Introduction. Functionally graded materials are of great use, because heterogeneity of properties enables to control the strength and rigidity of structures. This has caused great interest in the topic in the world scientific literature. The construction of solutions to such problems depends significantly on the type of boundary conditions. In this paper, we consider the equilibrium of a thin-walled circular cylinder whose mechanical properties change along the radius. Homogeneous boundary conditions were set on cylindrical surfaces that had not been considered before, the effect was on the ends. The mathematical formulation of the problem was carried out in the linear theory of elasticity in the framework of axisymmetric deformation. Expressions were constructed for the components of the stress-strain state of the cylinder, in which some coefficients were found from the solution to the resulting system of linear algebraic equations.

Materials and Methods. The material of the cylinder was linearly elastic, the elastic modulus of which depended linearly on the radial coordinate. The basic research method was the asymptotic method, in which half the logarithm of the ratio of the outer and inner radii acted as a small parameter. Iterative processes were used to construct the characteristics of the stress-strain state of the cylinder.

Results. Homogeneous solutions to the boundary value problem were obtained for a linearly elastic functionally gradient hollow thin-walled cylinder. An analysis of these solutions made it possible to reveal the nature of the stress-strain state in the cylinder wall. For this purpose, an asymptotic analysis of the solutions was carried out, relations for displacements and stresses were obtained. It was determined that those solutions corresponded to the boundary layer, while their first terms determined Saint-Venant edge effect similar to the plate theory.

Discussion and Conclusion. The analytical solution to the equilibrium problem of a thin-walled cylinder inhomogeneous in radius constructed by asymptotic expansion can be used for numerical solution to a specific problem. For this, it is required to solve the obtained systems of linear algebraic equations and determine the corresponding coefficients. The resulting asymptotic representations provide analyzing the three-dimensional stress-strain state. The selection of the number of expansion terms makes it possible to calculate displacements and stresses with a given degree of accuracy. This analysis can be useful in assessing the adequacy of applied calculation methods used in engineering practice.

Introduction. The widespread use of piezoelectric materials in various industries stimulates the study of their physical characteristics and determines the urgency of such research. In this case, modal analysis makes it possible to determine the operating frequency and the coefficient of electromechanical coupling of piezoelectric elements of various devices. These indicators are of serious theoretical and applied interest. The study was aimed at the development of numerical methods for solving the problem of determining resonance frequencies in a system of elastic bodies. To achieve this goal, we needed new approaches to the discretization of the problem based on the finite element method and the execution of the software implementation of the selected method in C# on the .net platform. Current solutions were created in the context of the ACELAN-COMPOS class library. The known methods of solving the generalized eigenvalue problem based on matrix inversion are not applicable to large-dimensional matrices. To overcome this limitation, the presented scientific work implemented the logic of constructing mass matrices and created software interfaces for exchanging data on eigenvalue problems with pre- and postprocessing modules.

Materials and Methods. A platform was used to implement numerical methods .net and the C# programming language. Validation of the research results was carried out through comparing the values found with solutions obtained in well-known SAE packages (computer-aided engineering). The created routines were evaluated in terms of performance and applicability for large-scale tasks. Numerical experiments were carried out to validate new algorithms in small-dimensional problems that were solved by known methods in MATLAB. Next, the approach was tested on tasks with a large number of unknowns and taking into account the parallelization of individual operations. To avoid finding the inverse matrix, a modified Lanczos method was programmatically implemented. We examined the formats for storing matrices in RAM: triplets, CSR, СSC, Skyline. To solve a system of linear algebraic equations (SLAE), an iterative symmetric LQ method adapted to these storage formats was used.

Results. New calculation modules integrated into the class library of the ACELAN-COMPOS complex were developed. Calculations were carried out to determine the applicability of various formats for storing sparse matrices in RAM and various methods for implementing operations with sparse matrices. The structure of stiffness matrices constructed for the same task, but with different renumbering of nodes of a finite element grid, was graphically visualized. In relation to the problem of the theory of electroelasticity, data on the time required to perform basic operations with stiffness matrices in various storage formats were summarized and presented in the form of a table. It has been established that the renumbering of grid nodes gives a significant increase in performance even without changing the internal structure of the matrix in memory. Taking into account the objectives of the study, the advantages and weaknesses of known matrix storage formats were named. Thus, CSR was optimal when multiplying a matrix by a vector, SKS was optimal when inverting a matrix. In problems with the number of unknowns of the order of 10³, iterative methods for solving a generalized eigenvalue problem won in speed. The performance of the software implementation of the Lanczos method was evaluated. The contribution of all operations to the total solution time was measured. It has been found that the operation of solving SLAE takes up to 95% of the total time of the algorithm. When solving the SLAE by symmetric LQ method, the greatest computational costs were needed to multiply the matrix by a vector. To increase the performance of the algorithm, parallelization with shared memory was resorted to. When using eight threads, the performance gain increased by 40–50%.

Discussion and Conclusion. The software modules obtained as part of the scientific work were implemented in the ACELAN-COMPOS package. Their performance for model problems with quasi-regular finite element grids was estimated. Taking into account the features of the structures of the stiffness and mass matrices obtained through solving the generalized eigenvalue problem for an electroelastic body, the preferred methods for their processing were determined.

Introduction. The formation of the quality parameters of the surface layer and the operational properties of the parts occurs throughout all stages of their manufacture. However, the decisive impact is most often exerted by the stages of finishing. Therefore, in modern digital engineering, the task of process support of high quality of the surface layer of the part is one of the challenges in solving the problem of improving the quality and reliability and increasing the life cycle of manufactured machines. Surface plastic deformation treatment is instrumental in improving the performance characteristics of machine parts. Its essence is that the required quality parameters of parts are obtained not by removing a layer of material, but by plastic deformation. During the processing, both the dimensions of the parts and the physical and mechanical properties of the surface layers are changed. In this case, the technologist has the opportunity to significantly increase the life cycle of the manufactured products through controlling the process. These studies are aimed at providing the required quality parameters of the surface layer under processing with an eccentric hardener.

Materials and Methods. The article presents the results of research on a new method of surface plastic deformation treatment – with an oscillating eccentric hardener. The considered processing method enables to obtain high quality of the treated surface, to process large-sized parts in places that are stress concentrators, to process welds, small areas of surfaces, whose hardening is needed for the part to fulfill its intended service. A set of theoretical studies was carried out; their results provided determining the parameters of a single interaction of the indenter and the surface of the part, the diameter of the plastic imprint and its depth.

Results. Dependences for determining the surface roughness, the depth of the hardened layer and the degree of deformation were obtained. The resulting formulas were tested for adequacy by experimental studies.

Discussion and Conclusion. The obtained research results can be used in the technological design of surface plastic deformation treatment processes. Further tasks for the study of the considered processing method are determined.

Introduction. In the areas of power engineering where the thermal energy of superheated steam is used, an important aspect of providing the reliability and safety of equipment is the heat resistance of the materials employed. In the manufacture of induction superheaters, the optimal material for the steam pipe (coil) is copper. However, its ultimate resistance to oxidation does not exceed 400 °C, which significantly limits the efficiency of steam generators. Therefore, the objective of the work was to study the kinetics of oxidation of the combined galvanic coating of the Mo-Ni-Cr system applied to copper tubular samples and intended for thermal protection of steam generator coils.

Materials and Methods. A combined electroplating of the Mo-Ni-Cr system with a total thickness of 12–35 μm was formed on the experimental copper tubular samples. A Mo sublayer with a thickness of about 1.5 μm on the surface of the copper tube was formed to prevent the diffusion of Cu into the Ni coating. A 1.5 μm thick chromium layer on the coating surface acted as an indicator of the oxidation process. A comparative analysis of the oxidation processes of the copper surface and the combined coating of the Mo-Ni-Cr system on a copper substrate was carried out using the methods of optical and electron microscopy, energy dispersive analysis, and precision determination of the growth parameters of oxide films.

Results. The intervals of thermal stability of the copper substrate and nickel coating were experimentally determined. The obtained experimental dependences characterized the parabolic law of copper oxidation with the formation of a single-phase diffusion zone of CuO at temperatures above 350 °C, and nickel at temperatures above 750 °C, when the transition of NiO monoxide into oxide Ni₂O₃ began. The growth of oxide films according to quadratic laws provided a rapid increase in the thickness of the films, the accumulation of stresses in them, cracking, and chipping.

Discussion and Conclusion. It is shown that the Mo-Ni-Cr electroplating is resistant to heating during long-term operation up to temperatures of 750–800 °C. The functional roles of Mo and Cr in the coating architecture were described. The work focused on the applied aspect of using the coating under study to increase the thermal stability of the steam pipelines of industrial induction superheaters with low and medium power.

Introduction. The assessment of the manufacturability of products – as a stage of production planning and a key aspect of the development of modern industrial machining systems — is an urgent task of modern mechanical engineering. In this regard, theoretical and practical research on the development of methodological approaches to determining the weight significance of quantitative indicators in assessing the manufacturability of parts is highly relevant. The objective of the presented work was to develop an evaluation method aimed at improving the quality of part processing and the effectiveness of the performance of multiproduct manufacturing systems based on the development of additional quantitative indicators for assessing production manufacturability.

Materials and Methods. To assess the impact of quantitative production indicators associated with time spent during equipment downtime, a model was created. It was aimed at predicting event flows of delivery of batches of parts for manufacturing for a specific operation and flows of processed parts using the queuing theory apparatus. This approach makes it possible to take into account both the design-engineering characteristics of parts, the features of a particular production system, and the emerging manufacturing situation.

Results. The degree of influence of the manufacturability indicators at the level of the process operation was determined by assessing the possible impact on the components when calculating piece-calculation time (Тшт.к.). The interrelations between the manufacturability indicators and expenses for all items of the production cost of part processing (СОП), as well as costs associated with organizational downtime of equipment (Спр.о.i) were established. The degree of influence of the indicators of manufacturability relative to other indicators was determined by using the apparatus of paired comparisons in decision-making in relation to all structural elements of production costs.

Discussion and Conclusion. The approach to the implementation of this design procedure was described, which provided taking into account the composition and capabilities of processing equipment of a particular production and the actual production situation. The developed formalized models make it possible to comprehensively predict the impact of the manufacturability indicators of parts on the performance effectiveness of machining systems during their manufacture.

Introduction. The prediction module generates possible future trajectories of dynamic objects that enables a self-driving vehicle to move safely on public roads. However, all modern prediction methods evaluate their performance only under urban conditions and do not consider their applicability to the domain of rural roads. This work examined the adaptability of existing methods to work under rural unstructured conditions and suggested a new, improved approach.

Materials and Methods. As a solution, we propose to use a neural network that includes the following submodules: a graph-based scene encoder, a multimodal trajectory decoder, and a trajectory filtering module. Another proposed feature is to use an adapted loss function that penalizes the network for generating trajectories that go beyond the drivable area. These elements use standard practices for solving the prediction problem and adapting it to the domain of rural roads.

Results. The presented analysis described the basic features of the prediction module in the rural road domain, showed a comparison of popular models, and discussed its applicability to new conditions. The paper describes the new approach that is more adaptive to the considered domain of study. A simulation of the new domain was performed by modifying existing public datasets.

Discussion and Conclusion. Comparison to other popular methods has shown that the proposed approach provides more accurate results. The disadvantages of the proposed approach were also identified and possible solutions were described.

Introduction. The 2020s were marked by the emergence of a new generation of computer simulators using augmented reality. One of the promising advantages of augmented reality technology is the ability to safely simulate hazardous situations real-world. A prerequisite for realizing this advantage is to provide the visual coherence of augmented reality scenes: virtual objects must be indistinguishable from real ones. All IT leaders consider augmented reality as a next “big wave”; thus, the visual coherence is becoming a key issue for IT in general. However, it is in aerospace applications that the visual coherence has already acquired practical significance. An example is Boeing's development of an augmented reality flight simulator, which began in 2022. Visual coherence is a complex problem, one of the aspects of which is to provide the correct overall coloration of virtual objects in an augmented reality scene. The objective of the research was to develop a new method of such tinting.

Materials and Methods. The developed method (called spectral transplantation) uses two-dimensional spectral image transformations.

Results. A spectral transplantation technology is proposed that provides direct transfer of color, brightness, and contrast characteristics from the real background to virtual objects. An algorithm for automatic selection of the optimal type of spectral transformation has been developed.

Discussion and Conclusion. Being a fully automatic process without recording lighting conditions, spectral transplantation solves a number of complex problems of visual coherence. Spectral transplantation can be a valuable addition to other methods of providing visual coherence.

Introduction. It is obvious that in the near future, the issues of equipping moving robotic systems with autonomous control elements will remain relevant. This requires the development of models of group pursuit. Note that optimization in pursuit tasks is reduced to the construction of optimal trajectories (shortest trajectories, trajectories with differential constraints, fuel consumption indicators). At the same time, the aspects of automated distribution by goals in group pursuit were not considered. To fill this gap, the presented piece of research has been carried out. Its result should be the construction of a model of automated distribution of pursuers by goals in group pursuit.

Materials and Methods. A matrix was formed to study the multiple goal group pursuit. The control parameters for the movement of the pursuers were modified according to the minimum curvature of the trajectory. The methods of pursuit and approach were considered in detail. The possibilities of modifying the method of parallel approach were shown. Matrix simulation was used to build a scheme of multiple goal group pursuit. The listed processes were illustrated by functions in the given coordinate systems and animation. Block diagrams of the phase coordinates of the pursuer at the next step, the time and distance of the pursuer reaching the goal were constructed as a base of functions. In some cases, the location of targets and pursuers was defined as points on the circle of Apollonius. The matrix was formed by samples corresponding to the distribution of pursuers by goals.

Results. Nine variants of the pursuit, parallel, proportional and three-point approach on the plane and in space were considered. The maximum value of the goal achievement time was calculated. There were cases when the speed vector of the pursuer was directed arbitrarily and to a point on the Apollonius circle. It was noted that the three-point approach method was convenient if the target was moving along a ballistic trajectory. To modify the method of parallel approach, a network of parallel lines was built on the plane. Here, the length of the arc of the line (which can be of any shape) and the array of reference points of the target trajectory were taken into account. An equation was compiled and solved with these elements. On an array of samples with corresponding time values, the minimum time was found, i.e., the optimal time for simultaneous group achievement of multiple goals was determined. For unified access to the library, the control vector was expressed through a one-parameter family of parallel planes. A library of calculations of control vectors was formed. An example of applying matrix simulation to group pursuit was shown. A scheme of group pursuit of multiple goals was presented. For two goals and three pursuers, six samples corresponding to the distribution of pursuers by goals were considered. The data was presented in the form of a matrix. Based on the research results, the computer program was created and registered – “Parallel Approach on Plane of Group of Pursuers with Simultaneous Achievement of the Goal”.

Discussions and Conclusion. The methods of using matrices in modeling group pursuit were investigated. The possibility of modifying the method of parallel approach was shown. Matrix simulation of group pursuit enabled to build its scheme for a set of purposes. The matrix of the distribution of pursuers by goals would be generated at each moment of time. Methods of forming matrices of the distribution of pursuers and targets are of interest in the design of virtual reality systems, for tasks with simulating the process of group pursuit, escape, evasion. The dynamic programming method opens up the possibility of automating the distribution with optimization according to the specified parameters under the formation of the matrix of the distribution of pursuers by goals.

Introduction. The challenges of placing virtual objects in a real-world environment limit the potential of augmented reality (AR) technology. This situation identifies a gap in scientific knowledge that requires additional research. Therefore, the main task of this study was to develop a method for optimal placement of virtual objects, in which the objective function of comfort was minimized. This approach is aimed at improving AR systems and developing the corresponding theory.

Materials and Methods. The conducted research was based on the analysis of the placement of virtual objects in AR/VR applications with particular emphasis on optimization. The concept of comfort of placement was proposed, taking into account the size of the object and the distance to the boundaries of free space in X, Y, Z coordinates.

Results. As part of the study, formulas were obtained for the optimal placement of objects with an arbitrary comfort function. The basic criterion was to minimize the difference between comfort levels from different sides of the object. It was found that a successful placement of objects required taking into account their size and comfort zones, as well as solving a system of n linear equations.

Discussion and Conclusion. The results obtained make an important contribution to the study of the problem of placing virtual objects in AR/VR/MR. They open up new opportunities for improving user interaction and conducting further research in the field of spatial computing. Possible directions for further development are dynamic adjustments and integration of the results into various XR scenarios.

Introduction. In the field of computational mathematics, there are many ways to approximate the model of fluid mechanics. Methods and estimates of approximation quality criteria, such as stability and convergence, are developed, while a combination of approaches to constructing economical difference schemes, such as splitting by physical processes, regularization by B. N. Chetverushkin, a linear combination of the Upwind and Standard Leapfrog difference schemes in aggregate has not been implemented and evaluated before. The authors were faced with the task of approximating each part of the hydrodynamic model split by physical processes with the most adequate scheme and further investigating the correctness of this approach.

Materials and Methods. The mathematical model of hydrophysical processes is closed by the empirical equation of the state of salt water. Significant properties were selected, a mathematical model was built. Difference operators approximated differential operators. An algorithm for layer-by-layer modeling of transients was constructed. The algorithm has been implemented in the form of the program, which mainly contains elementwise (massively-parallel) operations.

Results. Mathematical models of hydrodynamic processes in reservoirs were obtained, taking into account three equations of motion in the presence of a density gradient of the aqueous medium when hydrostatic approximation was abandoned. A new method of calculating the pressure field using B. N. Chetverushkin’s regularizers in the continuity equation was tested. A software module for numerical simulation of hydrophysical processes of water movement with different salinity and density was developed. This is open-source software that provides not only the redefinition of empirical dependences (as algebraic functions), but also the connection of external simulating modules to display dependences algorithmically.

Discussion and Conclusion. The developed model of hydrophysics, taking into account the properties of salt water and the dynamic relationship of the mechanical movement of water with salinity, can be used to study the formation of a nonequilibrium distribution of parameters and identify the most stable parameters of the aquatic environment. The model explains the downward movement of oxygen. That will help in the future to estimate the values of the parameters of the aquatic environment, which are difficult to measure directly. It can be used in the procedure of parametric identification of hard-to-measure parameters of the aquatic environment.