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The work is dedicated to the use of the software package Turbulence Problem Solver (TPS) for numerical simulation of a wide range of laser problems. The capabilities of the package are demonstrated by the example of numerical simulation of the interaction of femtosecond laser pulses with thin metal bonds. The software package TPS developed by the authors is intended for numerical solution of hyperbolic systems of differential equations on multiprocessor computing systems with distributed memory. The package is a modern and expandable software product. The architecture of the package gives the researcher the opportunity to model different physical processes in a uniform way, using different numerical methods and program blocks containing specific initial conditions, boundary conditions and source terms for each problem. The package provides the the opportunity to expand the functionality of the package by adding new classes of problems, computational methods, initial and boundary conditions, as well as equations of state of matter. The numerical methods implemented in the software package were tested on test problems in one-dimensional, two-dimensional and three-dimensional geometry, which included Riemann's problems on the decay of an arbitrary discontinuity with different configurations of the exact solution.

Thin films on substrates are an important class of targets for nanomodification of surfaces in plasmonics or sensor applications. Many articles are devoted to this subject. Most of them, however, focus on the dynamics of the film itself, paying little attention to the substrate, considering it simply as an object that absorbs the first compression wave and does not affect the surface structures that arise as a result of irradiation. The paper describes in detail a computational experiment on the numerical simulation of the interaction of a single ultrashort laser pulse with a gold film deposited on a thick glass substrate. The uniform rectangular grid and the first-order Godunov numerical method were used. The presented results of calculations allowed to confirm the theory of the shock-wave mechanism of holes formation in the metal under femtosecond laser action for the case of a thin gold film with a thickness of about 50 nm on a thick glass substrate.

Mathematical models of many natural processes are described by partial differential equations with singular solutions. Classical numerical methods for determination of approximate solution to such problems are inefficient. In the present paper a boundary value problem for vector wave equation in L-shaped domain is considered. The presence of reentrant corner of size $3\pi/2$ on the boundary of computational domain leads to the strong singularity of the solution, i.e. it does not belong to the Sobolev space $H^1$ so classical and special numerical methods have a convergence rate less than $O(h)$. Therefore in the present paper a special weighted set of vector-functions is introduced. In this set the solution of considered boundary value problem is defined as $R_ν$-generalized one.

For numerical determination of the $R_ν$-generalized solution a weighted vector finite element method is constructed. The basic difference of this method is that the basis functions contain as a factor a special weight function in a degree depending on the properties of the solution of initial problem. This allows to significantly raise a convergence speed of approximate solution to the exact one when the mesh is refined. Moreover, introduced basis functions are solenoidal, therefore the solenoidal condition for the solution is taken into account precisely, so the spurious numerical solutions are prevented.

Results of numerical experiments are presented for series of different type model problems: some of them have a solution containing only singular component and some of them have a solution containing a singular and regular components. Results of numerical experiment showed that when a finite element mesh is refined a convergence rate of the constructed weighted vector finite element method is $O(h)$, that is more than one and a half times better in comparison with special methods developed for described problem, namely singular complement method and regularization method. Another features of constructed method are algorithmic simplicity and naturalness of the solution determination that is beneficial for numerical computations.

The article deals with the development of the noise-reduction algorithm based on anisotropic nonlinear data filtering of computed tomography (CT). Analysis of domestic and foreign literature has shown that the most effective algorithms for noise reduction of CT data use complex methods for analyzing and processing data, such as bilateral, adaptive, three-dimensional and other types of filtrations. However, a combination of such techniques is rarely used in practice due to long processing time per slice. In this regard, it was decided to develop an efficient and fast algorithm for noise-reduction based on simplified bilateral filtration method with three-dimensional data accumulation. The algorithm was developed on C ++11 programming language in Microsoft Visual Studio 2015. The main difference of the developed noise reduction algorithm is the use an improved mathematical model of CT noise, based on the distribution of Poisson and Gauss from the logarithmic value, developed earlier by our team. This allows a more accurate determination of the noise level and, thus, the threshold of data processing. As the result of the noise reduction algorithm, processed CT data with lower noise level were obtained. Visual evaluation of the data showed the increased information content of the processed data, compared to original data, the clarity of the mapping of homogeneous regions, and a significant reduction in noise in processing areas. Assessing the numerical results of the algorithm showed a decrease in the standard deviation (SD) level by more than 6 times in the processed areas, and high rates of the determination coefficient showed that the data were not distorted and changed only due to the removal of noise. Usage of newly developed context dynamic threshold made it possible to decrease SD level on every area of data. The main difference of the developed threshold is its simplicity and speed, achieved by preliminary estimation of the data array and derivation of the threshold values that are put in correspondence with each pixel of the CT. The principle of its work is based on threshold criteria, which fits well both into the developed noise reduction algorithm based on anisotropic nonlinear filtration, and another algorithm of noise-reduction. The algorithm successfully functions as part of the MultiVox workstation and is being prepared for implementation in a single radiological network of the city of Moscow.

A classical for nonlinear dynamics model, Brusselator, is considered, being augmented by addition of a third variable, which plays the role of a fast-diffusing inhibitor. The model is investigated in one-dimensional case in the parametric domain, where two types of diffusive instabilities of system’s homogeneous stationary state are manifested: wave instability, which leads to spontaneous formation of autowaves, and Turing instability, which leads to spontaneous formation of stationary dissipative structures, or Turing structures. It is shown that, due to the subcritical nature of Turing bifurcation, the interaction of two instabilities in this system results in spontaneous formation of stationary dissipative structures already before the passage of Turing bifurcation. In response to different perturbations of spatially uniform stationary state, different stable regimes are manifested in the vicinity of the double bifurcation point in the parametric region under study: both pure regimes, which consist of either stationary or autowave dissipative structures; and mixed regimes, in which different modes dominate in different areas of the computational space. In the considered region of the parametric space, the system is multistable and exhibits high sensitivity to initial noise conditions, which leads to blurring of the boundaries between qualitatively different regimes in the parametric region. At that, even in the area of dominance of mixed modes with prevalence of Turing structures, the establishment of a pure autowave regime has significant probability. In the case of stable mixed regimes, a sufficiently strong local perturbation in the area of the computational space, where autowave mode is manifested, can initiate local formation of new stationary dissipative structures. Local perturbation of the stationary homogeneous state in the parametric region under investidation leads to a qualitatively similar map of established modes, the zone of dominance of pure autowave regimes being expanded with the increase of local perturbation amplitude. In two-dimensional case, mixed regimes turn out to be only transient — upon the appearance of localized Turing structures under the influence of wave regime, they eventually occupy all available space.

A hierarchical method of mathematical and computer modeling of interval-stochastic thermal processes in complex electronic systems for various purposes is developed. The developed concept of hierarchical structuring reflects both the constructive hierarchy of a complex electronic system and the hierarchy of mathematical models of heat exchange processes. Thermal processes that take into account various physical phenomena in complex electronic systems are described by systems of stochastic, unsteady, and nonlinear partial differential equations and, therefore, their computer simulation encounters considerable computational difficulties even with the use of supercomputers. The hierarchical method avoids these difficulties. The hierarchical structure of the electronic system design, in general, is characterized by five levels: Level 1 — the active elements of the ES (microcircuits, electro-radio-elements); Level 2 — electronic module; Level 3 — a panel that combines a variety of electronic modules; Level 4 — a block of panels; Level 5 — stand installed in a stationary or mobile room. The hierarchy of models and modeling of stochastic thermal processes is constructed in the reverse order of the hierarchical structure of the electronic system design, while the modeling of interval-stochastic thermal processes is carried out by obtaining equations for statistical measures. The hierarchical method developed in the article allows to take into account the principal features of thermal processes, such as the stochastic nature of thermal, electrical and design factors in the production, assembly and installation of electronic systems, stochastic scatter of operating conditions and the environment, non-linear temperature dependencies of heat exchange factors, unsteady nature of thermal processes. The equations obtained in the article for statistical measures of stochastic thermal processes are a system of 14 non-stationary nonlinear differential equations of the first order in ordinary derivatives, whose solution is easily implemented on modern computers by existing numerical methods. The results of applying the method for computer simulation of stochastic thermal processes in electron systems are considered. The hierarchical method is applied in practice for the thermal design of real electronic systems and the creation of modern competitive devices.

Correct modeling of complex dynamics of urban transportation flows requires the collection of large volumes of empirical data to specify types of the modes and their identification. At the same time, setting a large number of observation posts is expensive and technically not always feasible. All this results in insufficient factographic support for the traffic control systems as well as for urban planners with the obvious consequences for the quality of their decisions. As one of the means to provide large-scale data collection at least for the qualitative situation analysis, the wide-area video cameras are used in different situation centers. There they are analyzed by human operators who are responsible for observation and control. Some video cameras provided their videos for common access, which makes them a valuable resource for transportation studies. However, there are significant problems with getting qualitative data from such cameras, which relate to the theory and practice of image processing. This study is devoted to the practical application of certain mainstream neuro-networking technologies for the estimation of essential characteristics of actual transportation flows. The problems arising in processing these data are analyzed, and their solutions are suggested. The convolution neural networks are used for tracking, and the methods for obtaining basic parameters of transportation flows from these observations are studied. The simplified neural networks are used for the preparation of training sets for the deep learning neural network YOLOv4 which is later used for the estimation of speed and density of automobile flows.

In this paper, a gravitational system is understood as a set of point bodies that interact according to Newton's law of attraction and have a negative value of the total energy. The question of the stability (nonstability) of a gravitational system of general position is discussed by direct computational experiment. A gravitational system of general position is a system in which the masses, initial positions, and velocities of bodies are chosen randomly from given ranges. A new method for the numerical solution of ordinary differential equations at large time intervals has been developed for the computational experiment. The proposed method allowed, on the one hand, to ensure the fulfillment of all conservation laws by a suitable correction of solutions, on the other hand, to use standard methods for the numerical solution of systems of differential equations of low approximation order. Within the framework of this method, the trajectory of a gravitational system in phase space is assembled from parts, the duration of each of which can be macroscopic. The constructed trajectory, generally speaking, is discontinuous, and the points of joining of individual pieces of the trajectory act as branch points. In connection with the latter circumstance, the proposed method, in part, can be attributed to the class of Monte Carlo methods. The general conclusion of a series of computational experiments has shown that gravitational systems of general position with a number of bodies of 3 or more, generally speaking, are unstable. In the framework of the proposed method, special cases of zero-equal angular momentum of a gravitational system with a number of bodies of 3 or more, as well as the problem of motion of two bodies, are specially considered. The case of numerical modeling of the dynamics of the solar system in time is considered separately. From the standpoint of computational experiments based on analytical methods, as well as direct numerical methods of high-order approximation (10 and higher), the stability of the solar system was previously demonstrated at an interval of five billion years or more. Due to the limitations on the available computational resources, the stability of the dynamics of the planets of the solar system within the framework of the proposed method was confirmed for a period of ten million years. With the help of a computational experiment, one of the possible scenarios for the disintegration of the solar systems is also considered.

We consider a model of spontaneous formation of a computational structure in the human brain for solving a given class of tasks in the process of performing a series of similar tasks. The model is based on a special definition of a numerical measure of the complexity of the solution algorithm. This measure has an informational property: the complexity of a computational structure consisting of two independent structures is equal to the sum of the complexities of these structures. Then the probability of spontaneous occurrence of the structure depends exponentially on the complexity of the structure. The exponential coefficient requires experimental determination for each type of problem. It may depend on the form of presentation of the source data and the procedure for issuing the result. This estimation method was applied to the results of a series of experiments that determined the strategy for solving a series of similar problems with a growing number of initial data. These experiments were described in previously published papers. Two main strategies were considered: sequential execution of the computational algorithm, or the use of parallel computing in those tasks where it is effective. These strategies differ in how calculations are performed. Using an estimate of the complexity of schemes, you can use the empirical probability of one of the strategies to calculate the probability of the other. The calculations performed showed a good match between the calculated and empirical probabilities. This confirms the hypothesis about the spontaneous formation of structures that solve the problem during the initial training of a person. The paper contains a brief description of experiments, detailed computational schemes and a strict definition of the complexity measure of computational structures and the conclusion of the dependence of the probability of structure formation on its complexity.

The work is devoted to the structural analysis of complex technical systems. Mechanical structures are considered, the properties of which affect the behavior of products during assembly, repair and operation. The main source of data on parts and mechanical connections between them is a hypergraph. This model formalizes the multidimensional basing relation. The hypergraph correctly describes the connectivity and mutual coordination of parts, which is achieved during the assembly of the product. When developing complex products in CAD systems, an engineer often makes serious design mistakes: overbasing of parts and non-sequential assembly operations. Effective ways of identifying these structural defects have been proposed. It is shown that the property of independent assembly can be represented as a closure operator whose domain is the boolean of the set of product parts. The images of this operator are connected and coordinated subsets of parts that can be assembled independently. A lattice model is described, which is the state space of the product during assembly, disassembly and decomposition into assembly units. The lattice model serves as a source of various structural information about the project. Numerical estimates of the cardinality of the set of admissible alternatives in the problems of choosing an assembly sequence and decomposition into assembly units are proposed. For many technical operations (for example, control, testing, etc.), it is necessary to mount all the operand parts in one assembly unit. A simple formalization of the technical conditions requiring the inclusion (exclusion) of parts in the assembly unit (from the assembly unit) has been developed. A theorem that gives an mathematical description of product decomposition into assembly units in exact lattice terms is given. A method for numerical evaluation of the robustness of the mechanical structure of a complex technical system is proposed.

The work analyzes known classes of convolutional neural network models and studies selected from them promising models for detecting flying objects in images. Object detection here refers to the detection, localization in space and classification of flying objects. The work conducts a comprehensive study of selected promising convolutional neural network models in order to identify the most effective ones from them for creating mobile real-time computer vision systems. It is shown that the most suitable models for detecting flying objects in images, taking into account the formulated requirements for mobile real-time computer vision systems, are models of the YOLO family, and five models from this family should be considered: YOLOv4, YOLOv4-Tiny, YOLOv4-CSP, YOLOv7 and YOLOv7-Tiny. An appropriate dataset has been developed for training, validation and comprehensive research of these models. Each labeled image of the dataset includes from one to several flying objects of four classes: “bird”, “aircraft-type unmanned aerial vehicle”, “helicopter-type unmanned aerial vehicle”, and “unknown object” (objects in airspace not included in the first three classes). Research has shown that all convolutional neural network models exceed the specified threshold value by the speed of detecting objects in the image, however, only the YOLOv4-CSP and YOLOv7 models partially satisfy the requirements of the accuracy of detection of flying objects. It was shown that most difficult object class to detect is the “bird” class. At the same time, it was revealed that the most effective model is YOLOv7, the YOLOv4-CSP model is in second place. Both models are recommended for use as part of a mobile real-time computer vision system with condition of additional training of these models on increased number of images with objects of the “bird” class so that they satisfy the requirement for the accuracy of detecting flying objects of each four classes.