Dervishi PAPER

Faculty of Architecture and
Engineering
Department of Computer Engineering
APPLICATION OF PROBABILITY
DISTRIBUTIONS AND NUMERICAL METHODS.
OPTIMIZATION PROBLEM,
APPROXIMATIONS,
ERRORS,
INTERPOLATION
Megi Dervishi
Computer Engineering Department ,Epoka University ,Tirana ,Albania
Abstract
This article explores the application of probability distributions and numerical methods in
various fields, including mechanical and electrical oscillations theory, electromagnetic wave
theory, wave theory in general, constructive mathematics, mathematics, engineering, computer
science, and civil engineering. The purpose of the research is to demonstrate the usefulness and
effectiveness of probability distributions and numerical methods in solving optimization
problems, approximations, errors, interpolation, and other related problems .The article
provides a detailed explanation of the different probability distributions and numerical methods
used in each of these fields, but is focused mostly on application of numerical methods in civil
engineering. The data used in this study are based on real-life scenarios and case studies from
each of these fields. The main results of the research demonstrate that probability distributions
and numerical methods are essential tools in solving complex problems in various fields. They
provide accurate approximations and predictions, as well as solutions to optimization problems
with minimal errors. The article highlights the importance of these methods in real-life
applications, such as finite element analysis computational fluid dynamics and the analysis and
design of geotechnical structures such as tunnels, retaining walls, and foundations .So
numerical methods are essential in civil engineering as they provide efficient solutions to
complex problems in designing and analyzing civil structures. These methods allow engineers
to evaluate the behavior of structures under different loading conditions, optimize their designs,
and ensure their safety and efficiency.
Keywords: numerical methods, purpose, real-life scenarios ,civil engineering ,structures
Faculty of Architecture and
Engineering
Department of Computer Engineering
Introduction
The article is divided into three parts, each of which highlights the importance of these methods
in different aspects of civil engineering.
1.0 General Knowledge
The first part of the article provides a general introduction to the topic of numerical methods.
The article explains how these methods are used to solve complex problems, such as
optimization problems, approximations, errors, and interpolation. The article provides a brief
overview of the different methods used and highlights the most commonly used applications of
these methods.
1.1 Numerical methods are powerful tools that are widely used in many fields of science and
engineering.. In this section, we will provide an overview of these methods and explain their
importance in solving complex problems.
They are computational techniques used to solve mathematical problems that cannot be solved
analytically. They involve the use of algorithms and computer programs to obtain approximate
solutions to complex problems.
1.1.1 Optimization problems are a common application of probability distributions and
numerical methods in civil engineering. These problems involve finding the best solution to a
given problem subject to certain constraints. For example, an engineer may need to optimize
the design of a building to minimize the cost of construction while ensuring that the structure
is strong enough to withstand the forces it will be subjected to.
Example The famous diet problem (the perhaps first application in linear optimisation) how to design a meal with enough nutritions for a school kid, as cheap as possible.(
Dehghan & Ghanavati,2018)[2]
Figure 1:linear programming to optimize diets
1.1.2 Approximations are another application of probability distributions and numerical
methods in civil engineering. These involve approximating the solution to a complex problem
by using simpler, more manageable models.
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1.1.3 Errors are an important consideration when using probability distributions and
numerical methods. These methods always involve some level of approximation, and the
accuracy of the solution depends on the choice of method and the level of detail in the model.
Engineers must carefully assess the level of error in their models and determine the sensitivity
of the results to changes in the input parameters.
Example Meteorologists use statistical measures such as confidence intervals and error margins
to convey the level of uncertainty in their forecasts. For example, a forecast for a high
probability of rain with an error margin of plus or minus two hours means that there is a high
chance of rain, but the exact timing of the rain may vary by up to two hours.(Xue & Shen &
Chou,2013)[1]
Figure 2:NWP model(Numerical Weather Prediction)
1.1.4 Interpolation is another common application of probability distributions and numerical
methods in civil engineering. This involves estimating the value of a variable at a point within
a given range based on the known values at other points. For example, an engineer may need
to interpolate the strength of a material based on the results of laboratory tests.
Example Interpolation is used in financial modeling to estimate the value of an asset or liability
between two known values.
Figure 3:finding IRR(Internal Rate of Return)
2.0 Application in Civil Engineering
The second part of the article focuses on the applications of probability distributions and
numerical methods in civil engineering. This section highlights how these methods are used to
design and analyze civil structures, including buildings, bridges, dams, and hydraulic structures.
The article explains how probability distributions and numerical methods are used to evaluate
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the behavior of structures under different loading conditions, optimize their designs, and ensure
their safety and efficiency.
2.1 Buildings: In the design of a high-rise building, engineers use probability distributions to
model the variability in the wind loads that the building will experience. By using numerical
methods, such as wind tunnel testing or computational fluid dynamics, engineers can evaluate
the structural response of the building to wind loads and optimize its design to ensure that it
can withstand strong winds.
Figure 4:wind loading on tall buildings
2.2 Bridges: In the design of a suspension bridge, engineers use probability distributions to
model the variability in the traffic loads that the bridge will experience over its lifetime. By
using numerical methods, such as finite element analysis or dynamic response analysis,
engineers can evaluate the dynamic behavior of the bridge under different loading conditions
and optimize its design to ensure that it meets the required safety and efficiency standards.
2.3 Dams: In the design of a concrete dam, engineers use probability distributions to model
the variability in the concrete strength and in the hydrological conditions, such as river flow
and rainfall. By using numerical methods, such as finite element analysis or computational
fluid dynamics, engineers can evaluate the structural and hydraulic performance of the dam
under different loading conditions and optimize its design to ensure that it can withstand
extreme events, such as floods and earthquakes.(Singh,2020)[6]
2.4 Hydraulic structures: In the design of a stormwater drainage system, engineers use
probability distributions to model the variability in the rainfall intensity and duration. By
using numerical methods, such as hydrologic modeling or hydraulic modeling, engineers can
evaluate the capacity of the drainage system to handle stormwater runoff and optimize its
design to prevent flooding and other damages. (Cruz & Lepech ,2015)[7]
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2.4.1 Example-Heavy rainfall modelling in southern Brazil
Figure 5:rain gauges
Table 1: descriptive statistics of the probability distribution parameters
3.0 Detailed Discussion
The third part of the article provides a more detailed discussion of the different numerical
methods used in civil engineering, including finite element analysis (FEA), computational fluid
dynamics (CFD), and geotechnical analysis. The article explains how each of these methods is
used to analyze and design different types of civil structures and highlights their most
significant applications.
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3.1 Application of Numerical Methods in Design of Hydraulic Structures(CFD)
Numerical methods can assist in the design of hydraulic structures in the evaluation of the
energy loss, calculated discharge coefficient and cavitation phenomena investigated by
examining the pressure field and flow field. Designers are challenged to use numerical
methods for the design of hydraulic structures due to the complexity of the specific flow field
of hydraulic structures, such as the free surface flow, two-phase and multi-phase flows,
turbulent flow and turbulence. (Rad,2016)[3]
.Computational Fluid Dynamics (CFD) is a type of numerical model that can be used to solve
problems involving fluid flow. CFD can provide a significant amount of computation time
and more economical solution than a physical model.
Generally, a CFD model study consists of the following steps: i) obtaining the data of the
physical model for grid development; ii) selecting or developing appropriate model method;
iii) define the boundary conditions based on available field information; iv) develop the
computational grids; v) calibrate and verify the model; and vi) analysis various parameters or
scenarios.
Figure 6:Types of spillways
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Figure 7:Moving fluid element
Body force:
Net Force:
Total force:
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3.2 FEA (Finite element analysis)
Finite element analysis (FEA) refers to the process of using mathematical calculations,
models, and simulations to understand, analyze, and predict the behavior of an object, part,
assembly, or structure under various physical conditions. FEA is typically used by engineers
to model physical forces that help them identify vulnerabilities in the design of their product
prototypes. It allows them to localize the design problems and optimize the components,
thereby streamlining the design phase of the product. Potential design issues may refer to
identifying overheating parts, weak spots, or areas of tension. (Kanade,2022)[4]
Technically, FEA uses the finite element method (FEM) to break down the structure of an
object into multiple elements and manageable portions, which are later reconnected through
points called nodes. FEM generates a series of algebraic equations that engineers, scientists,
developers, and designers can use to perform the FEA of any structure.
3.2.1 Example
One example of using FEA techniques with a formula is the analysis of a beam under bending
loads. Consider a simply supported beam with a length of L, a rectangular cross-section with
a width of b and a height of h, and a material with a Young's modulus of E and a Poisson's
ratio of ν.(Desai,2021)[5]
The bending moment in the beam can be calculated using the formula:
M = 1/4 * w * L^2
where w is the distributed load applied to the beam. This formula assumes that the beam is
uniformly loaded and that the maximum bending moment occurs at the center of the beam.
To analyze the stress and deformation in the beam under bending, FEA techniques can be
used to divide the beam into a series of finite elements. Each element is then analyzed
separately using the principles of elasticity to calculate the stresses and deformations.
The stress in each element can be calculated using the formula:
σ=M*y/I
where y is the distance from the neutral axis to the point where the stress is being calculated,
and I is the moment of inertia of the cross-section.
Similarly, the deformation in each element can be calculated using the formula:
δ = M * L^3 / (3 * E * I)
where δ is the deformation at the point where the deformation is being calculated.
By combining the results from all of the finite elements, the overall stress and deformation in
the beam can be determined. This information can then be used to optimize the design of the
beam for maximum safety and performance.
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Department of Computer Engineering
Conclusion
In conclusion, probability distributions and numerical methods are powerful tools that are
widely used in various fields of science and engineering, including civil engineering. These
methods allow engineers to solve complex problems, such as optimization problems,
approximations, errors, and interpolation, with high accuracy and efficiency. In civil
engineering, numerical methods are essential in designing and analyzing civil structures,
optimizing their designs, and ensuring their safety and efficiency under different loading
conditions. Therefore, engineers must have a good understanding of the principles and
applications of probability distributions and numerical methods to provide efficient solutions
to the challenges of modern civil engineering.
References
1.Jin, T., & Wang, X. (2013). Forecast error correction method in numerical weather
prediction by using recent multiple-time evolution data. Meteorology and Atmospheric
Physics
2.Dehghan, P., & Ghanavati, M. (2018). Nutritional composition and anti-nutritional factors
of legume seeds with emphasis on chickpea
3.Naderi Rad, I. (2016). Application of numerical methods in design of hydraulic structures.
Department of Civil Engineering, Malayer Branch, Islamic Azad University, Malayer, Iran.
4.Kanade (2022)."What Is Finite Element Analysis (FEA)?"Spiceworks
5.Vandit Desai (2021)."FEA Analysis for Bending in Beams." Skill-Lync
6."Forces Acting on Gravity Dam." (2020, October 18). Civil Engineering Web.
7.Cruz, R. A., & Lepech, M. D. (2015). Resilience-based design of engineered and
infrastructure systems. Procedia Engineering