Preface

The relentless pursuit of understanding the complexities of our world has always been a driving force in the evolution of human knowledge. From the early days of mathematical breakthroughs to the current age of advanced computing, our quest to analyze, predict, and control natural and humanmade phenomena has led us to develop increasingly sophisticated tools and techniques. The advent of numerical simulation has been a key milestone in this journey, empowering scientists and engineers to tackle problems once thought out of reach.

*Numerical Simulation – Advanced Techniques for Science and Engineering* provides a comprehensive and accessible introduction to this ever-expanding field. It equips readers with the knowledge and skills to understand, design, and implement numerical simulations in various disciplines, such as engineering, physics, chemistry, and biology.

This book caters to a wide readership, including students, researchers, and professionals in science and engineering. It begins with the history and principles of numerical simulation, then explores mathematical modeling, numerical methods, and computational algorithms. Further chapters delve into various applications, illustrating the potential of numerical simulations. The book emphasizes the significance of accuracy, stability, and efficiency in simulations, striving to blend theory with practice. It includes examples, exercises, software, and code snippets, encouraging readers to apply what they learn to practical scenarios.

This book contains different use cases for numerical simulation in different industries. Practical examples and scientific details support all the information. The chapters contain enough information for beginners to get familiar with the high technology and science applications to solve business problems and more detailed technical information for advanced readers.

**Chapter 1:** "Introductory Chapter: Numerical Simulation"

Chapter 1 is about numerical simulation, a computational technique used across numerous scientific and engineering disciplines to study complex systems that are otherwise difficult to explore physically. Numerical simulation creates approximate models of real-world behavior, helping to predict outcomes under various scenarios. This method involves discretizing continuous variables into finite points and converting differential equations into algebraic ones that computers can solve. Its diverse applications include weather prediction, fluid flow simulation in aerodynamics, stress analysis in mechanical structures, and biological system simulation in medicine. While computationally intensive, the development of high-performance computing has significantly broadened the use of numerical simulation, establishing it as a crucial tool in contemporary research and development.

**Chapter 2:** "Mathematical Basics as a Prerequisite to Artificial Intelligence in Forensic Analysis"

This chapter delves into the confluence of mathematics, statistics, and AI, emphasizing their application in image processing and forensics. It underscores the importance of math in these areas and the progression of image processing techniques. Key notions like neural networks are examined, setting the groundwork for understanding artificial neural networks. The chapter tackles hurdles in understanding prerequisites and explores niche areas like steganographic security and image forensic detection. It proposes a score-based likelihood ratio over traditional statistical methods. The chapter is divided into two sections, tackling math prerequisites for image processing and connecting these to forensic sciences, facilitating an efficient overview of related concepts across multiple specializations.

**Chapter 3:** "Computer Vision: Anthropology of Algorithmic Bias in Facial Analysis Tool"

This chapter examines bias in Computer Vision (CV), notably in autonomous machines, due to human influence during data labeling for machine learning, leading to an unequal representation of social groups. Referencing Russell and Lee, it emphasizes the need for broad and relevant datasets for effective recognition, highlighting those biases from disproportionate representation can result in algorithmic decisions that marginalize specific social groups. The authors scrutinize Amazon's facial recognition tool's identification and categorization of non-conventional or dissenting genders, questioning these machines' capacity to recognize beyond binary gender classifications.

**Chapter 4:** "Numerical Simulations and Validation of Engine Performance Parameters Using Chemical Kinetics"

This chapter promotes computer modeling and simulations for enhancing fuels and engines, critiquing the conventional dependence on global reactions in combustion simulations due to its impact on engine performance predictions. The authors suggest a refined combustion model combining a 3D turbulent Navier–Stokes solver with detailed kinetic reactions and fluid dynamics for improved accuracy. They advocate for reduced chemical reaction mechanisms to expedite simulations, aiding efficient engine performance analysis. The chapter underscores sensitivity analysis and the computational singular perturbation method to hone the reaction mechanism. An interface for surrogate fuels study is proposed, emphasizing the need for simulation validation through experimental data. Comprehensive studies are urged for validating performance parameters across all mixtures. The necessity of a standard reduced mechanisms library for varied engines and combustion systems is highlighted, ending with a validated reduced reaction mechanism for premixed and direct injection spark ignition engines.

**Chapter 5:** "Bayesian Methods and Monte Carlo Simulations"

This chapter discusses Bayesian methods and tools for studying probabilistic models of linear and non-linear stochastic systems. They allow the tracking of probability distribution changes using Bayes' theorem and the chain rule, but their complexity often requires numerical statistical and causal inference methods. The chapter introduces

various Bayesian techniques for managing intractable distributions, including sampling, filtering, approximation, and likelihood-free methods, explaining their principles and key challenges. These methods find applications in various areas: Bayesian experiment design maximizes information gain and is usually combined with optimal model selection; Bayesian hypothesis testing improves data-driven decision-making; Bayesian machine learning treats data labels as random variables; and a Bayesian optimization is a powerful tool for configuring and optimizing large-scale complex systems. The chapter discusses Bayesian Monte-Carlo simulations, proposing that augmented Monte-Carlo simulations can better explain capability and information efficiency.

**Chapter 6:** "Numerical Simulation on Sand Accumulation behind Artificial Reefs and Enhancement of Windblown Sand to Hinterland"

This study investigates the impact of artificial reefs on Kimigahama Beach in Chiba Prefecture, Japan. Due to their wave-sheltering effect, the reefs formed Salients, or protrusions, in the shoreline, leading to a significant amount of fine sand being transported inland by the wind. To analyze these effects, the study employed a model that combines the Beta-Geometric (BG) model (for predicting three-dimensional beach changes due to waves) with a cellular automaton method. This model was used to forecast shoreline changes after the installation of reefs, beach changes induced by windblown sand, beach changes after reef removal, and the impact of beach nourishment. The findings indicate that constructing wave-sheltering structures like artificial reefs on fine sand coasts speeds up the wind-driven landward sand transport. The model used was successful in predicting such effects.

**Chapter 7:** "Numerical Simulation of Land and Sea Breeze (LSB) Circulation along the Guinean Coast of West Africa"

This chapter examines the dynamics of land and sea-breeze rotation along the West African Guinean Coast using observed and simulated data. The Weather Research and Forecasting (WRF) model, modified with ERA-Interim (a global atmospheric reanalysis) and Climate Forecast System (CFS) forcing data, was used to simulate the local circulation, displaying accurate results aligned with observed data. The research reveals that pressure gradients, advection, and diffusion forces shape wind rotation direction. An hourly breakdown indicated surface gradient forces dominate the ocean, while diffusion terms impact more on land due to variations in surface roughness caused by landscape and urbanization. The study highlights a connection between urbanization and local circulation in major cities along the Guinean Coast.

**Chapter 8:** "Fluid Dynamics Simulation of an NREL-S Series Wind Turbine Blade"

This chapter focuses on the detailed study of the theory, design, modeling, and simulation of a 1.2-MW wind turbine blade that measures 35 meters. Given the wind turbine blade geometry's complexity and unpredictable characteristics, the chapter employs Computational Fluid Dynamics (CFD) to simulate the blade. The design's central focus is the Tip Speed Ratio (TSR), optimally set at 7 for this study. The chapter then juxtaposes the simulation results with those from the Blade Element & Momentum (BEM) theory. Finally, the results from Q Blade and X Foils are compared with a more precise CFD Simulation. The chapter concludes by comparing and evaluating the accuracy of the various methods used in the study.

**Chapter 9:** "Methods of the Perturbation Theory for Fundamental Solutions to the Generalization of the Fractional Laplaciane"

**Chapter 12:** "Modeling of Thermal Conductivity in Gas Field Rocks"

This chapter provides a comprehensive examination of the significance of understanding the thermal conductivity of rocks in petroleum engineering, specifically during the initial stages of oil and gas deposit exploitation. The information is essential for devising secondary and tertiary extraction methods, including hot water and steam injection, CO2 injection, flue gas injection, and initiation of underground combustion. The authors introduce an innovative method to measure the thermal conductivity of rocks, improving understanding of heat transfer processes in subsurface reservoirs and aiding the development of efficient extraction strategies. They also analyze the relationships between thermal conductivity and properties of oil and gas collector rocks, particularly density and porosity, offering insights that could optimize extraction processes. The chapter is a valuable resource blending theory and application, beneficial for researchers, professionals, and students in petroleum engineering and related fields.

**Chapter 13:** "Simulation Study of Microwave Heating of Hematite and Coal Mixture"

This chapter presents a computational approach to predict the temperature distribution in a hematite ore mixed with 7.5% coal. Using MATLAB 2018a software, a 1D heat conduction equation was solved via an implicit finite difference method. The study focused on a 20 cm x 20 cm square slab, where coal was assumed to be uniformly mixed with the ore. The model considered convective and radiative boundary conditions, microwave heating time, thermal conductivity, heat capacity, carbon percentage, sample dimensions, and other factors like penetration depth, permittivity, and permeability of the ore-coal mixture. The temperature profile derived from this model could optimize the microwave-assisted carbothermal reduction process for hematite. The model was also extended to slabs of varying sizes, and the predictions

**Chapter 14:** "Perspective Chapter: Computational Modeling for Predicting the Optical

This chapter investigates the impact of aero-optical effects (AOE) on interceptor systems equipped with infrared detectors. As these interceptors move at supersonic speeds, they create a variable density field, altering the optical properties, particularly the index of refraction. This alteration can distort the incoming light, causing blur, shift, jitter, intensity loss, and resolution loss, collectively known as AOE. These aberrations can severely degrade the imaging quality of onboard optical sensors, compromising guidance accuracy and potentially leading to mission failure. Given the importance of achieving high guidance accuracy for endo-atmospheric flight vehicles, it is essential to understand the principles of AOE and evaluate these aberrations. Therefore, this chapter studies the influence of supersonic flow fields on optical propagation and imaging, which holds both theoretical value and practical implications in the design of optical systems and restoration of turbulence-degraded images.

This chapter presents a novel method using moving nodes in computing, promising improved accuracy and efficiency in complex numerical calculations. It uses a

aligned well with experimental results.

Distortions through the Hypersonic Flow Fields"

**Chapter 15:** "Moving Node Method for Differential Equations"

This chapter delves into the regularity properties of solutions to the fractional Laplacian equation with perturbations, a model significant in various fields of mathematics and physics. The authors utilize semigroup theory to illuminate the dynamics of solutions, highlighting the effects of perturbations. They establish the Harnack inequality for a weak solution to the fractional Laplacian problem, which is a crucial tool for analyzing elliptic and parabolic partial differential equations and provides vital information about the interior regularity of solutions. Furthermore, the authors estimate the oscillation of the solution to the fractional Laplacian, a necessary step for understanding solutions' qualitative behavior and developing numerical methods for the equation. Overall, this chapter provides a comprehensive exploration of the regularity properties of the fractional Laplacian equation with perturbations, aiming to spur further research and development in this crucial field.

**Chapter 10:** "The Analysis on the Effects of COMT, DRD2, PER3, eNOS, NR3C1 Functional Gene Variants and Methylation Differences on Behavoiral Inclinations in Addicts through the Decision Tree Algorythm"

This chapter presents a study exploring the influence of functional gene variants (COMT, DRD2, PER3, eNOS, and NR3C1) on individuals with substance use disorder (SUD). A decision tree algorithm is used to analyze and compare the impacts of these gene variants, guided by the influences of genetic and epigenetic sequences. This classification system is evaluated through a 10-fold cross-validation considering various factors, such as criminal history, continuity of substance use, previous polysubstance abuse, suicide attempts, and inpatient treatment. Performance criteria are gauged based on accuracy, sensitivity, and precision values, consistent with earlier research. The gene variants branching structure resulting from the tree classification aligns with existing literature. This research highlights the potential of machine learning in predicting the effect of gene variants on behavior, emphasizing the need for more extensive studies that include data from diverse ethnic groups to improve predictive accuracy rates.

**Chapter 11:** "Mathematical Modeling of a Porous Medium in Diesel Engines"

This chapter discusses the issue of particulate matter (PM) emissions in directinjection diesel engines. Despite their high-power density and low exhaust emissions, these engines face challenges with PM emissions due to the simultaneous fuel injection and combustion process. This process results in a non-homogeneous mixture in the cylinder, contributing to emissions. The chapter proposes separating fuel injection and combustion processes to create a homogeneous mixture, using porous media in diesel engine combustion chambers as a practical approach to enable stable ultra-lean combustion and reduce emissions. The chapter presents a thorough overview of the mathematical modeling of PM diesel engines, divided into three parts: thermodynamic modeling, zero-dimensional modeling with chemical kinetics, and three-dimensional computational fluid dynamics (CFD) modeling with chemical kinetics.

**Chapter 12:** "Modeling of Thermal Conductivity in Gas Field Rocks"

This chapter provides a comprehensive examination of the significance of understanding the thermal conductivity of rocks in petroleum engineering, specifically during the initial stages of oil and gas deposit exploitation. The information is essential for devising secondary and tertiary extraction methods, including hot water and steam injection, CO2 injection, flue gas injection, and initiation of underground combustion. The authors introduce an innovative method to measure the thermal conductivity of rocks, improving understanding of heat transfer processes in subsurface reservoirs and aiding the development of efficient extraction strategies. They also analyze the relationships between thermal conductivity and properties of oil and gas collector rocks, particularly density and porosity, offering insights that could optimize extraction processes. The chapter is a valuable resource blending theory and application, beneficial for researchers, professionals, and students in petroleum engineering and related fields.

**Chapter 13:** "Simulation Study of Microwave Heating of Hematite and Coal Mixture"

This chapter presents a computational approach to predict the temperature distribution in a hematite ore mixed with 7.5% coal. Using MATLAB 2018a software, a 1D heat conduction equation was solved via an implicit finite difference method. The study focused on a 20 cm x 20 cm square slab, where coal was assumed to be uniformly mixed with the ore. The model considered convective and radiative boundary conditions, microwave heating time, thermal conductivity, heat capacity, carbon percentage, sample dimensions, and other factors like penetration depth, permittivity, and permeability of the ore-coal mixture. The temperature profile derived from this model could optimize the microwave-assisted carbothermal reduction process for hematite. The model was also extended to slabs of varying sizes, and the predictions aligned well with experimental results.

**Chapter 14:** "Perspective Chapter: Computational Modeling for Predicting the Optical Distortions through the Hypersonic Flow Fields"

This chapter investigates the impact of aero-optical effects (AOE) on interceptor systems equipped with infrared detectors. As these interceptors move at supersonic speeds, they create a variable density field, altering the optical properties, particularly the index of refraction. This alteration can distort the incoming light, causing blur, shift, jitter, intensity loss, and resolution loss, collectively known as AOE. These aberrations can severely degrade the imaging quality of onboard optical sensors, compromising guidance accuracy and potentially leading to mission failure. Given the importance of achieving high guidance accuracy for endo-atmospheric flight vehicles, it is essential to understand the principles of AOE and evaluate these aberrations. Therefore, this chapter studies the influence of supersonic flow fields on optical propagation and imaging, which holds both theoretical value and practical implications in the design of optical systems and restoration of turbulence-degraded images.

**Chapter 15:** "Moving Node Method for Differential Equations"

This chapter presents a novel method using moving nodes in computing, promising improved accuracy and efficiency in complex numerical calculations. It uses a

common fluid mechanics and heat transfer problem to demonstrate the method's proficiency in solving convective-diffusion issues. The goal is to highlight how this innovative method can tackle such problems more effectively than existing approaches. Validation through test examples illustrates the method's benefits, encouraging its broader use in computing technology. This work aims to stimulate fresh insights and further exploration in this crucial field.

**Chapter 16:** "On the Analytical Properties of Prime Numbers"

Prime numbers, unique in their properties and fundamental to number theory, intrigue mathematicians due to their unpredictability and fundamental role in mathematics. Despite exhaustive research, their seemingly random distribution remains an enigma encapsulated in the Prime Number Theorem. As the Fundamental Theorem of Arithmetic highlights, their multiplicative properties emphasize their role as the "building blocks" of the number system. Furthermore, unresolved mysteries like the Twin Prime Conjecture and the Riemann Hypothesis, which are deeply connected to the distribution of primes, add to the intrigue. Beyond their theoretical interest, primes have significant practical applications, particularly in cryptography, where their complex factorization properties underpin secure data transmission systems like Rivest–Shamir–Adleman (RSA). In this chapter, the authors focus on the prime numbers and their analytical properties.

This book helps readers understand numerical simulation applications in different areas, and we hope it will be a valuable resource for industry professionals and researchers. The chapters discuss the state of the art of critical topics in numerical simulation. Furthermore, their coverage and depth make this book a helpful tool for all managers and engineers interested in the new generation of data analytics applications. Above all, the editor hopes this volume will spur further discussions on all aspects of numerical simulation applications in different industries.

As you embark on this journey, we encourage you to embrace the spirit of curiosity, perseverance, and innovation that has fueled the development of numerical simulation throughout history. We hope this book will be invaluable and ignite a passion for lifelong learning and discovery in numerical simulations.

Happy reading, and best of luck on your journey into the world of numerical simulation!

> **Dr. Ali Soofastaei** AI Program Leader, Artificial Intelligence Center, Vale, Brisbane, Australia

Section 1

Intersecting Mathematics,

Statistics, and AI for

Improving the Numerical

Simulation
