Mathematical Modeling
Vol. 8 (2024), Issue 2

Table of Contents

  • THEORETICAL FOUNDATIONS AND SPECIFICITY OF MATHEMATICAL MODELLING

    • Determination of the average layer pressure in the case of radial flow.

      pg(s) 42-43

      Oil and gas accumulations located in the porous spaces of the formation exist as a single hydraulically connected system and are factors of many physico-chemical processes that occur, depending on the different conditions that develop in the layer. The study of fluid filtration in porous media, their type, the condition in which the flow occurs and its geometric form, are the main key in which the testing of a well begins and then the determination of various hydrodynamic parameters. In reservoir engineering, the stages of filtration that develop and the changes that occur in the phase state of the fluid are closely related to the changes that occur in the pressure of the reservoir, which characterizes the energy of the layer for the production of fluids on the surface. Radial flow is used in many practical applications in solving various problems encountered in reservoir engineering and precisely the main objective of this paper, is to present a solution of the average pressure of the reservoir in the case of radial flow using a mathematical approach.

    • Key aspects of Markov Chain Monte Carlo simulations in Bayesian statistical analysis

      pg(s) 44-46

      This paper focuses on the simulation aspects of Bayesian hypothesis testing applied on ophthalmic data. Bayesian statistical analysis often relies on Markov Chain Monte Carlo (MCMC) methods for estimation, when analytical solutions are not possible. We highlight the key aspects of MCMC including model specification, details on the simulation, MCMC diagnostics as well as its limitations and advantages. The simulations are done using R and JAGS.

    • Particle-in-cell modeling of intense terahertz emission from gaseous targets ionized by a two-color circularly polarized laser pulse

      pg(s) 47-50

      In this work we study numerically the generation of intense terahertz radiation in the interaction of two-color circularly polarized laser pulses of optical or infrared wavelength range with argon. The terahertz pulses obtained in such a scheme can be used to generate strong slowly changing electric and magnetic fields of a given configuration. To investigate the terahertz emission by a plasma source arising from the ionization of a gas medium by femtosecond laser field with a peak intensity of 10¹⁴-10¹⁵ W/cm² a fully kinetic plasma model consisting of the Vlasov equations for the plasma distribution function and the Maxwell equations for the self-consistent electromagnetic field has been used. Our numerical code is based on the particle-in-cell method and employs state-of-the-art widely used algorithms, such as finite difference time domain method for modeling the electromagnetic fields, Boris pusher to update the particle positions and velocities, and the charge conservation scheme to satisfy the continuity equation. The field ionization is implemented using the tunneling ionization rate formula. Our simulations have shown that at a sufficiently small interaction volume the plasma oscillations excited by asymmetric ionization are almost homogeneous in space and lead to an efficient conversion of electron energy into the energy of emitted terahertz radiation.

  • MATHEMATICAL MODELLING OF TECHNOLOGICAL PROCESSES AND SYSTEMS

    • Modified Split Hopkinson Pressure Bar adapted to porous and low strength materials testing

      pg(s) 51-55

      A modified version of the Split Hopkinson-Kolsky Pressure Bar is described, intended for impact testing of porous metals and other low-strength materials. In it, one bar is made of aluminum alloy and the other bar is a steel tube. Several variants of the mathematical model for working with such devices are described. The stress-strain diagram of the specimen is extracted from the first impact signal. Attention has been paid to the approach to choosing the initial moments from which the calculations must start. Several examples are shown with solid and porous aluminum alloy samples. Despite the lack of high accuracy of the diagram at small stresses, the facility is suitable for quickly obtaining the overall appearance of the diagram after crushing small cylindrical specimens (up to 10 mm in diameter and height) and estimating the stresses at which the material deforms plastically.

    • Effect of the limiting deformation zone under conditions of asymmetric loading during rolling

      pg(s) 56-58

      In this article, is to develop a physical and mathematical model of the process under complex asymmetric loading in conditions of single-area and two-area deformation zone during plastic processing of medium-thickness strips. The stress state in case of loss of stability during rolling of strips of medium thickness was investigated. The patterns of changes in the stress state of the strip under conditions of reach of the limiting deformation zone, as well as the effects of plastic shaping determined by a decrease in contact stresses under conditions of increasing deformation loading, are revealed. The described method is a visual approach to assessing the stress state of a plastic medium under conditions of complex interaction and asymmetric loading.

    • Analysis and Modelling of the Nonwoven Waste Fabric Cutting Unit

      pg(s) 59-61

      In this study, the design and modeling of a shredding unit intended to fragment nonwoven waste fabrics and prepare them for recycling were carried out. Stress analyses were performed on the shaft attached to the blade, along with motor power and torque analyses required for smooth operation, and mass flow calculations. Mathematical models were developed based on these data. Additionally, structural analyses of the chassis, necessary for stable system operation, were conducted, resulting in a comprehensive mathematical model of the entire system.

    • Key Parameters in Zipline Design: An Analytical Approach

      pg(s) 62-66

      Zipline is an adventure activity that involves a steel cable suspended between two anchor points at varying heights and distances. Participants are secured to a trolley that moves along the cable, primarily driven by gravitational force. This sport, often classified as an adrenaline activity, has become increasingly popular for recreational use.
      This paper presents a comprehensive analysis of the key parameters influencing zipline systems, including the modelling of movement dynamics and the selection of essential components. Utilizing the principles of the catenary curve, our approach draws parallels with established methodologies in cable car and cable crane systems.
      To ensure the quality design and safe operation of zipline systems, it is crucial to analyse various factors, such as participant weight, rope anchoring methods, tension force, and angle of inclination, movement speed, release position, trolley wheel resistance, air resistance, temperature effects, and geographical considerations.
      As a case study, we detail the findings from a theoretical model, validated through computer simulations, focusing on the specific conditions of a zipline constructed in Theth, Albania. Our results provide valuable insights into the design and safety protocols necessary for effective zipline operations.

    • Control-relevant identification of the DC engine coupled with a reaction wheel

      pg(s) 67-70

      This paper proposes to estimate the mathematical model of the DC motor coupled with a reaction wheel in two ways: open-loop identification and closed-loop identification. For open-loop identification, the experimental curve is approximated with the model of object with inertia first order, and inertia second order. In the case of closed-loop identification, it was obtained the model with inertia third order, that approximates the dynamic of the process, where the coefficients are calculated based on the simple analytical expressions according to the values that are extracted from the undamped step response of the closed-loop system with P controller. The results of the experimental model estimation were compared with the results obtained using the System Identification Toolbox from MATLAB. In addition, to the identified object model, it was proposed to tune a PID controller based on the maximum stability degree criterion.

  • MATHEMATICAL MODELLING OF SOCIO-ECONOMIC PROCESSES AND SYSTEMS

    • Mathematical Processing and Analysis of Sleep Signals Using a Portable and Cost-Effective Oculograph

      pg(s) 71-74

      The study of sleep is crucial for understanding various physiological and neurological processes, yet research on different sleep phases often comes with high costs and requires specialized equipment. To address these challenges, we developed a portable and relatively inexpensive oculograph, which enables more accessible sleep studies. A critical technical aspect of using the device is the necessity for mathematical transformations to interpret the signals generated by eye movements, which are often complex and prone to noise. We implemented several mathematical procedures for noise reduction, signal filtering, and the extraction of key signal features. To assess the accuracy of the oculograph, we conducted 10 daytime experiments with predefined protocols involving specific eye movements. The results indicate that the oculograph successfully measures eye movements with high precision, which was further validated through comparison with graphical signal representations. Moreover, we performed tests for nighttime use of the device, and validation of REM sleep signals is planned using a camera to record the subject during sleep. These promising outcomes suggest significant potential for the oculograph to help sleep research by offering a more affordable and mobile solution, suitable for both laboratory and home environments. The mathematical procedures and signal processing techniques presented here are tailored to the needs of psychological and medical sleep studies. Additionally, practical applications of the oculograph for targeted sleep research, including tracking eye movements during various sleep stages, are proposed.

    • Classical Optimization methods for an Ornstein-Uhlenbeck process-based model in pair trading

      pg(s) 75-80

      Mathematical Optimization or Mathematical Programming is a set of theoretical and applied methods originating from applied mathematics to computer science, used to find the optimal value, given some predefined criterion for optimality and based on some input parameters. More precisely, it concerns finding the input parameters of a function (called the objective function) which maximize/minimize its output. Mathematical Optimization is widely used for various quantitative or computational problems that arise everywhere in science and engineering, ranging from statistics and applied math to aerospace engineering and finance. In the following paper, we demonstrate the use of two classical optimization algorithms – Gradient Descent and Lagrangian Multipliers method – in model development in finance and trading. The specific stochastic process on which the trading model is based is called the Ornstein-Uhlenbeck (O-U) process and the point of the use of these two optimization approaches is
      to find the parameters that minimize the log likelihood function of this process. This fits the O-U process to the historical data and in the context of finance and trading, maximizes our profits and helps us hedge against losses

  • MATHEMATICAL MODELLING OF MEDICAL-BIOLOGICAL PROCESSES AND SYSTEMS

    • Aortic Elasticity versus Aortic Valve Elasticity – Structural Aspects, Preliminary works

      pg(s) 82-84

      This elaboration proposes results obtained during initial stages of numerical modeling of the function of aortic heart valve depending on its elastic module and the aortic elastic module.
      The actuality of such an elaboration could be based on the advancing progress in the material sciences, as well as on the increasing opportunities given by the development of the medical diagnostic technics for observation of processes in the human organism and the development of computational technics.
      The work and the interactions between the aortic valve and the aorta is here structurally modeled according to Finite Element Method by help of Ansys commercial product. Elastic material models are assumed for the materials of aorta and aortic valve. The geometry of the aortic valve and the aorta are designed as averaged ones according to suggested reference data. The boundary conditions are assumed according to reference data about left-ventricle blood pressure and the aortic blood pressure. These material models, geometry and boundary conditions could be reformulated according each separately given case. That will give possibilities of development of the subject of this work following as called patient-orientated model.
      The displacements, the stress and strain distributions in both aorta and valve are established depending on both blood pressures aortic and left-ventricle depending on two aortic elastic modules 0.476 and 1MPa under elastic modules of the sinotubular junction 15.34MPa and sinus of Valsalva 20.24MPa. Pumping functions of the heart and the aorta are numerically observed depending on the two aortic elasticmodules. Some variations in the elastic modules of the sinotubular junction and sinuses of Valsalva are commented.
      The proposed finite element model and the obtained results could open future possibilities to enlarge this work, for example, through variations in the four elastic modules, variations in the aortic geometry also including additional branches to the aorta or modeling of interactions between pulsating blood flow and arterial-valve structures, also not excluding real experimental formulation or design and synthesis of new biological materials.

    • Modeling of skin/sensor contact impact and neuron loss effect on EEG signals of voltage sensor

      pg(s) 85-88

      The present work concerns the development of a model which have the capability to represent accurately the skin/sensor contact in an electromagnetic problem study. The later consist of elctro-encyphalography (EEG ) signals encountered in neuroscience health diagnosis. The electromagnetic problem studied is governed by Poisson’s equation and solved using finite element method. The skin/sensor contact is modeled by an analytical solution, which is coupled to finite element resolution. The signals induced by neuronal equivalent charges are presented and investigated. A comparison with existing results is then performed. Neuron loss effect is also investigated.