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RUSSIAN FEDERAL NUCLEAR CENTER 
ALLRUSSIAN RESEARCH INSTITUTE OF EXPERIMENTAL PHYSICS 

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Issue N^{o} 4, 2021  A DIFFERENCE SCHEME TO SOLVE 3D EQUATION OF RADIATION THERMAL CONDUCTIVITY ON HEXAHEDRAL CELLS WITH SINGLECURVED FACETS
A. M. Stenin VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 4. P. 323.
A difference scheme to solve 3D equation of radiation thermal conductivity on structured mesh is presented; it consists of random hexahedral cells with singlecurved facets. Keeping to the presumption of initially accepted agreement on the ruled character of facets of a cell, the formulas for the volume of the cell and normal vector in the center of its facets are produced. The definition of the facet area through which the cell of the mesh has heat exchange with neighboring cells is provided; it is based on the concept of vector area, or oriented area, of a ruled surface in 3D. An algorithm for computing heat flows at the facets of cells of the mesh is described. A linearized system of difference equations for iteration solution of nonlinear equations of thermal balance is produced. Key words: 3D equation of radiation thermal conductivity, hexahedral cells of the mesh, singlecurved facets, a normal to a singlecurved facet, area of the facet, heat flows at the facets of cells.
 A NUMERICAL METHOD TO SOLVE 3D PROBLEMS OF HIGHVELOCITY GAS JET INTERACTION WITH ELASTOPLASTIC BARRIERS M. Kh. Abuzyarov, E. G. Glazova, A. V. Kochetkov, S. V. Krylov VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 4. P. 2440.
A numerical method to solve 3D problems of highvelocity gas jet interaction with elastoplastic barriers is described. The method is based on the unified modified difference method by S. K. Godunov to compute both energy release at detonation and gas motion and dynamic deformation of elastoplastic barriers. The method realizes EulerianLangrangian approach with explicit identification of moving contact surfaces using multimesh algorithms. Results of numerical research on the process of highvelocity jet formation in Пshaped charges with a small elongation and its interaction with elastoplastic steel barrier are provided. Thee numerical results agree well with the known experimental data. Key words: numerical simulation, Godunov scheme, improved accuracy, multimesh approach, 3D problem, explosion, detonation, highvelocity jets, elastoplastic barrier, interaction, comparison with experiment.
 ON ONE COUNT EFFECT OF UNPHYSICAL WARMINGUP OF THE MATERIAL WHEN MODELLING THERMAL RADIATION TRANSFER A. I. Bochkov, V. Yu. Rezchikov, V. V. Suchkova VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 4. P. 4149.
Many different mesh and numerical effects can appear when solving thermal radiation transfer problems numerically. They are not related to the physics of the processes being modeled and impede correct interpretation of the computation results. The paper considers a computing effect that the authors have not found in scientific publications. This computing effect of not physical (it is not the solution of a system of integrodifferential equations for radiation transfer and does not fit the frames of description of a physical process) accelerated warmingup of the material is caused by the combination of two factors: strong anisotropy of the incoming radiation flow in the directions of the particles travel and specific features of some difference schemes of the second order of accuracy. We describe 1D model problem where the computing effect reveals itself demonstrably; the causes why it occurs are found. Several modifications of computational scheme are offered to fight with the revealed effect. They allow getting rid of artefacts in numerical solution almost completely. Key words: radiation transfer, numerical simulation, computing effect.
 "LOGOSWAVE" ("LOGOSVOLNA") METHOD TO COMPUTE 2D GASDYNAMIC PROBLEMS WITH ACCOUNT FOR THERMAL CONDUCTIVITY ON MOVING UNSTRUCTURED GRIDS E. A. Veselova, Yu. N. Deryugin, D. K. Zelenskiy VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 4. P. 5066.
A method for parallel computations on 2D gasdynamic problems with account for thermal conductivity on geometrically adaptive unstructured grids is described. Geometrical adaptation is related to identification of basic specific features in the solution such as shock waves and contact discontinuities. The motion velocity at discontinuities and parameters at discontinuities are found from solution of Riemann problem. Displacement of internal nodes of the mesh is determined with boundarynodedisplacement interpolation method. The numerical method is based on split method, on the solution of Eulerian equations explicitly on moving mesh and on the solution of heatconductivity equation implicitly on immobile mesh. Difference equations are produced with discretization of initial equations in integral form with quadrature formulas of rectangles. When solving Eulerian equations, numerical convective flows are found on the basis of Riemann problem solution. Prediscontinuity parameters of the flow are found using either linear or quadratic reconstruction of the solution to improve simulation accuracy. Algorithm of additional turn of the velocity vector of prediscontinuity parameters of the flow is used in problems with spherical symmetry to decrease nonmonotonic character of numerical solution. Heat flows at the facets are approximated by the upper time layer with central differences. Iteration Newton method is used to solve implicit equations. The system of linear algebraic equations resulted from approximation is solved using parallel solvers from PMLP library when temperature increment is regarded. Possibilities of the method are illustrated on a number of benchmarks. Key words: gas dynamics, heat conductivity, splitting scheme, difference scheme, moving meshes, computation parallelization, testing computations.
 DEFORMABILITY MODEL OF SUBSIDENCE OF SOIL FOUNDATION OF THE RAILWAY TRACK WITH THROUGHPUT OF A LARGE NUMBER OF TRAINS OF DIFFERENT CAPACITY A. V. Anisin, I. M. Anisina, S. S. Nadyezhin, V. O. Pevzner, V. P. Solovyev, V. V. Tretyakov, I. V. Tretyakov VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 4. P. 6775.
A model of standard linear solid is studied as applied to the description of roadbed deformation when loaded with trains. Earlier it was shown that in a particular case of cyclic sinusoidal loads on the rail the model of standard linear solid describes well both the deformation of the ground under the load and relaxation of the ground after unloading. The paper analyzes experimental data obtained by specialists from VNIIZhT at KovdorPinozero railway haul. The model of standard linear solid is extended to the case of a random form of loading function. Applicability of this model is proved in case of the load nonuniform in depth. Simple evaluating formulas are suggested for ground deformation when one or several trains pass there as a function of the number and the mass of passing train sets of cars, and for the railway track relaxation after the trains have passed. The produced method makes it possible to predict the growth of dynamic retreats in a vertical plane when long train sets of cars pass, which is necessary for planning lining works. Key words: a railway track, subballast foundation, track subsidence, a model of a standard linear solid, viscosity.
 VECTOR OPTIMIZATION OF "LOGOSAEROHYDRO" PROGRAM MODULE WITH TOOLS OF IAL LIBRARIES I. P. Ryzhachkin VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 4. P. 7682.
Method of finite volumes on unstructured meshes is used for discretization of NavierStocks equation within "LogosAeroHydro" module of "Logos" software package. It results into solution of SLAE with a smallblock matrix to find several interdependent indeterminates in each cell of the mesh. During parallelization of computing algorithms to solve SLAE, SSE2SSE4.2 (Intel®LegacySSE), Intel®AVX, Intel®AVX512 techniques are used. IAL libraries developed in RFNCVNIIEF are described; the issues related to creation of optimized libraries are discussed. Implementation results for IAL libraries for vector optimization of "LogosAeroHydro" module are provided. Key words: vector optimization, SSE, AVX, "Logos" software package.
 USE OF FULLSCALE EXPERIMENT DATA TO BUILD A SEMIEMPIRICAL MODEL WHEN THE ICEBREAKER MOVES WITH INCURSIONS E. M. Gramuzov, V. A. Zuev, N. V. Kalinina, A. A. Kurkin VANT. Ser.: Mat. Mod. Fiz. Proc. 2021. No 4. P. 8390.
Attention to arctic maritime traffic is growing. The most universal means to fight ice hurdles is icebreaking fleet. Heavy ice conditions are frequent at operation of icebreakers when it is impossible to move continuously. So, icebreakers have to move in incursions. This type of motion is currently studied most poorly. Theoretical and experimental research and generalization of the operational experience for icebreakers play an important role in the development of the study. When the icebreaker operates in incursions (irruptions), it is a cyclic movement, and each cycle consists of stages. Nonlinear differential equations of the second order are presented; they describe all motion stages when the icebreaker operates with incursions. Their earlier produced analytical solutions are provided. Methods to use data produced in the course of fullscale experiments and accumulated earlier are offered to obtain coefficients of semiempirical models of cyclic motion in incursions. We show the implementation efficiency of the proposed methods. Key words: ice, movement of the icebreaker in incursions, mathematical simulation, semiempirical model, data of fullscale experiments.
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