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А discrete element model is applied to the problem of a spherical projectile penetration into a massive obstacle. According to the model both indenter and obstacle are described by a set of densely packed particles. To model the interaction between the particles the two-parameter Lennard–Jones potential is used. Computer implementation of the model has been carried out using parallelism on GPUs, which resulted in high spatial — temporal resolution. Based on the comparison of the results of numerical simulation with experimental data the binding energy has been identified as a function of the dynamic hardness of materials. It is shown that the use of this approach allows to accurately describe the penetration process in the range of projectile velocities 500–2500 m/c.

The paper is devoted to the analysis of the proximity of the population system to dangerous boundaries. An intersection of these boundaries results in the collapse of the stable coexistence of interacting populations. As a reason of such destruction one can consider random perturbations inevitably presented in any living system. This study is carried out on the example of the well-known model of interaction between predator and prey populations, taking into account both a stabilizing factor of the competition of predators for another than prey resources, and also a destabilizing saturation factor for predators. To describe the saturation of predators, we use the second type Holling trophic function. The dynamics of the system is studied as a function of the predator saturation, and the coefficient of predator competition for resources other than prey. The paper presents a parametric description of the possible dynamic regimes of the deterministic model. Here, local and global bifurcations are studied, and areas of sustainable coexistence of populations in equilibrium and the oscillation modes are described. An interesting feature of this mathematical model, firstly considered by Bazykin, is a global bifurcation of the birth of limit cycle from the separatrix loop. We study the effects of noise on the equilibrium and oscillatory regimes of coexistence of predator and prey populations. It is shown that an increase of the intensity of random disturbances can lead to significant deformations of these regimes right up to their destruction. The aim of this work is to develop a constructive probabilistic criterion for the proximity of the population stochastic system to the dangerous boundaries. The proposed approach is based on the mathematical technique of stochastic sensitivity functions, and the method of confidence domains. In the case of a stable equilibrium, this confidence domain is an ellipse. For the stable cycle, this domain is a confidence band. The size of the confidence domain is proportional to the intensity of the noise and stochastic sensitivity of the initial deterministic attractor. A geometric criterion of the exit of the population system from sustainable coexistence mode is the intersection of the confidence domain and the corresponding separatrix of the unforced deterministic model. An effectiveness of this analytical approach is confirmed by the good agreement of theoretical estimates and results of direct numerical simulations.

The application of a conservative numerical method of fluxes is examined for studying the vortex structures in the massive, fast-turned compact astrophysical objects, which are in self-gravity conditions. The simulation is accomplished for the objects with different mass and rotational speed. The pictures of the vortex structure of objects are visualized. In the calculations the gas-dynamic model is used, in which gas is accepted perfected and nonviscous. Numerical procedure is based on the finite-difference approximation of the conservation laws of the additive characteristics of medium for the finite volume. The “upwind” approximations of the densities of distribution of mass, components of momentum and total energy are applied. For the simulation of the objects, which possess fast-spin motion, the control of conservation for the component of moment of momentun is carried out during calculation. Evolutionary calculation is carried out on the basis of the parallel algorithms, realized on the computer complex of cluster architecture. Algorithms are based on the standardized system of message transfer Message Passing Interface (MPI). The blocking procedures of exchange and non-blocking procedures of exchange with control of the completion of operation are used. The parallelization on the space in two or three directions is carried out depending on the size of integration area and parameters of computational grid. For each subarea the parallelization based on the physical factors is carried out also: the calculations of gas dynamics part and gravitational forces are realized on the different processors, that allows to raise the efficiency of algorithms. The real possibility of the direct calculation of gravitational forces by means of the summation of interaction between all finite volumes in the integration area is shown. For the finite volume methods this approach seems to more consecutive than the solution of Poisson’s equation for the gravitational potential. Numerical calculations were carried out on the computer complex of cluster architecture with the peak productivity 523 TFlops. In the calculations up to thousand processors was used.

In this article, a meshfree-spectral method for numerical investigation of dynamics of planar geophysical flows is proposed. We investigate inviscid incompressible fluid flows with the presence of planetary rotation. Mathematically this problem is described by the non-steady system of two partial differential equations in terms of stream and vorticity functions with different boundary conditions (closed flow region and periodic conditions). The proposed method is based on several assumptions. First of all, the vorticity field is given by its values on the set of particles. The function of vorticity distribution is approximated by piecewise cubic polynomials. Coefficients of polynomials are found by least squares method. The stream function is calculated by using the spectral global Bubnov –Galerkin method at each time step.

The dynamics of fluid particles is calculated by pseudo-symplectic Runge –Kutta method. A detailed version of the method for periodic boundary conditions is described in this article for the first time. The adequacy of numerical scheme was examined on test examples. The dynamics of the configuration of four identical circular vortex patches with constant vorticity located at the vertices of a square with a center at the pole is investigated by numerical experiments. The effect of planetary rotation and the radius of patches on the dynamics and formation of vortex structures is studied. It is shown that, depending on the direction of rotation, the Coriolis force can enhance or slow down the processes of interaction and mixing of the distributed vortices. At large radii the vortex structure does not stabilize.

The work is devoted to the development of a computational algorithm of the Cartesian grid method for studying the interaction of a shock wave with moving bodies with a piecewise linear boundary. The interest in such problems is connected with direct numerical simulation of two-phase media flows. The effect of the particle shape can be important in the problem of dust layer dispersion behind a passing shock wave. Experimental data on the coefficient of aerodynamic drag of non-spherical particles are practically absent.

Mathematical model is based on the two-dimensional Euler equations, which are solved in a region with varying boundaries. The defining system of equations is integrated using an explicit scheme and the Cartesian grid method. The computational algorithm at the time integration step includes: determining the step value, calculating the dynamics of the body movement (determining the force and moment acting on the body; determining the linear and angular velocities of the body; calculating the new coordinates of the body), calculating the gas parameters. At each time step, all cells are divided into two classes – external (inside the body or intersected by its boundaries) and internal (completely filled with gas). The solution of the Euler equations is constructed only in the internal ones. The main difficulty is the calculation of the numerical flux through the edges common to the internal and external cells intersected by the moving boundaries of the bodies. To calculate this flux, we use a two-wave approximation for solving the Riemann problem and the Steger-Warming scheme. A detailed description of the numerical algorithm is presented.

The efficiency of the algorithm is demonstrated on the problem of lifting a cylinder with a base in the form of a circle, ellipse and rectangle behind a passing shock wave. A circular cylinder test was considered in many papers devoted to the immersed boundary methods development. A qualitative and quantitative analysis of the trajectory of the cylinder center mass is carried out on the basis of comparison with the results of simulations presented in eight other works. For a cylinder with a base in the form of an ellipse and a rectangle, a satisfactory agreement was obtained on the dynamics of its movement and rotation in comparison with the available few literary sources. Grid convergence of the results is investigated for the rectangle. It is shown that the relative error of mass conservation law fulfillment decreases with a linear rate.

The gradual incorporation of automated vehicles into the global transport networks leads to the need to develop tools to assess the impact of this process on various aspects of traffic. This implies a more organized movement of automated vehicles which can form uniformly moving platoons. The influence of the formation and movement of these platoons on the dynamics of traffic flow is of great interest. The currently most developed traffic flow models are based on the cellular automaton approach. They are mainly developed in the direction of increasing accuracy. This inevitably leads to the complication of models, which in their modern form have significantly moved away from the original philosophy of cellular automata, which implies simplicity and schematicity of models at the level of evolution rules, leading, however, to a complex organized behavior of the system. In the present paper, a simulation model of connected automated vehicles platoon dynamics in a heterogeneous transport system is proposed, consisting of two types of agents (vehicles): human-driven and automated. The description of the temporal evolution of the system is based on modified rules 184 and 240 for elementary cellular automata. Human-driven vehicles move according to rule 184 with the addition of accidental braking, the probability of which depends on the distance to the vehicle in front. For automated vehicles, a combination of rules is used depending on the type of nearest neighbors, regardless of the distance to them, which brings non-local interaction to the model. At the same time, it is considered that a group of sequentially moving connected automated vehicles can form an organized platoon. The influence of the ratio of types of vehicles in the system on the characteristics of the traffic flow during free movement on a circular one-lane and two-lane roads, as well as in the presence of a traffic light, is studied. The simulation results show that the effect of platoon formation is significant for a freeway traffic flow; the presence of a traffic light reduces the positive effect by about half. The movement of platoons of connected automated vehicles on two-lane roads with the possibility of lane changing was also studied. It is shown that considering the types of neighboring vehicles (automated or human-driven) when changing lanes for automated vehicles has a positive effect on the characteristics of the traffic flow.

In this paper, the structure of a shock wave in a binary gas mixture is studied on the basis of direct solution of the Boltzmann kinetic equation. The conservative projection method is used to evaluate the collision integral in the kinetic equation. The applied evaluation formulas and numerical methods are described in detail. The model of hard spheres is used as an interaction potential of molecules. Numerical simulation is performed using the developed simulation environment software, which makes it possible to study both steady and non-steady flows of gas mixtures in various flow regimes and for an arbitrary geometry of the problem. Modeling is performed on a cluster architecture. Due to the use of code parallelization technologies, a significant acceleration of computations is achieved. With a fixed accuracy controlled by the simulation parameters, the distributions of macroscopic characteristics of the mixture components through the shock wave front were obtained. Computations were conducted for various ratios of molecular masses and Mach numbers. The total accuracy of at least 1% for the local values of molecular density and temperature and 3% for the shock front width was achieved. The obtained results were compared with existing computation data. The results presented in this paper are of theoretical significance, and can serve as a test computation, since they are obtained using the exact Boltzmann equation.

The study of the development of π-periodic perturbations of a converging spherical shock wave leading to cumulation limitation is performed. The study is based on 3D hydrodynamic calculations with the Carnahan – Starling equation of state for hard sphere fluid. The method of solving the Euler equations on moving (compressing) grids allows one to trace the evolution of the converging shock wave front with high accuracy in a wide range of its radius. The compression rate of the computational grid is adapted to the motion of the shock wave front, while the motion of the boundaries of the computational domain satisfy the condition of its supersonic velocity relative to the medium. This leads to the fact that the solution is determined only by the initial data at the grid compression stage. The second order TVD scheme is used to reconstruct the vector of conservative variables at the boundaries of the computational cells in combination with the Rusanov scheme for calculating the numerical vector of flows. The choice is due to a strong tendency for the manifestation of carbuncle-type numerical instability in the calculations, which is known for other classes of flows. In the three-dimensional case of the observed force, the carbuncle effect was obtained for the first time, which is explained by the specific nature of the flow: the concavity of the shock wave front in the direction of motion, the unlimited (in the symmetric case) growth of the Mach number, and the stationarity of the front on the computational grid. The applied numerical method made it possible to study the detailed flow pattern on the scale of cumulation termination, which is impossible within the framework of the Whitham method of geometric shock wave dynamics, which was previously used to calculate converging shock waves. The study showed that the limitation of cumulation is associated with the transition from the Mach interaction of converging shock wave segments to a regular one due to the progressive increase in the ratio of the azimuthal velocity at the shock wave front to the radial velocity with a decrease in its radius. It was found that this ratio is represented as a product of a limited oscillating function of the radius and a power function of the radius with an exponent depending on the initial packing density in the hard sphere model. It is shown that increasing the packing density parameter in the hard sphere model leads to a significant increase in the pressures achieved in a shock wave with broken symmetry. For the first time in the calculation, it is shown that at the scale of cumulation termination, the flow is accompanied by the formation of high-energy vortices, which involve the substance that has undergone the greatest shock-wave compression. Influencing heat and mass transfer in the region of greatest compression, this circumstance is important for current practical applications of converging shock waves for the purpose of initiating reactions (detonation, phase transitions, controlled thermonuclear fusion).

In this paper, we consider the interaction of a kink of the $\varphi^4$ equation with two identical extended impurities. An extended impurity is described using a rectangular function. The case of an attractive impurity is analyzed. Using analytical methods, we consider the case of small amplitudes of localized waves, when it is possible to linearize the equations of motion. For the numerical solution, the method of lines for partial differential equations was used. To find the oscillation frequencies of waves localized on impurities, the discrete Fourier transform is used. The kink was launched in the direction of the impurities with different initial velocities. The distance between the two impurities was also varied. It is shown that when a kink interacts with impurities, long-lived localized breather-type waves are excited on them. Their structure and coupled dynamics are investigated. It is determined how, by changing the parameters of the impurities and the distance between them, it is possible to control the type and dynamic parameters of the coupled oscillations of the waves localized on the impurities. Possible solutions in the form of in-phase, antiphase oscillations, in the form of beats are found. The oscillations of localized waves occur with the emission of small-amplitude waves. The spectrum of these emissions consists of two frequencies. The first is approximately equal to $\sqrt{2}$, which corresponds to the frequency value for the wobbling breather tail of the $\varphi^4$ equation. The second is approximately equal to the doubled frequency of impurity mode oscillations. The presence of two possible frequencies for coupled localized oscillations is found both analytically and numerically. It is shown that the frequencies strongly depend on the distance between impurities. With increasing distance between impurities, the frequencies merge into one — frequency obtained for the case of a single impurity. The dependences of the frequencies on the distance between impurities found numerically and analytically coincide well for large distances, when the interaction between impurities is weak, and begin to differ noticeably at small distances, when the interaction between impurities is strong. The analytical value of the obtained frequencies is always greater than the numerical ones. It is shown that the dependence of the amplitude of localized waves on the initial kink velocity has several minima and maxima.

The mathematical model for aeroelastic oscillations of a narrow channel wall with a nonlinear-elastic suspension and interacting with a pulsating viscous gas layer is proposed. Within the framework of this model, the aeroelastic response of the channel wall and its phase response were determined and investigated. The authors simultaneously studied the influence of the nonlinear stiffness elastic suspension of the wall, compressibility and dissipative properties of gas, as well as the inertia of its motion on the wall oscillations. The model was elaborated based on the formulation and solution of the initial boundary-value plane problem of mathematical physics. The problem governing equations include the equations of dynamics for barotropic viscous gas, equation of dynamics for the rigid wall as the spring-mass nonlinear oscillator. Using the perturbation method, the asymptotic analysis of the problem was carried out. The solution of the equations of dynamics for the thin layer of viscous gas was obtained by the iteration method. As a result, the law of gas pressure distribution in the channel was determined and the initial problem of aeroelasticity was reduced to the study of the generalized Duffing equation. Its solution was realized by the harmonic balance method, which allowed us to determine the aeroelastic and phase responses of the channel wall in the form of implicit functions. The numerical study of these responses was carried out to evaluate the influence for inertia of gas motion and its compressibility, as well as a comparison of the results obtained with the special cases of creeping motion of viscous gas and incompressible viscous fluid. The results of this study have shown the importance of simultaneous consideration of compressibility and inertia of viscous gas motion when modeling aeroelastic oscillations of the considered channel wall.