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RUSSIAN FEDERAL NUCLEAR CENTER 
ALLRUSSIAN RESEARCH INSTITUTE OF EXPERIMENTAL PHYSICS 

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Issue N^{o} 3, 2023  EXACT SOLUTIONS TO THE STATIONARY SYSTEM OF RADIATION AND ENERGY TRANSPORT EQUATIONS IN MULTIDIMENSIONAL CASE
A. A. Shestakov VANT. Ser.: Mat. Mod. Fiz. Proc. 2023. No 3. P. 316.
For code testing purposes, it is desirable to select model problems having exact solutions. In spite of the advances achieved in constructing analytical solutions to the radiation transport equation these solutions are not always sufficient for different classes of the transport problems. In a stationary case, one can find the analytical solution to the radiation transport problem by expanding in von Neumann series the transport operator resolvent. The expansion in von Neumann series allows obtaining the spectralangular characteristics of the radiation field using the known temperature profiles. The paper presents the developed analytical formulas for finding the thermal radiation parameters in a multidimensional geometry. These formulas allow explicitly finding the analytical expressions for the main spectral quantities, such as intensity, density and radiation flux, using the known equilibrium intensity determined by the temperature distribution only and the given absorption and dispersion factors. The grey and spectral approximation solutions for 1D, 2D, and 3D geometries are presented. Key words: exact solutions, system of thermal radiation transport equations.
 THE HISTOGRAM DATA APPROXIMATION WITH THE CONDITIONAL MINIMIZATION METHOD FOR THE LENGTH OF THE CUBIC C^{1}CLASS SPLINE WITH THE NONNEGATIVENESS AND LOCAL MONOTONICITY PROPERTIES. PART 2 S. V. Mzhachikh, N. V. Kolobyanina, Yu. N. Lapshina VANT. Ser.: Mat. Mod. Fiz. Proc. 2023. No 3. P. 1733.
This paper is the continuation of the previous paper by the authors, which has the same name. Some issues of the theory of building a cubic spline approximating the histogram data are considered, the recommended multistage tactics of solving problems is presented, and the calculated results for some test problems demonstrating the algorithm efficiency are given. The final section of the paper discusses the benefits and specific features of the method. Key words: cubic spline, conditional minimization, locally monotone approximation, nonnegative approximation.
 COMPARISON BETWEEN THE WPH AND DISPH METHODS OF PARTICLES IN SIMULATION OF SHOCK WAVES F. A. Sapozhnikov, V. S. Rykovanov VANT. Ser.: Mat. Mod. Fiz. Proc. 2023. No 3. P. 3454.
The paper presents theoretical grounds of the smoothed particles hydrodynamics (SPH) method with consideration of its two modifications  the WPH and DISPH methods. Results of the method applicability tests are given for the shock wave simulation problems. The following problems are considered: 1D problems, such as the Sod problem, the Noh problem, and the WoodwardColella problem, as well as the 3D Sedov blast wave problem. While comparing the WPH and DISPH methods, the L_{1} norm error and astronomical computation time are taken into account. For 3D tests, the solution asymmetry is assessed. All tests with the WPH method demonstrate a less error and a 1.5 to 2 times higher computation speed, as compared to the DISPH method. So, the conclusion of the WPH method expediency for the simulation of problems with shock waves is made. Key words: smoothed particles hydrodynamics, SPH, MOLOKh code, shock waves, Sod problem, Noh problem, WoodwardColella problem, Sedov blast wave.
 THE SPECIFICS OF CALCULATING CONVECTIVE FLOWS IN AERODYNAMICS PROBLEMS USING MESHES WITH MOVING NODES A. V. Sarazov VANT. Ser.: Mat. Mod. Fiz. Proc. 2023. No 3. P. 5566.
The paper considers the 3D numerical simulation of aerodynamics (hydrodynamics) with moving boundaries. The approaches ensuring the numerical scheme conservativeness without diminution of accuracy are discussed. A formula is proposed for the velocity calculation of the moving faces of computational cells, which allows correctly calculating the face motion velocity vector in case of arbitrary distortions of the computational mesh. This formula is based on the idea of geometric conservativeness with the use of the normal reconstruction method with corrections in the direction. To demonstrate the performance of the implemented algorithms, direct unsteady problems are solved. The proposed scheme for calculating the moving face velocity demonstrates a good qualitative agreement of the calculated results with the real physics of the flow. Key words: velocity of a face, the geometrical conservativeness condition, the NavierStokes equation system, NACA0012, AGARD 445.6, computational mesh distortions.
 SYMMETRIC CONFIGURATIONS OF LASER SOURCES FOR DIRECTDRIVE TARGET. TETRAHEDRAL SYMMETRY S. A. Belkov, S. V. Bondarenko, L. V. Solnyshkova VANT. Ser.: Mat. Mod. Fiz. Proc. 2023. No 3. P. 6779.
The paper describes the method of building symmetric laser irradiation systems with a higher level of illuminating a hohlraum. Optimal configurations of laser sources having tetrahedron rotation symmetries were found for directly driving a spherical target and capabilities of these configurations were examined from the viewpoint of achievable levels of uniformly illuminating the hohlraum by laser beams. A criterion is proposed to estimate the efficiency of configurations of laser sources. The comparison has been made between the efficiency of reducing the lowest modes in the laser illumination structure for a spherical hohlarum in zonal configurations of laser sources constructed on the base of GaussLegendre quadrature and in configurations with tetrahedral symmetry. Key words: inertial confinement fusion, directdrive target, target illumination uniformity, zonal and symmetric irradiation systems, tetrahedral symmetry.
 MOMENTARY FINITE ELEMENT FOR SOLVING 3D DYNAMIC ELASTICITY AND PLASTICITY PROBLEMS D. T. Chekmarev, Abu Dawwas Yasser VANT. Ser.: Mat. Mod. Fiz. Proc. 2023. No 3. P. 8090.
A description of a new 8node finite element for solving 3D dynamic elasticity and plasticity problems is given. This 8node finite element in the form of a hexahedron has the following features: 1) stresses and their moments (three bending and one torsional moments) are constant within the element; 2) it has one integration point; 3) the element has four parameters, by setting them one can control the numerical solution convergence. The finite element construction method is based on a combination of the two ideas: the rare mesh FEM scheme with a finite element in the form of a simplex inscribed in an $n$dimensional cube is used and a mesh problem of high dimensionality is projected onto a lower dimension mesh. The implementation of the numerical solution technique to solve 3D nonstationary elasticity and plasticity problems based on a given finite element is described. The paper presents solutions for a number of test elasticity and plasticity problems and compares them with those based on the other numerical schemes. Key words: the finite element method, the hourglass instability, a 3D problem, a nonstationary elasticity problem
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