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A 2D numerical model of the immersion of a cold oceanic plate into the thickness of the Earth’s upper mantle has been developed, where the stage of the initial immersion of the plate is preceded by the establishment of a regime of thermogravitational convection of the mantle substance. The model approximation of the mantle is a two-dimensional image of an incompressible Newtonian quasi-liquid in a Cartesian coordinate system, where, due to the high viscosity of the medium, the equations of mantle convection are accepted in the Stokes approximation. It is assumed that seawater that has leaked here enters the first horizons of the mantle together with the plate. With depth, the increase in pressure and temperature leads to certain losses of its light fractions and fluids, losses of water and gases of water-containing minerals of the plate, restructuring of their crystal lattice and, as a consequence, phase transformations. These losses cause an increase in the plate density and an uneven distribution of stresses along the plate (the initial sections of the plate are denser), which subsequently, together with the effect of mantle currents on the plate, causes its fragmentation. The state of mantle convection is considered when the plate and its individual fragments have descended to the bottom of the upper mantle. Computational schemes for solving the model equations have been developed. Mantle convection calculations are performed in terms of the Stokes approximation for vorticity and the stream function, and SPH is used to calculate the state and subsidence of the plate. A number of computational experiments have been performed. It is shown that fragmentation of the plate occurs due to the effect of mantle convection on the plate and the development of inhomogeneous stress fields along the plate. Following the equations of the model, the time of the final stage of subduction is estimated, i.e. the time of the entire oceanic plate reaching the bottom of the upper mantle. In geodynamics, this process is determined by the collision of plates that immediately follows subduction and is usually considered as the final stage of the Wilson cycle (i. e., the cycle of development of folded belts).

The article contains results of modeling a meteor entering dense atmosphere layers using Galerkin’s method and smoother particle hydrodynamics. Numerical simulations were run using experimental data gathered for the Chelyabinsk meteor while varying the meteor material characteristics and its orientation when entering the atmosphere.

The problem discrete statement by the smoothed particle hydrodynamics method (SPH) include a discretization constants parameters set. Of them particular note is the model sound speed $c_0$, which relates the SPH-particle instantaneous density to the resulting pressure through the equation of state.

The paper describes an approach to the exact determination of the model sound speed required value. It is on the analysis based, how SPH-particle density changes with their relative shift. An example of the continuous medium motion taken the plane shear flow problem; the analysis object is the relative compaction function $\varepsilon_\rho$ in the SPH-particle. For various smoothing kernels was research the functions of $\varepsilon_\rho$, that allowed the pulsating nature of the pressures occurrence in particles to establish. Also the neighbors uniform distribution in the smoothing domain was determined, at which shaping the maximum of compaction in the particle.

Through comparison the function $\varepsilon_\rho$ with the SPH-approximation of motion equation is defined associate the discretization parameter $c_0$ with the smoothing kernel shape and other problem parameters. As a result, an equation is formulated that the necessary and sufficient model sound speed value provides finding. For such equation the expressions of root $c_0$ are given for three different smoothing kernels, that simplified from polynomials to numerical coefficients for the plane shear flow problem parameters.

The two significant geometric parameters are proposed that affect the physical quantities interpolation in the smoothed particle hydrodynamics method (SPH). They are: the smoothing coefficient which the particle size and the smoothing radius are connecting and the volume coefficient which determine correctly the particle mass for a given particles distribution in the medium.

In paper proposes a technique for these parameters influence assessing on the SPH method interpolations accuracy when the hydrostatic problem solving. The analytical functions of the relative error for the density and pressure gradient in the medium are introduced for the accuracy estimate. The relative error functions are dependent on the smoothing factor and the volume factor. Designating a specific interpolation form in SPH method allows the differential form of the relative error functions to the algebraic polynomial form converting. The root of this polynomial gives the smoothing coefficient values that provide the minimum interpolation error for an assigned volume coefficient.

In this work, the derivation and analysis of density and pressure gradient relative errors functions on a sample of popular nuclei with different smoothing radius was carried out. There is no common the smoothing coefficient value for all the considered kernels that provides the minimum error for both SPH interpolations. The nuclei representatives with different smoothing radius are identified which make it possible the smallest errors of SPH interpolations to provide when the hydrostatic problem solving. As well, certain kernels with different smoothing radius was determined which correct interpolation do not allow provide when the hydrostatic problem solving by the SPH method.

The paper analyzes the methods of studying the process of interaction of soil environments with the tillage tools of soil processing machines. The mathematical methods of numerical modeling are considered in detail, which make it possible to overcome the disadvantages of analytical and empirical approaches. A classification and overview of the possibilities the continuous (FEM — finite element method, CFD — computational fluid dynamics) and discrete (DEM — discrete element method, SPH — hydrodynamics of smoothed particles) numerical methods is presented. Based on the discrete element method, a mathematical model has been developed that represents the soil in the form of a set of interacting small spherical elements. The working surfaces of the tillage tool are presented in the framework of the finite element approximation in the form of a combination of many elementary triangles. The model calculates the movement of soil elements under the action of contact forces of soil elements with each other and with the working surfaces of the tillage tool (elastic forces, dry and viscous friction forces). This makes it possible to assess the influence of the geometric parameters of the tillage tools, technological parameters of the process and soil parameters on the geometric indicators of soil displacement, indicators of the self-installation of tools, power loads, quality indicators of loosening and spatial distribution of indicators. A total of 22 indicators were investigated (or the distribution of the indicator in space). This makes it possible to reproduce changes in the state of the system of elements of the soil (soil cultivation process) and determine the total mechanical effect of the elements on the moving tillage tools of the implement. A demonstration of the capabilities of the mathematical model is given by the example of a study of soil cultivation with a disk cultivator battery. In the computer experiment, a virtual soil channel of 5×1.4 m in size and a 3D model of a disk cultivator battery were used. The radius of the soil particles was taken to be 18 mm, the speed of the tillage tool was 1 m/s, the total simulation time was 5 s. The processing depth was 10 cm at angles of attack of 10, 15, 20, 25 and 30°. The verification of the reliability of the simulation results was carried out on a laboratory stand for volumetric dynamometry by examining a full-scale sample, made in full accordance with the investigated 3D-model. The control was carried out according to three components of the traction resistance vector: $F_x$, $F_y$ and $F_z$. Comparison of the data obtained experimentally with the simulation data showed that the discrepancy is not more than 22.2%, while in all cases the maximum discrepancy was observed at angles of attack of the disk battery of 30°. Good consistency of data on three key power parameters confirms the reliability of the whole complex of studied indicators.