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RUSSIAN FEDERAL NUCLEAR CENTER 
ALLRUSSIAN RESEARCH INSTITUTE OF EXPERIMENTAL PHYSICS 

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Issue N^{o} 2, 2022  METHOD OF ASSOCIATED INVARIANT SUBSPACES TO SOLVE NEUTRON DISTRIBUTION PROBLEMS IN LOOSELY COUPLED SYSTEMS
E. A. Biberdorf, E. F. Mitenkova, T. V. Semyenova, E. V. Solovyeva VANT. Ser.: Mat. Mod. Fiz. Proc. 2022. No 2. P. 316.
When simulating the neutron transport in complex heterogeneous media, the most accurate distribution of neutrons is achieved with statistical Monte Carlo algorithms.However, in loosely coupled systems the algorithmic features of the power iteration method used in calculations may lead to significant errors and even incorrect results. The method of associated invariant subspaces based on the fission matrix generated using the TDMCC code is proposed to calculate the neutron distribution. It is based on the matrix spectrum dichotomy method developed at the Sobolev Institute of Mathematics of the Siberian Branch of RAS. A new formulation of the asymmetric spectral problem is considered for solving application problems. Key words: loosely coupled system, fission matrix, spectral problem, matrix spectrum dichotomy.
 2D NUMERICAL SIMULATION OF MIXING EXPERIMENT FOR A THREELAYER GAS SYSTEM WITH ADHESION TO SHOCK TUBE’S WALLS Yu. V. Yanilkin VANT. Ser.: Mat. Mod. Fiz. Proc. 2022. No 2. P. 1726.
The paper presents results of the 2D numerical simulation for a flow generated in a tube with a fixed crosssection by a shock wave propagating through a threelayer gas system. Both the direct numerical simulation (the solution of 2D Eulerian, or NavierStokes equations without any models of turbulence) and simulation with the k, ε model of turbulence were performed. Computations were carried out with and without consideration of a boundary layer on side walls of the shock tube. At initial time, gases were at rest under atmospheric pressure and separated by two thin films. The first interface was inclined at an angle of 45º to the shock wave front, the second one was parallel to the front. The central layer of this system was filled out with xenon, air was in front of the first and second interfaces. The shock wave was generated at one end of the tube and moved towards the first interface.Comparison was performed between results of the numerical simulation for different formulations of the problem, as well as comparison of the calculated results with experimental data. Key words: model experiments, a threelayer gas system, numerical simulation, KelvinHelmholtz instability, RichtmyerMeshkov instability.
 A MATHEMATICAL DESCRIPTION USING SUCHKHOV’S SOLUTION OF THE TWO METHODS OF GASDYNAMICALLY IMPACTING A TARGET E. I. Ponkin VANT. Ser.: Mat. Mod. Fiz. Proc. 2022. No 2. P. 2739.
It is proposed to use a Suchkov’s double wave as a partial solution to the system of equations describing a 2D gas dynamic flow adjoining an inclined wall for the mathematical modeling of the two methods of impacting a target in controlled thermonuclear fusion experiments. It has been found that the first method of impacting the target by a moving compression impermeable piston causes the generation of a large local cumulation region in the vicinity of the point of piston abutment to the inclined wall. The more acute is the angle between the impermeable piston surface and the oblique wall at the initial time, the larger are this region and cumulation parameters. The second method of impacting the target leads to the gas compression in Suchkov’s double wave by the permeable piston with a given pressure. The line of this piston is always orthogonal to the inclined wall and, naturally, the effect of large local cumulation is not observed. For both methods, the values of gas dynamic parameters and integral characteristics of compression flows were found, as well as gas masses compressed to different densities and energy contributions from various sections of compressing pistons to the total compression process were determined. Key words: a centered wave, a Suchkov’s double wave, the piston motion law, gas dynamic parameters.
 PREEXPERIMENTAL NUMERICAL SIMULATION OF THE QUASIISENTROPIC COMPRESSIBILITY OF DEUTERIUM AND HELIUM IN REGION OF HIGH PRESSURES USING THE LEGAK CODE A. O. Blikov, M. A. Mochalov, E. V. Shuvalova, E. A. Bakulina, E. A. Pronin VANT. Ser.: Mat. Mod. Fiz. Proc. 2022. No 2. P. 4052.
The paper presents results of the numerical simulation with the LEGAK code for the deuterium and helium compression in the region of pressures up to 12 TPa and densities up to 11 g/cm^{3}, which was performed prior to the obtainment of experimental results. Problem setups for simulations correspond to the three types of experimental designs used for recording of the quasiisentropic compressibility of gases and their mockups, in which kinematic parameters of the motion of shell boundaries are recorded. Main objectives of the numerical study prior to experiments are the calculation of gas compression parameters, including those which cannot be measured in experiments, and optimization, if necessary, of experimental setup for such experiment to be efficiently conducted.Results of the preexperimental numerical simulation are compared with results of subsequent experiments. A good agreement between them allows expanding the LEGAK code application for a new class of problems and using LEGAK results to optimize the procedure of experimentally recording the quasiisentropic compressibility parameters of gases. Key words: helium, deuterium, compressibility, a twostage spherical device, 2D problem simulations, the LEGAK code.
 ON THE USE OF MACHINE LEARNING TO MAINTAIN THE SPATIAL GRID QUALITY IN SOLVING GAS DYNAMICS PROBLEMS A. V. Babanov, A. V. Voevodin, A. N. Shcherbakov VANT. Ser.: Mat. Mod. Fiz. Proc. 2022. No 2. P. 5360.
The problem of automatically correcting a difference grid while numerically solving gas dynamics problems is considered. In the EulerianLagrangian code MIMOZA, a spatial grid has an unsatisfactory quality because of distorted difference grid’s lines having the Lagrangian flag which prevents from the reconstruction of nodes on these lines upon completion of the Lagrangian stage in the numerical solution of gas dynamic equations. Such feature of nodes is used, as a rule, to select contact boundaries of materials coinciding with lines of the difference grid. If the difference grid deformation is strong, there occurs the time point during the problem solution, when it is required to cancel Lagrangian flag of nodes. This operation in the MIMOZA code is manually performed, as usual, and introduces to the numerical solution the factor of uncertainty of the Lagrangian flag cancellation time. The paper authors resolve the problem of identifying defective nodes of the difference grid having Lagrangian flag and timely selecting another type of the grid reconstruction by using the machine learning technology. A template input dataset is presented for training the neural network, which characterizes premises for the generation of defective configurations for nodes with the Lagrangian flag. Results of the verification of these new procedure algorithms using two test problems with strong deformations of Lagrangian lines are presented. Key words: artificial neural network, multilayer perceptron, maintenance of the spatial grid quality, gas dynamics.
 "LOGOS" SOFTWARE PACKAGE: QUALITATIVE IMPROVEMENT OF VOLUMETRIC CELLS BY ELIMINATING SMALL EDGES WITH THE CUTOFF METHOD DURING THE GENERATION OF MESHES D. N. Smolkina VANT. Ser.: Mat. Mod. Fiz. Proc. 2022. No 2. P. 6171.
The paper describes an approach used in the "Logos" software package to eliminate small edges in polyhedral cells with the cutoff method during the generation of unstructured meshes. Small edges are eliminated by cutting off cells of the stencil mesh using triangles of the surface mesh. A small edge is a polyhedral mesh’s edge which length is 30% less than the edge length in cells of the stencil mesh. The cut off cells are of the two types: those containing characteristic features of the model and simples cells.Approaches used to eliminate small edges for each type of cells are significantly different. For simple cells, an algorithm based on the method of marching cubes is used. To eliminate small edges in cells containing characteristic features of the model, the analysis of the set of tetrahedral volumes built by partitioning a convex cut off cell and its vicinity is performed. The way of building tetrahedrons and the convexity condition for a cell and its vicinity ensure positive volumes of all generated tetrahedrons and this is the necessary condition for elimination of small edges. The performed testing of these algorithms demonstrates that this approach allows eliminating from the mesh up to 70% of small edges and is an automatic and generalpurpose approach, because it does not depend on the class of problems in consideration. This allows using the approach to build polyhedral meshes for geometric models of an arbitrary level of complexity. Key words: "Logos" software package, an unstructured mesh, mesh quality improvement, small edges, the cutoff method.
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