tie models in prestressed concrete structures
and reinforced concrete shear walls may also be found using this
approach. More research is required in both the theoretical and
practical realms to make topology optimization a user-friendly,
routine design tool for the concrete industry.
When designing any kind of industrial facility, it is crucial to
take into account the complicated nature of the plan. This article
presents a mathematical strategy for creating flexible industrial
designs. The model considers the interdependence between the
block and detailed layout problems. This cutting-edge model
considers not only the dissimilar inputs and outputs of each part
of the system, but also their dissimilar geometries. By
determining where, in a two-dimensional (2D) space, all of the
plant’s equipment and pipes may be placed to achieve the lowest
feasible connection costs, an optimum plant design can be
determined. A variety of topological factors are considered,
including the position and orientation of the apparatus, its
proximity to other components, the non-overlapping limitations
of those components, the input/output connection, the shape, and
the available space over a 2D continuous region. Here,
production and operations are modeled simultaneously within
the bounds of safety and operability.
The challenge of maximizing production within a continuous
three-dimensional space has been investigated. The process of
developing a broad model that incorporates key features of the
existing reality. Rectangular and asymmetrical equipment
shapes, input and output connections, equipment orientations,
distance constraints, space availability, and allocations over
many floors are only some of the topological features that have
been represented. We analyzed two cases that included multitiered allocations. There are two possible strategies for arranging

a building’s residential units: scattering them over an unspecified
number of floors, each of which may have a different height, and
scattering them across a specified number of levels with uniform
heights throughout the building. The model also accounts for
potential constraints on safety and operability, as well as the
presence of production or operational segments. Ultimately, the
model is distinguished by its mixed-integer linear programming
(MILP) formulation. The optimal plant layout is guaranteed by
solving this issue, which takes into consideration both the capital
costs of connection and the operating expenditures, the latter of
which are principally related with the costs of flow pumping. As
such, the model’s primary objective was cost reduction. This
ensures that the final plant structure is economical both initially
and over time. Their responses to the numerous set-up problems
displayed a wide range of design characteristics. With the use of
computers, we were able to quickly determine the best course of
action for fixing this sort of issue. While this approach may be
time-consuming in more complex cases, it is still an alternative
worth exploring. As a result, it’s feasible that future research
may want to investigate whether or not modifying the problem’s
features or using novel solution strategies can help to speed up
the computation time.
Future Work
It is recommended that future theoretical work investigate the
impact of material property on the best strut-and-tie models,
while also limiting the impact of element mesh size and removal
ratios. Furthermore, experimental research is required to learn
about the ultimate load capacity of reinforced concrete members
constructed using optimum strut-and-tie models created by the
current topology optimization method. Results from the tests
will be compared to industry standards.
This is a model for addressing design and layout concerns as a
single, overarching difficulty at the same time. This formulation
is very helpful for modeling design challenges that entail

analyzing the particulars of a given plant, and it can be altered
with very little effort on the user’s part. The proposed model for
the layout problem in a 2D continuous area takes into account
important topological aspects such as the orientation of the
equipment, the availability of space, the diversity of the inputs
and outputs, the diversity of the shapes (both rectangular and
irregular), and the safety and operational constraints that are
translated into distance constraints (minimum and maximum).
The model generates a mixed integer formulation that, given a
certain economic goal, zeroes down on the most suitable plant
layout. In the context of this discussion, “minimization” refers to
the reduction of the total cost of the connection structure as well
as the decrease of the unit capital cost of the equipment. In
conclusion, we are able to make the observation that a
formulation that is rather generic is provided, and that the layout
outcomes that are generated from this formulation reflect actual
circumstances with a high degree of precision. In further
investigations, we want to make advantage of the model’s
potential by applying it to the planning of industrial
multifunctional batch facilities at the same time as the process of
design is being initiated.

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