Mathematical Model Of Process Research Articles

Расчет времени промерзания надземного водовода в условиях отрицательных температур

Purpose: When the water movement in the pipeline is stopped in conditions of negative temperatures, its freezing and destruction are possible, which leads to the failure of water supply systems for a long time. For accident-free operation of the water pipeline, it is important to know the time during which its complete or partial freezing occurs. The purpose of this work is to study the physical processes that occur during freezing, to formulate the basic mathematical model and its numerical solution, to build an analytical solution in a quasistationary approximation, to obtain formulas that are convenient for freezing time calculation and to determine the range of parameter values at which they are valid. Methods: The mathematical model of the freezing process relies on the use of energy conservation law. When constructing a difference scheme, the integrointerpolation method is used to numerically solve the nonlinear differential equations of the model. To obtain an approximate analytical solution, the method of separating equations describing processes that occur at different speeds, is used. Results: The mathematical model of pipeline freezing during water movement shutdown under conditions of constant negative temperature of the surrounding atmosphere has been formulated and substantiated. Within the framework of the quasi-stationary approximation, simple formulas for the freezing time of the water pipeline have been obtained. The criteria for the applicability of these formulas have been established. The numerical solution of equations of initial mathematical model is compared with the results obtained within quasi-stationary approximation. Practical significance: The ratios obtained in the work make it possible to estimate the time during which repair work should be carried out and the movement of water in the pipeline should be restored before its destruction due to freezing occurs.

Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator

Differential equations containing fractional derivatives, for both time and spatial variables, have now begun to attract the attention of mathematicians and physicists; they are used in connection with these equations as mathematical models of various processes. The fractional derivative equation tool plays a crucial role in describing plenty of natural processes concerning physics, biology, geology, and so on. In this paper, we studied a loaded equation in relation to a spatial variable for a linear pseudoparabolic equation, with an initial and second boundary value condition (the Neumann condition), and a fractional Caputo derivative. A distinctive feature of the considered problem is that the load at the point is in the higher partial derivatives of the solution. The problem is reduced to a loaded equation with a nonlocal boundary value condition. A way to solve the considered problem is by using the method of energy inequalities, so that a priori estimates of solutions for non-local boundary value problems are obtained. To prove that this nonlocal problem is solvable, we used the method of continuation with parameters. The existence and uniqueness theorems for regular solutions are proven.

BOUNDARY VALUE PROBLEM FOR THE EQUATION OF UNSTEADY THERMAL CONDUCTIVITY IN A NON-CYLINDRICAL REGION

The application of the method of decomposition by eigenfunctions of a self-adjoint differential operator to solving a non-stationary heat transfer problem with a phase transition in a non-automatic formulation under special initial conditions is presented for the example of the solidification process in a continuous medium. The one-dimensional problem is solved in spherical coordinates. Solving of the problem begins with its transformation to a problem in a domain with fixed boundaries, then a finite integral transformation with an unknown kernel is constructed to solve the transformed problem, the finding of which is associated with the formulation and solving of the corresponding spectral problem through degenerate hypergeometric functions. The eigenvalues and eigenfunctions are found, as well as the inversion formula for the introduced integral transformation, which makes it possible to write out an analytical solution to the problem. In the course of solving the problem, the parabolic law of motion of the interface of the two phases is established. Problems of this type arise in the mathematical modeling of heat transfer processes in construction, especially in permafrost areas, in oil and gas production during drilling and operation of wells, in metallurgy, etc.

BOUNDARY VALUE PROBLEM FOR THE EQUATION OF UNSTEADY THERMAL CONDUCTIVITY IN A NON-CYLINDRICAL REGION

The application of the method of decomposition by eigenfunctions of a self-adjoint differential operator to solving a non-stationary heat transfer problem with a phase transition in a non-automatic formulation under special initial conditions is presented for the example of the solidification process in a continuous medium. The one-dimensional problem is solved in spherical coordinates. Solving of the problem begins with its transformation to a problem in a domain with fixed boundaries, then a finite integral transformation with an unknown kernel is constructed to solve the transformed problem, the finding of which is associated with the formulation and solving of the corresponding spectral problem through degenerate hypergeometric functions. The eigenvalues and eigenfunctions are found, as well as the inversion formula for the introduced integral transformation, which makes it possible to write out an analytical solution to the problem. In the course of solving the problem, the parabolic law of motion of the interface of the two phases is established. Problems of this type arise in the mathematical modeling of heat transfer processes in construction, especially in permafrost areas, in oil and gas production during drilling and operation of wells, in metallurgy, etc.

The collapse problem as a consistency problem. Is the quantum measurement hypothesis consistent with conservation laws?

In his book Mathematical Foundations of Quantum Mechanics von Neumann asserted that the state vector must collapse upon measurement of any self-adjoint operator. This assertion was based on his reading of the Compton-Simon experiment. However, comparing von Neumann’s account with the Compton-Simon paper itself, we find that von Neumann had completely misinterpreted the experiment. Compton and Simon had measured two angles on the same photographic plate; in von Neumann’s reading it became two successive measurements which gave identical results, where the experimenter could choose which measurement to perform first. In short, von Neumann’s case for collapse simply did not exist!In quantum mechanics, superpositions of different eigenvalues of conserved observables are freely admissible, but conservation laws require successive measurements of a conserved quantity upon the same object to return the same value (if the object is isolated and moves under a Schrödinger equation). This can be guaranteed only if the first measurement has caused the state vector to collapse. Thus collapse is a consistency condition which the theory must satisfy.Consistency problems are solved by displaying examples, or models, which fulfil the conditions in question. We therefore have to construct a mathematical model of the measurement process. This requires taking a stand on whether the measuring instrument does (Bohr) or does not (von Neumann–Wigner) have a classical description. If it does not, the ‘quantum measurement problem’ is insoluble. If it does, the problem – for conserved quantities – does have a solution, given by Sewell in 2005.The model displays collapse while evolving under the Schrö- dinger equation with a time-dependent hamiltonian; it does not have to call upon the observer’s ‘conscious ego’. Its existence shows that additive conservation laws are consistent with the superposition principle and Schrödinger evolution. Measurement of a conserved observable returns an eigenvalue of the observable; measurement of a non-conserved observable returns a diagonal matrix element of the observable in the basis of energy eigenvectors.If ‘classical descriptions’ of the apparatus are essential for these consistency conditions to be satisfied, what does it imply about the relation between classical and quantum mechanics? If the two are on the same footing, is it not possible that ‘dark matter’ is purely classical?

ENHANCED DIAGNOSTIC TECHNIQUES FOR MARINE DIESEL ENGINES: ACHIEVING IMO DECARBONIZATION COMPLIANCE

This paper discusses hardware-based methods for monitoring the operational parameters of marine diesel engines and online mathematical modeling techniques (digital twins) for calculating the emissions of CO2, NOx, soot, and other harmful pollutants. It is demonstrated that measuring and analyzing the engine's vibrational diagrams in parallel with gas pressure diagrams in the cylinders allows the evaluation of the actual fuel injection and gas distribution timings. The obtained data is used to refine the mathematical model of the engine's working process, which determines the engine's indicator parameters and power, as well as the emission of harmful substances that need to be monitored in compliance with current IMO requirements. The authors have accumulated experience in using the discussed hardware-based methods for diagnosing marine engines during operation. The hardware and software methods discussed have been implemented into a real-time system for in-service diagnostics of marine engines. The system is designed based on a modern dual-core controller with high performance and low power consumption, incorporating a high-speed ADC with sufficient capability to monitor the working process with a 0.1-degree crankshaft rotation resolution for all types of marine main and auxiliary engines. The system also utilizes wireless data transmission technology. A contemporary Android/iOS smartphone or tablet serves as the computational and graphical component of the system. This real-time system enables the utilization of all the advantages of parallel pressure and vibroacoustic analysis, including real-time determination of key operating parameters, identification of top dead center position, and evaluation of fuel delivery and gas distribution phases. Additionally, it harnesses the benefits of employing a digital twin—an online mathematical model of the engine cylinder's working process. These solutions will enhance diagnostic quality and, ultimately, improve the operational efficiency of marine engines by reducing operational costs and extending the period of reliable, trouble-free operation.

A new method for determining strength parameters of rock using digital drilling technology

Fast and accurate prediction for mechanical parameters of rock mass is an important prerequisite for guiding the safe construction of rock engineering. At present, the rock strength parameters test requires well-prepared samples; sample preparation is time consuming, costly, and difficult as well as destructive, in particular the polishing and rectification of the sample ends. In this paper, the mathematical model of the drilling process is established based on the force characteristics of the drill bit and the principle of mechanical limit equilibrium. The digital drilling tests are carried out on four types of rocks to drilling characteristics using the digital drilling system. The concept of cutting strength ratio is proposed to analyze the correlation between cutting strength ratio and rock internal friction angle and cohesion. A new method is proposed for determining the compressive and tensile strength of rock. The results show that the cutting strength ratio increases with the increasing of internal friction angle and cohesion. The uniaxial compressive strength and tensile strength of rock increase linearly with the cutting strength ratio. The average error of rock tensile strength is 12.29%, and the average error of uniaxial compressive strength is 20.44%. The internal friction angle of rock obtained based on the digital drilling technology is consistent with the value measured by the laboratory standard test. This method can provide a new way to determine rock strength parameters in real time.

A Reliability Evaluation Method for Gamma Processes with Multiple Random Effects

The multi-random-effects gamma process has a better characterization effect for degraded data with individual differences. In this paper, a reliability evaluation method for gamma stochastic processes with multiple random effects is studied. The mathematical model of multiple random effects gamma process was established. The model parameters estimation method was established based on the Bayesian approach. The prior distribution acquisition method was discussed, and the parameters of the multiple randomeffects gamma process were estimated by the MCMC-Gibbs method. The correctness of the model and method was verified by numerical simulation, the influence of algorithm parameters on the algorithm solving process was studied. In the fourth part, the reliability of aviation hydraulic rotary joints was studied by using multiple random effects gamma processes.

Pathological Study of Light Cupula Syndrome on a Visual Bionic Semicircular Canal.

A type of persistent direction-changing positional nystagmus with a null point during head position deflection is known as light cupula syndrome (LCS) in the clinic. To date, the pathogenesis and biomechanical response of human semicircular canals with light cupula syndrome (LCS) (HSCs-LCS) are still unclear. In this study, based on the anatomical structure and size of the one-dimensional human semicircular canal (HSC) and imitating the pathological changes of the endolymph in HSC with LCS, a visual bionic semicircular canal (BSC) with LCS was fabricated using three-dimensional printing technology, hydrogel modification, and target tracking technology. Through theoretical derivation, mathematical models of the HSC-LCS perception process were established. By conducting in vitro experiments on the bionic model, the biomechanical response process of HSC-LCS was studied, and the mathematical models were validated. The results of pulse acceleration stimulation showed that the pathological changes in the density and viscosity of the endolymph could reduce the deformation of the cupula of the BSC-LCS and increase the time constant. The results of the sinusoidal acceleration stimulation showed that the amplitude-frequency gain of the BSC-LCS decreased and the phase difference increased. The BSC-LCS can be used as a tool for pathological research of the HSC-LCS. The results of this study can provide a theoretical basis for clinical diagnosis.

Mathematical model of technological process of manufacturing thin films taking into account their anisotropy and heterogeneity

Mathematical model of technological process of manufacturing thin films taking into account their anisotropy and heterogeneity

Electrosurgical resection of colorectal polyps – mathematical modelling of processes during medical treatment

Electrosurgical resection of colorectal polyps – mathematical modelling of processes during medical treatment

Parametric identification of the mathematical model of the micro-arc oxidation process

The article is aimed at solving the problem of parametric identification of non-linear object models using the example of a mathematical model of the micro-arc oxidation process. An algorithm for parametric identification, based on an experiment in the micro-arc oxidation process, the results of which form a training and control sample is proposed; sequential training of neural networks and calculation of the parameters estimates of the nonlinear model according to experimental data are performed. Experimental testing of the proposed method of neural network parametric identification on the example of the micro-arc oxidation process confirmed that the standard deviation of current and voltage from the nominal values does not exceed ±4%. The obtained results were used in the development of an intelligent hardware-software complex for the production of protective coatings by the micro-arc oxidation method.

Критика и совершенствование нормативной базы по расчету сооружений на сейсмические воздействия

In the work of Kurbatsky E.N., Mondrus V.L. «Dynamic Coefficients or Response Spectra of Structures to Earthquake?» with the reference to the norms of the technically developed countries, as well as to the monographs of the famous foreign scientists, an erroneous statement was put forward that during earthquakes the ground parts of structures are not subject to the influence of any external forces, and that the internal stresses and deformations in the elements of structures are created exclusively due to dynamic reactions to movement of their bases. It does not state that the cancellation of the action of gravity is associated with the inadequacy (incompleteness) of the mathematical models of the interaction of foundations and structures, but argues that if the force of gravity is clearly absent in the vibration equations, then it does not act on the structure. This incompleteness of ideas is present in the equation of horizontal vibrations in translated and domestic works, but in them, unlike the reviewed works, it is not stated that during earthquakes the ground parts of structures are not subject to the influence of any external forces. The main reasons for the incompleteness (inadequacy) of the mathematical models of foundations and structures and their interaction was identified, which are that the center of gravity of the structure is located at the level of the base, and also that when deriving the equations for horizontal and vertical vibrations of the structure, the incomplete (selective) deformability of the base is used. But the selective deformability of the base when deriving the classical equations of horizontal and vertical vibrations does not allow rocking vibrations of the structure to occur under only horizontal or only vertical seismic influences, and this is a significant drawback of the mathematical modeling of the vibration processes of structures on a completely deformable base. To eliminate this shortcoming, the work presents differential equations for translational and rocking plane-parallel vibrations of a rigid structure on a completely yielding foundation, including the effects of gravity. From them, in particular, it follows that as the height of the center of gravity tends to zero, these equations turn into classical linear equations of oscillations in the horizontal and vertical directions.

Mathematical modeling of the heat process in closure electrical contacts with a heat source

This study presents a mathematical model that describes the heat process during the vaporization of metal at electrical contacts. The arc erosion model for the micro-asperity in the liquid state is established using a one-phase Stefan problem. Two problems involving Joule heating effects on the generalized heat equation are considered. The solution of these two problems is based on the similarity principle, which allows for the reduction of the problem to ordinary differential equations. The existence and uniqueness of the solutions are proven using fixed point theory in a Banach space. Additionally, a solution for the particular case of constant thermal coefficients is provided.

Mathematical Modeling of the Argon Arc Welding Process. Part 2. Welding of HP40NbTi Alloy Pipelines

Mathematical Modeling of the Argon Arc Welding Process. Part 2. Welding of HP40NbTi Alloy Pipelines

Mathematical Modeling of the Argon Arc Welding Process. Part 1. Thermomechanical Approach and Model Justification

Mathematical Modeling of the Argon Arc Welding Process. Part 1. Thermomechanical Approach and Model Justification

Mathematical Modeling of Deformation Processes of Medical Textile Elastomers for Implantology

Mathematical Modeling of Deformation Processes of Medical Textile Elastomers for Implantology

On Mathematical Models of Degradation Processes According to ISO 16204 and fib Model Code

AbstractThe main factors affecting the service life of concrete structures are the presence of chloride ions and carbon dioxide, alternating frost action and mechanical stresses on the structure. In practice, these effects can be taken into account by using mathematical modelling of these phenomena. In this contribution, the authors focus on the processes of chloride ions and carbon dioxide (carbonation process) diffusion through concrete and the associated subsequent corrosion of reinforcement. Mathematical models recommended by the fib Model Code and ISO 16204 are described and the effect of simultaneous environmental and mechanical load on the service life of the structure is discussed. The models are used to study the residual service life of a simple reinforced concrete frame bridge based on the percentage loss of reinforcement area analysis over time.

Developing a Comprehensive Mathematical Model for Aluminium Production in a Soderberg Electrolyser

The technological process of aluminium electrolysis is a complex scientific and technical task. This is due to a large number of internal, external and resultant factors. The aim of this work is to analyse these factors, assess them and their influence on the technological process of electrolysis and develop a comprehensive and mathematical model of aluminium production in the Soderberg electrolyser. The work analyses the technological process of primary aluminium production on the basis of the Bayer method and then on the basis of the Hall–Eru method. The existing methods and technologies for computer modelling of the technological process are analysed. The modern methods of analysis for thermal and electromagnetic fields in electrolysers are considered. On the basis of an in-depth analysis, a number of factors influencing the process of primary aluminium production are identified. Using the methods of system analysis to analyse the identified factors, a ranked list of factors according to the degree of influence is obtained. Using the Pareto diagram, we obtain a list of factors with maximum impact. A conceptual model of the technological process is derived. Based on the obtained conceptual model, the mathematical model of the technological process is derived. The conducted research may be useful to specialists in the field of metallurgy for the analysis of the technological processes of primary aluminium production.

Analysis of a mathematical model of the aggregation process of cellular slime mold within the frame of fractional calculus

ABSTRACT The pivotal aim of the present study is to find the solution for a nonlinear system describing the aggregation process of cellular slime mold by using -homotopy analysis transform method. The coupled system is considered within the frame of the Caputo fractional operator. We examine three different cases with distinct values of sensitivity function , and to exemplify the efficiency and applicability of the considered scheme. We capture the nature of the obtained results with respect to the fractional order, with distinct initial conditions, and illustrate them using 2D and 3D plots for particular values of the parameters. The considered scheme offers parameters, which help to adjust the convergence region, and we plotted the ℏ-curves to dissipate the effect in the present framework. Moreover, some simulating and important behavior of the considered model using attained results shows the prominence of the hired operator while analyzing the coupled equations and confirms the competence of the projected scheme.