Real Physical Processes Research Articles

Relations between timescales of stochastic thermodynamic observables

Abstract Any real physical process that produces entropy, dissipates energy as heat, or generates mechanical work must do so on a finite timescale. Recently derived thermodynamic speed limits place bounds on these observables using intrinsic timescales of the process. Here, we derive relationships for the thermodynamic speeds for any composite stochastic observable in terms of the timescales of its individual components. From these speed limits, we find bounds on thermal efficiency of stochastic processes exchanging energy as heat and work and bound the rate of entropy change in a system with entropy production and flow. Using the time set by an external clock, we find bounds on the first time to reach any value for the entropy production. As an illustration, we compute these bounds for Brownian particles diffusing in space subject to a constant-temperature heat bath and a time-dependent external force.

Online Prediction of Mechanical Properties of the Hot Rolled Steel Plate Using Time-series Deep Neural Network

The prediction of steel’s properties by the process parameters has been of great interest because they can effectively reduce production costs and improve product quality. At present, the extensively used artificial neural network (ANN) can only reveal the correlation between parameters and mechanical properties from the perspective of statistics but loses the critical information on time-series correlation in the steel production process. In this work, time-series neural networks based on long short-term memory (LSTM) were established to predict the steel plate’s yield strength (YS), ultimate tensile strength (UTS), and elongation (EL). The results verified that the proposed LSTM model exhibited sufficient accuracy and outperformed the classical algorithms (SVM, random forest, ANN), with mean squared error reduced by 30% when compared with ANN. Also, the sensitivity analysis proved that LSTM was more capable to make full use of the information contained in input parameters and achieved better generalization performance. Notably, the interpretation of the results was more consistent with the real physical metallurgical process of a hot-rolled steel plate and can provide a better understanding of the physical metallurgical process.

Three-Dimensional Finite Element Analysis of Maxillary Protraction Using Diverse Modes of Rapid Palatal Expansion.

Three-dimensional finite elemental analysis (FEA) is a contemporary research instrument for the numeric simulation of a real physical system's mechanical process. FEAcan be used as a very effective tool to analyze and compare various aspects of rapid palatal expanders and to determine the stress distribution in maxillofacial bones and displacement and the biomechanical effects it has on the circummaxillary sutures. This study evaluates the effects of different modes of rapid palatal expansion on maxillary protraction as a treatment modality in skeletal Class III malocclusion by determining the stress and displacement along the circummaxillary sutures using the FEA. Initially, a three-dimensional finite element simulation of the maxillofacial skeleton and sutures was obtained by Mimics software (Leuven, Belgium) from the cone-beam computed tomography (Dentsply Sirona, USA) images of a 30-year-old adult with normal occlusion. A geometrical preparation of the three expansion appliances, (A) hybrid MARPE (miniscrew-assisted rapid palatal expander)appliance (Fav anchor, India), (B) tooth-borne HYRAX (hygenic rapid expander)appliance (Welcare orthodontics, Kerela), and (C) bone-borne modified MARPE appliance (Biomaterials, Korea), was transferred to ANSYS WORKBENCH, 2020 R1 software (ANSYS, Inc., USA), and three finite element models with each appliance were prepared. A protraction force of 500g was applied to the occlusal plane that is directed 20 degrees inferiorly. The tensile stress, compressive stress, and the amount of displacement on the circummaxillary sutures were assessed and compared in all the three appliances. Young's modulus (kg/mm2) and Poisson's ratio (V) were used to calculate the stress and displacement in sutures adjacent to the maxilla in different aspects. On analyzing the stress distribution, the tensile stress was found to be maximum in the medial aspect of the frontomaxillary suture of the bone-borne modified MARPE appliance (C), and the minimum tensile stress was found in the lateral aspect of the sphenozygomatic suture in hybrid MARPE (A). Again, the compressive stress distribution was found to be maximum in the medial aspect of the frontomaxillary suture in all three simulations and the minimum compressive stress in the superior aspect of the internasal suture in hybrid MARPE (A) along with the frontonasal suture at its medial aspect for tooth-borneHYRAX (B) and bone-bornemodified MARPE (C). Displacement of the maxilla in all the planes was observed to be the largest for the bone-borne modified MARPE (C) appliance. On the contrary, the minimum displacement was found in the tooth-borne HYRAX (B) appliance. Conclusion: The findings reveal that all three modes of rapid palatal expanders produced stress and displacement along the circummaxillary sutures on the application of protraction force with bone-borne modified MARPE being more effective in treating posterior crossbites thereby correcting the skeletal Class III malocclusions successfully.

Bounce behaviors of double droplets simultaneously impact cold superhydrophobic surface

The real physical process of ice formation involves multiple droplets rather than a single isolated one. Clarifying the mechanism of multi-droplet impact-freezing is important for designing functional surfaces for anti-icing. The rebound behavior of double droplets simultaneously impinging on a cold superhydrophobic surface (SHS) was investigated experimentally and numerically, which is a classical simplified case in multi-droplet impingement. We found five impacting-freezing modes, including no-coalescence rebound (NCR), partial-coalescence rebound (PCR), complete-coalescence rebound (CCR), no-coalescence adhesion (NCA) and coalescence adhesion (CA). Moreover, the effects of horizontal spacing, impact velocity and surface temperature on the spreading factor and uprising sheet height during the impacting-freezing process were also analyzed. The contact time of PCR mode is even shorter than that of a single droplet impact at room temperature, which is verified and explained in our results. However, as the surface temperature decrease, the behavior of the coalescing droplet transitions from PCR to CCR and CA mode. The increase in Weber number could cause the adhesion of double droplets. Moreover, the horizontal spacing strongly affects the coalescing process. Multiple droplet impact-freezing will lead to severe icing on the substrate due to multiple droplets interaction involved. This study is beneficial for anti-icing and de-icing technologies.

CFD modeling of influenza virus diffusion during coughing and breathing in a ventilated room

The virus diffusion in a ventilated room with the droplets produced by coughing and breathing are presented by the Lagrangian model. When the human body is located in the middle of the room with two locations of AC, in front of and behind the human body, three angles of Air Conditioning (AC) gate are applied 0°, 30°, and 60° to show droplet particle diffusion in the room in these cases. Three types of coughing velocity profiles were selected, real human coughing, sinusoidal cough, and cough jet with one velocity profile of breathing as a step function to cover the inhaling and exhaling cycle. The simulation results show that the uncovered standing in the middle of the room, are more susceptible to infection for the bouncy and forced flow around the human body. Droplet particle moves in the room as a random diffusion and it is very sensitive to the thermal load inside the room, generally depends on the bouncy force and pressure force due to convection heat transfer. when the AC location at the opposite direction of coughing flow, the droplet travels a distance of about 3 m, 2.85 m, and 2.75 m for real cough, sinusoidal cough, and cough jet respectively. While the droplet travel distance is about 3.1 m, 3.2 m, and 2.9 m when the AC location is at the same direction of coughing flow. Finally, the adopted CFD modeling was also used to show the effects of different AC locations on coughing, breathing particle droplets distribution in different indoor spaces, such as buildings, hospitals, and public transports, Also, showed good visual demonstration and representation of the real physical processes.

МАТЕМАТИЧНІ МОДЕЛІ ВИЗНАЧЕННЯ ТЕМПЕРАТУРНИХ ПОЛІВ У НЕОДНОРІДНИХ ЕЛЕМЕНТАХ ЦИФРОВИХ ПРИСТРОЇВ ІЗ УРАХУВАННЯМ ТЕРМОЧУТЛИВОСТІ Primary tabs View(active tab) Edit Translate

Linear and nonlinear mathematical models for determining the temperature field and subsequently analyzing temperature regimes in isotropic spatial media with semi-through foreign inclusions subjected to internal and external thermal loads are developed. For this purpose, the heat transfer coefficient for such structures is described as a single unit using asymmetric unit functions, which makes it possible to consider boundary value problems of heat transfer with one linear and nonlinear differential equations of heat transfer with discontinuous and singular coefficients and linear and nonlinear boundary conditions on the boundary surfaces of the media. In the case of a nonlinear boundary value problem, the introduced linearizing function is used to linearize the original nonlinear heat conduction equation and nonlinear boundary conditions, and as a result, a partially linearized second-order differential equation with partial derivatives and discontinuous and singular coefficients is obtained relative to the linearizing function with partially linearized boundary conditions. For the final linearization of the partially linearized differential equation and boundary conditions, the temperature is approximated by one of the spatial coordinates on the boundary surfaces of the inclusion by piecewise linear functions, as a result of which both the differential equation and boundary conditions become fully linearized. To solve the resulting linear boundary value problem, the Hankel integral transformation method is used, which results in an analytical solution that determines the introduced linearizing function. As an example, the linear dependence of the thermal conductivity coefficient of structural materials of a structure on temperature, which is often used in many practical problems, is chosen. As a result, analytical relations in the form of quadratic equations were obtained to determine the temperature distribution in a thermally sensitive layer with a foreign semi-through inclusion under external heating in the form of a heat flux. A numerical analysis of the temperature behavior as a function of spatial coordinates for given values of geometric and thermophysical parameters is performed. The influence of a foreign inclusion on the temperature distribution is investigated if the VK94-I ceramic is chosen as the material of the medium and the inclusion is silver. To determine the numerical values of temperature in the above structures, as well as to analyze heat transfer processes inside these structures caused by internal and external thermal loads, software tools have been developed that have been used to perform a geometric image of the temperature distribution depending on spatial coordinates. The obtained numerical temperature values indicate that the developed mathematical models for analyzing heat transfer processes in spatially heterogeneous environments with internal and external heating correspond to a real physical process. The software also makes it possible to analyze such environments subjected to internal and external thermal loads in terms of their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual elements but also the entire structure.

Crack resistance of bar reinforced concrete elements during construction and reconstruction

The consequence of the well-established idea of the crack resistance of reinforced concrete structures are two types of engineering calculations: for the formation of cracks and for limiting their opening width. Such a view is based on certain assumptions, simplifying the calculation to the possibility of its implementation “manually” and neglecting some real physical processes occurring in reinforced concrete under load. Seven of the most significant assumptions are considered in the article. For each of them, an analysis was carried out to determine whether they correspond to reality throughout the entire life cycle of the structure. It was proposed to use only one single calculation at the level of standards to assess the crack resistance of structures - according to the crack opening width, acrc. At the same time, the calculations already available in the design codes for limiting permeability and the safety of reinforcement will still remain in demand. The compressed algorithm is proposed for the possible consideration of the effect of cracks at all scale levels of the concrete structure, the key for which is the normalization of the statistical parameters of the distribution of discontinuities by diameters, lengths, openings, depths, directions, distances between discontinuities, etc.

Математичні моделі визначення температурних полів у теплоактивних елементах цифрових пристроїв з локальним внутрішнім нагріванням та із урахуванням термочутливості

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic spatial heat-active media subjected to internal local heat load, have been developed. In the case of a nonlinear boundary-value problem, the Kirchhoff transformation is applied, using which the original nonlinear heat conduction equation and nonlinear boundary conditions are linearized, and as a result, a linearized second-order differential equation with partial derivatives and a discontinuous right-hand side and partially linearized boundary conditions is obtained. For the final linearization of the partially linearized boundary conditions, the approximation of the temperature by the radial spatial coordinate on the boundary surface of the thermosensitive medium was performed by a piecewise constant function, as a result of which the boundary value problem was obtained completely linearized. To solve the linear boundary value problem, as well as the obtained linearized boundary value problem with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, an analytical relationship was obtained for determining the temperature distribution in this medium. Numerical analysis of temperature behavior as a function of spatial coordinates for given values of geometric and thermophysical parameters was performed. The influence of the power of internal heat sources and environmental materials on the temperature distribution was studied. To determine the numerical values of the temperature in the given structure, as well as to analyze the heat exchange processes in the middle of these structures, caused by the internal heat load, software tools were developed, using which a geometric image of the temperature distribution depending on the spatial coordinates was made. The developed linear and nonlinear mathematical models for determining the temperature field in spatial heat-active environments with internal heating testify to their adequacy to a real physical process. They make it possible to analyze such environments for their thermal stability. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements, but also the entire structure.

DISTANCE EDUCATION METHODS: VIDEO ANALYSIS IN TEACHING PHYSICS

Formation of the skills of conducting an experiment and analyzing its results during laboratory work in natural science has always been an important didactic problem, which has significantly increased in the conditions of distance and mixed learning. The study of approaches to the effective use of software for the analysis of video recordings of observations of real physical processes and phenomena is one of the tasks of instrumental digital didactics. The affordable and regularly updated software Tracker: Video Analysis and Modeling Tool is a popular didactic tool for the analysis of physical quantities based on the processing of static and dynamic images followed by comparison with the corresponding mathematical model. The rules for creating educational videos suitable for analysis in a digital environment are summarized. On the examples of laboratory classes on many topics of physics (mechanics, hydrodynamics, molecular and atomic physics, and optics) and astronomy, the general features of creating video recordings, laboratory works, and problem tasks based on video analysis are shown. The STEM laboratory of the Junior Academy of Sciences of Ukraine has created numerous reference videos about physical experiments and the rules for their use; a collection of video tasks was also created; innovative methods of educational physical experiments were developed. The methods of video analysis were tested during distance and mixed education in the conditions of threats, also in formal and informal education formats, such as the summer science school for students and the All-Ukrainian natural science online tournament "Open Natural Science Demonstration". Instrumental digital didactics is a component of training courses for teachers and is regularly discussed at seminars and conferences on science education. The stemua.science source of the "MANU" NC is popular among Internet users.

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

After the discovery in early 1960s by E. Lorenz and Y. Ueda of the first example of a chaotic attractor in numerical simulation of a real physical process, a new scientific direction of analysis of chaotic behavior in dynamical systems arose. Despite the key role of this first discovery, later on a number of works have appeared supposing that chaotic attractors of the considered dynamical models are rather artificial, computer-induced objects, i.e., they are generated not due to the physical nature of the process, but only by errors arising from the application of approximate numerical methods and finite-precision computations. Further justification for the possibility of a real existence of chaos in the study of a physical system developed in two directions. Within the first direction, effective analytic-numerical methods were invented providing the so-called computer-assisted proof of the existence of a chaotic attractor. In the framework of the second direction, attempts were made to detect chaotic behavior directly in a physical experiment, by designing a proper experimental setup. The first remarkable result in this direction is the experiment of L. Chua, in which he designed a simple RLC circuit (Chua circuit) containing a nonlinear element (Chua diode), and managed to demonstrate the real evidence of chaotic behavior in this circuit on the screen of oscilloscope. The mathematical model of the Chua circuit (further, Chua system) is also known to be the first example of a system in which the existence of a chaotic hidden attractor was discovered and the bifurcation scenario of its birth was described. Despite the nontriviality of this discovery and cogency of the procedure for hidden attractor localization, the question of detecting this type of attractor in a physical experiment remained open. This article aims to give an exhaustive answer to this question, demonstrating both a detailed formulation of a radiophysical experiment on the localization of a hidden attractor in the Chua circuit, as well as a thorough description of the relationship between a physical experiment, mathematical modeling, and computer simulation.

Analysis of problems of small-angle approximation in mathematical models of projectile flight

The article deals with the topical issue of developing mathematical models of projectile flight, which accurately describe the projectile motion in the air. It is shown that the nature of the mathematical models presentation varies depending on the required reliability degree of the real physical projectile flight process representation by the mathematical model, the adequate consideration of certain forces (moments) acting on the projectile, as well as the level of information about the external flight conditions which include the parameters of the air in which the projectile moves.At the same time, the use of the shape coefficient - the agreement coefficient in the differential equation system leads to "rough" mathematical models, which does not allow to adequately describe the projectile flight trajectory and its individual elements. The solution to this problem is especially relevant during developing and implementing procedures, technical solutions in the interest of achieving the necessary level of interoperability with NATO forces, the gradual abandonment of the standard functions of air resistance, the transition to individual functions and mathematical models of projectile motion, which are currently accepted in the member states of the Alliance. The conducted analysis of modern mathematical models showed that their construction is based on an approximate approach, which was called the small-angle approximation, in which, for an axisymmetric rotating projectile, it is considered that the nutation angles are sufficiently small, the aerodynamic forces (moments) depend only on the speed of its flight and the nutation angle, and only the linear terms of their Taylor series expansion are used in the calculations. The nutation-precessional behavior of the projectile was considered and the nonlinear dependencies of the coefficients of the aerodynamic forces (moments) of the projectile on the angles of nutation were revealed.

Modeling in CupCarbon

Modeling is one of the key categories of modern cognitive theory and design-engineering practice. Simulation and visualization of real physical processes and phenomena are important technological tools and simulation results. The article presents a comparative review and analysis of existing emulators, such as Cisco Packet Tracer and UNetLab, as well as the rationale for choosing CupCarbon as a platform for modeling, in particular, wireless networks for the Internet of Things. The functional potential of today’s most common emulators like Cisco Packet Tracer and UNetLab are significant. Cisco Packet Tracer is a powerful software emulator, which gives the possibility to users to simulate networks, organizing them with the almost unlimited number of devices, using the equipment and adjusting it for the specific tasks of this or that environment. The program gives the opportunity to develop the quality of decision-making speed, creativity and critical thinking, configuration and troubleshooting networks with the use of virtual equipment and simulated connection. Despite the great advantages of this software emulator, it has several disadvantages, which make it incorrect for modeling IoT networks, such as the lack of simulation in a 2D/3D environment and adding urban and natural noise, not a complete emulation of IOS, almost everything that goes beyond the CCNA, it also cannot be built on it, the possible manifestation of a variety of bugs that are treated only by restarting the program.

Anomaly detection of industrial state quantity time-Series data based on correlation and long short-term memory

ABSTRACT Anomaly detection of multi-dimensional time-series data is a key research area, and the analysis of control, switching, and other state signals (i.e., industrial state quantity time series) is of particular importance to the operational sciences. When only the limited values of industrial state quantities are taken in the discrete set, there is no continuous change trend, making it difficult to achieve good results when applying analogue anomaly detection methods directly. In this study, assuming a correlation between the time series of state and analogue quantities in industrial systems, a model for anomaly detection in state quantity time-series data is built through a correlation supported by long short-term memory, and the model is verified using real physical process data. These results demonstrate that the proposed method is superior to extant industrial time-series models. Thus far, no studies focusing on the anomaly detection of two-state quantity time-series outliers have been performed. We believe that the research problem addressed herein and the proposed method contribute an interesting design methodology for the anomaly detection of time-series data in IIoT.

QoS ROUTING IN TELECOMMUNICATIONS NETWORKS

With the increasing demand for quality of service (QoS) in today's multiservice networks, more and more attention is paid to the routing facilities. With the transition to new generation networks, the issues of improving routing algorithms and protocols seem to be especially relevant. The existing routing algorithms have been developed quite fully and, on the whole, function effectively if the optimization of the route selection is carried out according to one parameter. In the case of multi-criteria routing, specialists are faced with the problem of reducing many criteria to a single one by means of additive convolution. Less commonly, multiplicative convolution is used, since it is not always possible to apply multiplicative transformation to real physical processes. The problem of applying multiplicative convolution to various QoS requirements is to choose the weight coefficients for each of the parameters. In this paper, we analyze the choice of the optimal path for multicriteria routing, taking into account the weight coefficients and cost parameters on the basis of the above twolevel approach. The results of numerical modeling of the search for the optimal path for various values of weight coefficients and cost coefficients are presented. It is shown that when choosing a path for multi-criteria optimization, it is necessary to choose the coefficients of the additive convolution as the product of the weight coefficients and the cost coefficients directly. It has been found that the value of the probability of packet transmission affects to a greater extent the choice of the optimal path than the values of the coefficients for the parameters of delay, delay variation, and bandwidth.

Multivariable Fractional Polynomials for lithium-ion batteries degradation models under dynamic conditions

Longevity and safety of lithium-ion batteries are facilitated by efficient monitoring and adjustment of the battery operating conditions. Hence, it is crucial to implement fast and accurate algorithms for State of Health (SoH) monitoring on the Battery Management System. The task is challenging due to the complexity and multitude of the factors contributing to the battery capacity degradation, especially because the different degradation processes occur at various timescales and their interactions play an important role. Data-driven methods bypass this issue by approximating the complex processes with statistical or machine learning models: they rely solely on the available cycling data, while remaining agnostic to the underlying real physical processes. This paper proposes a data-driven approach which is understudied in the context of battery degradation, despite being characterised by simplicity and ease of computation: the Multivariable Fractional Polynomial (MFP) regression. Models are trained from historical data of one exhausted cell and used to predict the SoH of other cells. The data are provided by the NASA Ames Prognostics Center of Excellence, and are characterised by varying loads which simulate dynamic operating conditions. Two hypothetical scenarios are considered: one assumes that a recent observed capacity measurement is known, the other is based only on the nominal capacity of the cell. It was shown that the degradation behaviour of the batteries under examination is influenced by their historical data, as supported by the low prediction errors achieved (root mean squared errors ranging from 1.2% to 7.22% when considering data up to the battery End of Life). Moreover, we offer a multi-factor perspective where the degree of impact of each different factor to ageing acceleration is analysed. Finally, we compare with a Long Short-Term Memory Neural Network and other works from the literature on the same dataset. We conclude that the MFP regression is effective and competitive with contemporary works, and provides several additional advantages e.g. in terms of interpretability, generalisability, and implementability.

Gaussian Process Surrogates for Modeling Uncertainties in a Use Case of Forging Superalloys

The avoidance of scrap and the adherence to tolerances is an important goal in manufacturing. This requires a good engineering understanding of the underlying process. To achieve this, real physical experiments can be conducted. However, they are expensive in time and resources, and can slow down production. A promising way to overcome these drawbacks is process exploration through simulation, where the finite element method (FEM) is a well-established and robust simulation method. While FEM simulation can provide high-resolution results, it requires extensive computing resources to do so. In addition, the simulation design often depends on unknown process properties. To circumvent these drawbacks, we present a Gaussian Process surrogate model approach that accounts for real physical manufacturing process uncertainties and acts as a substitute for expensive FEM simulation, resulting in a fast and robust method that adequately depicts reality. We demonstrate that active learning can be easily applied with our surrogate model to improve computational resources. On top of that, we present a novel optimization method that treats aleatoric and epistemic uncertainties separately, allowing for greater flexibility in solving inverse problems. We evaluate our model using a typical manufacturing use case, the preforming of an Inconel 625 superalloy billet on a forging press.

Построение хранилища обобщѐнных вычислительных экспериментов на основе онтологического подхода

The paper deals with the problem of organizing storage and access to information about generalized computational experiments related to the mathematical modeling of real physical processes. A generalized computational experiment implies a multiple solution of the numerical simulation problem for different sets of values of defining model parameters, and such an approach produces a solution at once for a certain class of problems of mathematical modeling given in a multidimensional space of defining parameters. An approach to building a data repository, describing generalized computational experiments, including the history of their planning and scenario corrections, is proposed. The features of storage of generalized computational experiments, which are the basis for the construction of logical and physical models of the repository, are discussed. To organize effective user interaction with the repository, an ontological model of a generalized computational experiment and the associated metadata structure are proposed, on the basis of which the mechanism for generating queries to the repository in terms of the subject area of modeling is implemented. The application of the developed approach for storing and analyzing data on a generalized computational experiment related to the estimation of the accuracy of numerical models of the OpenFOAM software platform for the three-dimensional problem of supersonic flow around a wedge is discussed.

МАТЕМАТИЧНІ ПРОСТОРОВІ МОДЕЛІ ВИЗНАЧЕННЯ ТЕМПЕРАТУРНОГО ПОЛЯ ІЗ ЛОКАЛЬНО ЗОСЕРЕДЖЕНИМ ТЕПЛОВИМ НАГРІВАННЯМ

Linear and nonlinear mathematical models for determining the temperature field, and later the analysis of temperature regimes in isotropic spatial inhomogeneous media exposed to internal and external thermal loads have been developed. To do this, the thermal conductivity for such structures is described as a whole using symmetric unit functions, which allows us to consider boundary thermal conductivity problems with one linear and nonlinear differential equation of thermal conductivity with discontinuous coefficients and linear and nonlinear boundary conditions on boundary surfaces. In the case of a nonlinear boundary value problem, the Kirchhoff transform is applied, which linearizes the initial nonlinear equation of thermal conductivity and nonlinear boundary conditions and results in a second-order linear differential equation with partial derivatives and singular coefficients with respect to the Kirchhoff function with linear conditions. To solve the obtained linear boundary value problem, the method of integral Fourier transform was used, as a result of which an analytical solution was obtained, which determines the Kirchhoff linearizing function. As an example, the linear and cubic dependences of the thermal conductivity of structural materials on the structure, which are often used in many practical problems, are chosen. As a result, analytical relations in the form of quadratic and biquadratic equations are obtained to determine the temperature distribution in the thermosensitive layer with foreign inclusion at external local heating. Numerical analysis of temperature behavior as a function of spatial coordinates for given values of geometric and thermophysical parameters is performed. The influence of foreign inclusion on the temperature distribution was studied if the material of the medium was selected ceramics VK94-I, and the inclusion – silver, aluminum and silicon. To determine the numerical values of temperature in these structures, as well as the analysis of heat transfer processes in the middle of these structures due to internal and external heat loads, developed software that uses a geometric representation of temperature distribution depending on spatial coordinates. The obtained numerical values of temperature testify to the correspondence of the developed mathematical models of the analysis of heat exchange processes in spatial inhomogeneous media with internal and external heating to the real physical process. Software also allows you to analyze this type of environment, which are exposed to internal and external heat loads, in terms of their heat resistance. As a result, it becomes possible to increase it and protect it from overheating, which can lead to the destruction of not only individual elements but also the entire structure.

Математичні моделі температурного поля в термочутливих елементах електронних пристроїв

Nonlinear mathematical models for the analysis of temperature regimes in a thermosensitive isotropic plate heated by locally concentrated heat sources have been developed. For this purpose, the heat-active zones of the plate are described using the theory of generalized functions. Given this equation of thermal conductivity and boundary conditions contain discontinuous and singular right parts. The original nonlinear equations of thermal conductivity and nonlinear boundary conditions are linearized by Kirchhoff transformation. To solve the obtained boundary value problems, the integral Fourier transform was used and, as a result, their analytical solutions in the images were determined. The inverse integral Fourier transform was applied to these solutions, which made it possible to obtain analytical expressions for determining the Kirchhoff variable. As an example, the linear dependence of the thermal conductivity on temperature is chosen, which is often used in many practical problems. As a result, analytical relations were obtained to determine the temperature in the heat-sensitive plate. The given analytical solutions are presented in the form of improper convergent integrals. According to Newton’s method (three eighths), numerical values of these integrals are obtained with a certain accuracy for given values of plate thickness, spatial coordinates, specific power of heat sources, thermal conductivity of structural materials of the plate and geometric parameters of the heat-active zone. The material of the plate is silicon and germanium. To determine the numerical values of temperature in the structure, as well as the analysis of heat transfer processes in the middle of the plate due to local heating, developed software, using which geometric mapping of temperature distribution depending on spatial coordinates, thermal conductivity, specific heat flux density. The obtained numerical values of temperature testify to the correspondence of the developed mathematical models of the analysis of heat exchange processes in the thermosensitive plate with local heating to the real physical process.

АЛГОРИТМІЧНІ ФОРМУВАННЯ СПРЯЖЕНИХ МОДЕЛЕЙ ДЛЯ ІНФОРМАЦІЙНО-КЕРУЮЧОЇ СИСТЕМИ В РАЗІ ПРОЄКТУВАННЯ КІНЕМАТИЧНИХ ПАР

In modern systems for the automatic design of complex products for military equipment, algorithmic construction of conjugate models of curved surfaces using computer technologies that exclude interference are increasingly being used. This gives a powerful impetus to the development of applied geometry of surfaces and stimulates the search for new ways to improve the technological processes of manufacturing parts, on machines with an information control system. In particular, this applies to technological operations on numerically controlled machines. One of the main directions in modeling kinematic pairs should be considered the study and design of the shapes of surfaces in close connection with the operating conditions of structures in military equipment and weapons in which they are to be used. The shapes of complex curved surfaces affect the reliability and durability of mechanisms in military equipment and weapons and therefore require more careful consideration of external conditions when designing. Much attention is paid to the design of surfaces, taking into account an increasing number of predetermined requirements, conditions for the formation of curved surfaces of kinematic pairs in military equipment. The use of algorithmic constructions of conjugated kinematic pairs for an information control system in the manufacture of real surfaces of military equipment obtained as a result of stamping, the creation of a geometric model of stamping, reflecting a real physical process, is an urgent problem. In recent years, in the manufacture of precision high-quality products of kinematic pairs and cutting tools in military equipment and weapons, complex curved surfaces have been widely used, which require the development of a geometric and mathematical apparatus for their modeling. Algorithmic modeling of curvilinear transformations for the formation of mating surfaces of kinematic pairs in military equipment, based on a parametric kinematic screw, is proposed, which will eliminate interference during their manufacture.