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RUSSIAN FEDERAL NUCLEAR CENTER 
ALLRUSSIAN RESEARCH INSTITUTE OF EXPERIMENTAL PHYSICS 

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Issue N^{o} 4, 2006  ANGLEADAPTIVE METHOD FOR SOLVING 2D TRANSPORT EQUATION USING NONORTHOGONAL SPACE GRID
R. M. Shagaliev, A. V. Alexeyev, A. I. Bochkov, A. V. Gichuk, V. V. Evdokimov, A. A. Nuzhdin, T. V. Shemyakina VANT. Ser.: Mat. Mod. Fiz. Proc. 2006. No 4. P. 314.
Angleadaptive method for solving 2D radiation transport equation and radiationmatter interaction equation is considered. The transport equation is written in a cylindrical frame of reference using non orthogonal space grid. The numerical solution for 2D transport equation is based on the discrete ordinate method using the approximation in DSn schemes. The main objective of the adaptive method is removing the problem of ray effects. The described here are the problems of the criterion development for automatic determination of the adaptivity level in terms of the grid solution analysis, the reinterpolation of the grid values, and the organization of sweep method to solve the transport equation using the adaptivity.
 ROMB IMPLICIT FINITEVOLUME METHOD TO SOLVE 2D HYDRODYNAMIC EQUATIONS ON UNREGULAR TRI AND QUADRANGULAR GRIDS E. M. Vaziev, A. D. Gadzhiev, S. Yu. Kuzmin VANT. Ser.: Mat. Mod. Fiz. Proc. 2006. No 4. P. 1528.
We studied the implicit finitevolume method ROMB to solve 2D hydrodynamic equations on unregular tri and quadrangular grids. The uniform grid can be used as a particular case. The investigated method is based on LagrangianEuler medium description with the implemented method of concentrations, grid remapping and recalculation of values using fractional design style. The state vector is stored at cell centers and the coordinates — at cell nodes. The difference scheme is based on the integral laws of conservation of mass, momentum and energy for cell; this makes the scheme conservative. The way of artificial dissipation introduction is one of the scheme characteristics. It is introduced in the equation of motion, as well as in the equation for pressure by the dissipation quantities , related to , as grad P. ROMB method in conjunction with other scheme characteristics demonstrates better monotony as compared with traditional schemes with diversity model, which appeared to be especially important for unregular grids.
 TIM2D TECHNIQUE FOR SOLVING CONTINUUM MECHANICS PROBLEMS USING UNREGULAR POLYGONAL GRIDS WITH A RANDOM NUMBER OF CONNECTIONS AT NODES S. S. Sokolov, A. A. Voropinov, I. G. Novikov, A. I. Panov, I. V. Sobolev, A. A. Pushkarev VANT. Ser.: Mat. Mod. Fiz. Proc. 2006. No 4. P. 2944.
Studied here is TIM2D technique intended for 2D continuum mechanics problems using unregular polygonal Lagrangian grids. The technique allows computations using grids with a random number of connections at nodes (cells and edges adjacent to the node). The single computational algorithm is used for all types of grids. The calculation of initial data and solution of continuum mechanics equations are carried out in cylindrical or 2D Cartesian coordinate systems. The technique is intended for gas dynamics, unsteady elastoplasticity, magnetic hydrodynamics and heat conduction problems. The explicit finitedifference schemes are used to solve gas dynamics, elastoplasticity and magnetic hydrodynamics problems; the kinematic quantities are stored in counting grid nodes, thermodynamic quantities — in cell centers. The implicit finitedifference scheme is used to solve heat conduction problems.
 AN EFFICIENT METHOD TO SOLVE POORLY CONDITIONED 3D ELLIPTICAL EQUATIONS FOR THE CASE OF STEPHAN MELT CRYSTALLIZATION PROBLEM V. R Ginkin, O. M. Ginkina, S. M. Ganina, K. G. Chernov, S. V. Pupko, A. B. Kagalenko VANT. Ser.: Mat. Mod. Fiz. Proc. 2006. No 4. P. 4557.
The new efficient technique called CGPIF has been developed to solve 3D finitedifference elliptical equations with symmetric poorly conditioned matrices of coefficients. Twolevel iterative technique has been proposed to solve asymmetric problems using CGPIF technique in inner iterations. The convergence rate of the proposed technique has been studied using Dirichlet and Neumann test problems for diffusion and diffusionconvection equations. The new mathematical model is proposed to solve Stephan problem in enthalpy variables along with convection heatandmass transfer in liquid phase. The results of experimental benchmarks of melt crystallization are given which demonstrate model adequacy.
 NONUNIFORM GRID EFFECTS ON SHOCK PARAMETERS IN LAGRANGIAN DIFFERENCE SCHEMES WITH ARTIFICIAL VISCOSITY. ID ASYMPTOTIC ANALYSIS Yu. A. Bondarenko, O. A. Moskalev VANT. Ser.: Mat. Mod. Fiz. Proc. 2006. No 4. P. 5865.
It is well known that in the calculation that use Lagrangian difference schemes with artificial viscosity using the nonuniform grid, the propagated shock wave is perturbed, here the errors in shock parameters are proportionate to the quantity . It is demonstrated that the effect is the property of differential gas dynamics equations with viscosity, inherited by difference schemes. Approximate solutions to describe the interaction of the steadystate shock structure with the viscosity factor gradient have been set up by the asymptotic method, assuming that the shock is propagating in a resting gas of constant density and pressure and that there are no perturbations catching up with the shock front. The variable viscosity factor is constant in time, and it is assumed to be small compared to the characteristic dimensions determined by the spatial derivatives of the viscosity factor. The dominant term perturbations of shock parameters are directly proportionate to viscosity factor gradient. The results of numerical tests using Lagrangian difference schemes with artificial viscosity agree with the asymptotic form obtained.
 GEOMETRICAL APPROACH TO THE SIMULATION OF PARTICLE FLIGHT DIRECTION. N. N. CHENTSOV FORMULAE AND THEIR GENERALIZATION FOR POLARIZED RADIATION COMPUTATIONS A. N. Subbotin VANT. Ser.: Mat. Mod. Fiz. Proc. 2006. No 4. P. 6679.
The method of polarized radiation transport description is proposed; it consists in the kinetic scalar equation with extended phase space of particles, and a specific form of collision term. The new equation is an ordinary Kolmogorov equation for transient probabilities of Markovian process. To substantiate the validity of Monte Carlo method for polarized radiation computation it is proved that vector Chandrasekhar equation for 4D Stokes vectorparameter of polarized radiation is the system of equations for fourdimensional functionals of the new scalar equation solution. Thus, Monte Carlo solution of vector equation is reduced to functional estimation using the path of Markovian process. We have considered the geometrical approach proposed by N. N. Chentsov to the derivation of failureproof algorithms of particle scattering simulation based on the use of discontinuous functions for arraying of the transfer from one coordinate system to another. The choice of the coordinate system for the count of Stokes parameters of polarized radiation agrees with construction of moving coordinate system for simulation of new flight direction during the collision event, which allows economical algorithms of simulation of polarized radiation transport paths.
 ON POSSIBLE JETLIKE BEHAVIOR OF THE INITIALLY PERTURBED FREE BOUNDARY OF CONDENSED MATTER WITH A SHOCK WAVE ARRIVAL AT THIS BOUNDARY  A. D. Kovaleva, S. S. Kosarim, G. B. Krasovsky, S. E. Kuratov, A. E. Levushov, E. E. Meshkov, I. Yu. Myshkina, O. V. Olkhov, A. A. Polovnikov, E. A. Polovnikov 
VANT. Ser.: Mat. Mod. Fiz. Proc. 2006. No 4. P. 8088.
Theoretical, computational and experimental results show that the surface becomes unstable with a shock wave arrival at a curved surface of condensed matter, with jetlike behavior of instability. Analyzed here are the main regularities evaluating the characteristics of the jetlike instability. Theoretically studied is the impact of the stabilizing factors on the evolution of jet formation processes on the free curved surface.
 V’UGA COMPUTATION TECHNIQUE FOR 3D PROBLEMS OF SLOW MHDWAVES PROPAGATION K. G. Gainullin, V. A. Zhmaylo, Yu. F. Kir’yanov, E. A. Pozdyaeva VANT. Ser.: Mat. Mod. Fiz. Proc. 2006. No 4. P. 8999.
The described here V’UGA technique is intended for problems of slow MHDwaves propagation in ionosphere, stratosphere and soil. The comparison of numerical computation results is carried out along with analytical solutions of test problems.
 CLOSURE METHOD OF LAGRANGIAN GAS DYNAMICS IN MIXED CELLS BASED ON THE EQUALITY OF COMPONENT VELOCITIES E. A. Goncharov, V. Yu. Kolobyanin, Yu. V. Yanilkin VANT. Ser.: Mat. Mod. Fiz. Proc. 2006. No 4. P. 100105.
The described here is the method of distribution of component divergence of mixed cell velocity based on the equalizing of component mass velocity after perturbation propagation in heterogeneous mixture. Test problems are used to study the accuracy of computational results using the method supplemented with the algorithm of component pressure equalizing.
 LAGRANGIAN GRID SURFACES SMOOTHING A. V. Yalozo VANT. Ser.: Mat. Mod. Fiz. Proc. 2006. No 4. P. 106114.
In addition to the algorithms of velocity field developed, this work considers the technique which allows smoothing the surfaces of polygonal and polyhedral Lagrangian grids. An attempt has been made to work out the mechanism which allows supporting the grid smoothness without changing the velocity of node movement. The surface smoothing is carried out by node shifting in some optimum position. The new position of smoothed node on the surface is chosen from the condition of volume conservation to prevent mass loss. Since the surface nodes are topologically connected with nodes inside the grid, internal node smoothing is realized.
 SOFTWARE TOOLS — SIMULATORS OF RESEARCH REACTOR CORES FOR SUPPORT OF REACTOR OPERATION Yu. E. Vaneev, N. Yu. Marikhin, R. R. Kudoyarov, A. P. Maikov VANT. Ser.: Mat. Mod. Fiz. Proc. 2006. No 4. P. 115121.
The results of software development are given for support of research reactors. Described here are the structures of this software, design features of objectoriented components and verification results using experimental data obtained in SM reactor. Demonstrated here are the possibilities of the reactor cycle duration prediction.
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