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RUSSIAN FEDERAL NUCLEAR CENTER 
ALLRUSSIAN RESEARCH INSTITUTE OF EXPERIMENTAL PHYSICS 

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Issue N^{o} 3, 2009  THE INCOMPLETE FACTORIZATION METHOD FOR ITERATIVELY SOLVING SYSTEMS OF DIFFERENCE EQUATIONS AND ADAPTATION OF THIS METHOD TO NONMMATRICES
V. P. Ginkin, K. G. Chernov, Yu. G. Bartenev, Yu. A. Bondarenko, R. M. Shagaliev, E. B. Shchanikova VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 317.
New effective preconditioners  DIF and PIF  of the stabilized method of biadjoint gradients have been developed to solve systems of 2D and 3D elliptic finitedifference equations with asymmetric poorly conditioned Mmatrices of coefficients. To solve problems with nonMmatrices, we offer a stable and effective method of eliminating positive nondiagonal elements from a preconditioned matrix with simultaneously increasing its diagonal elements by the summarized value of all the elements eliminated from a given matrix row. The rate of convergence of the methods has been tested using both Dirichlet and von Neumann test problems for diffusion and diffusion convection equations and real 9 and 27diagonal positively defined nonMmatrices. Key words: systems of linear equations, iterative methods, method of conjugate gradients, BICGSTAB method, preconditioner, incomplete factorization method, twodimensional problems, threedimensional problems, 9diagonal matrices, 27diagonal matrices, nonsymmetric matrices, nonMmatrices, paralleling.
 NUMERICAL MODEL OF FRAGMENTING THIN METAL SHELLS V. Yu. Meltsas, G. F. Portnyagina VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 1833.
The paper describes the model of fragmenting thin spherical metal shells under the impact of intense loads with generation of two fractions of fragments and with calculation of their mass spectrum. Further motion of fragments is calculated using the model of a heterogeneous medium consisting of a gas and solid particles. The paper gives examples of computations for separation of fragments in space depending on their size. Key words: thin metal shells, detonation, fracture, fragments, spectrum of fragments, gasparticle heterogeneous medium, numerical calculations.
 FUNCTION.ESTIMATES IN THE STATISTICAL MODELING METHODS D. G. Modestov, K. E. Khatuntsev VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 3444.
When solving various problems using the statistical modeling methods, it is useful to have some distributions as solutions to the problems, which are measures, or charges, in general, in the subspace of states within the probabilistic space of these problems. Such methods are intended for estimation of mathematical expectations and, therefore, estimation of distributions is not unambiguous, in general. The paper offers a criterion of approximating a charge with a system of linearindependent functions. The estimated errors of the criterion are also given, as well as some justification of using smooth functions to represent smooth distributions. Key words: statistical modeling methods, mathematical expectation, covariance matrix, dispersion, RadonNykodym derivative, charge density, function approximation.
 ON THE ONE FORM OF ARTIFICIAL VISCOSITY V. Yanilkin, O. O. Toporova, V. Yu. Kolobyanin, G. S. Firsova VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 4557.
The paper describes investigations of the impact of various von Neumann—Richtmyer viscosity forms on simulation results for a number of typical flows, for which exact solutions are available. A new 2D viscosity form developed within EGAK technique is presented. As test computations show, the viscosity proposed in the paper improves the simulation accuracy for a number of flows in comparison with the classic von Neumann—Richtmyer viscosity (both scalar and oriented in the moving shock wave direction). The specific feature of this new viscosity consists in; that it is simple and easy to implement in codes. Key words: EGAK technique, von Neumann—Richtmyer viscosity, characteristic mesh size, shock wave.
 METHODS OF MANUPULATING ENTITIES IN 3D VISUAL ENVIRONMENT V. L. Averbukh, A. V. Zyryanov VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 5869.
The paper describes approaches to the development of entity manipulation techniques for modern visual data representation systems, including those constructed on the base of virtual reality. The notion of actionoriented language is considered. The paper overviews some capabilities of threedimensional input, including 3D manipulators of various kinds and means for capture of motions. A prototype implementation of a simple and easytouse interface for entering gestures on the base of a webcamera and a simple torch is described. Such interface can be useful to arrange interactions both within the viewing systems of computers and management systems using large displays. Key words: visualization, entity manipulation, virtual reality.
 MODELING OF WATER EXCHANGE PROCESSES IN AGGREGATED NATURAL POROUS MEDIA G. S. Goncharova, M. G. Khramchenkov VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 7073.
The paper considers a mathematical model of water exchange in an aggregated natural porous environment (clays, soils and rocks) with regard to nonuniform contraction of its porous structure. The model allows predicting unfavorable environmental consequences of storing hazardous wastes of various industry fields in storage ponds, or longterm storing wastes in special storage facilities due to areas of nonuniform contraction that could result in evolving fissures in the isolating material layer (bentonitic clays, as a rule). The novelty of problem is in the use of a new equation describing the rate of contraction. Key words: unsaturated flow, aggregated porous medium, water exchange, deformation, nonuniform contraction.
 THE INCOMPLETE FACTORIZATION METHOD FOR ITERATIVELY SOLVING SYSTEMS OF DIFFERENCE EQUATIONS AND ADAPTATION OF THIS METHOD TO NONMMATRICES V. P. Ginkin, K. G. Chernov, Yu. G. Bartenev, Yu. A. Bondarenko, R. M. Shagaliev, E. B. Shchanikova VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 317.
New effective preconditioners  DIF and PIF  of the stabilized method of biadjoint gradients have been developed to solve systems of 2D and 3D elliptic finitedifference equations with asymmetric poorly conditioned Mmatrices of coefficients. To solve problems with nonMmatrices, we offer a stable and effective method of eliminating positive nondiagonal elements from a preconditioned matrix with simultaneously increasing its diagonal elements by the summarized value of all the elements eliminated from a given matrix row. The rate of convergence of the methods has been tested using both Dirichlet and von Neumann test problems for diffusion and diffusion convection equations and real 9 and 27diagonal positively defined nonMmatrices. Key words: systems of linear equations, iterative methods, method of conjugate gradients, BICGSTAB method, preconditioner, incomplete factorization method, twodimensional problems, threedimensional problems, 9diagonal matrices, 27diagonal matrices, nonsymmetric matrices, nonMmatrices, paralleling.
 NUMERICAL MODEL OF FRAGMENTING THIN METAL SHELLS V. Yu. Meltsas, G. F. Portnyagina VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 1833.
The paper describes the model of fragmenting thin spherical metal shells under the impact of intense loads with generation of two fractions of fragments and with calculation of their mass spectrum. Further motion of fragments is calculated using the model of a heterogeneous medium consisting of a gas and solid particles. The paper gives examples of computations for separation of fragments in space depending on their size. Key words: thin metal shells, detonation, fracture, fragments, spectrum of fragments, gasparticle heterogeneous medium, numerical calculations.
 FUNCTION.ESTIMATES IN THE STATISTICAL MODELING METHODS D. G. Modestov, K. E. Khatuntsev VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 3444.
When solving various problems using the statistical modeling methods, it is useful to have some distributions as solutions to the problems, which are measures, or charges, in general, in the subspace of states within the probabilistic space of these problems. Such methods are intended for estimation of mathematical expectations and, therefore, estimation of distributions is not unambiguous, in general. The paper offers a criterion of approximating a charge with a system of linearindependent functions. The estimated errors of the criterion are also given, as well as some justification of using smooth functions to represent smooth distributions. Key words: statistical modeling methods, mathematical expectation, covariance matrix, dispersion, RadonNykodym derivative, charge density, function approximation.
 ON THE ONE FORM OF ARTIFICIAL VISCOSITY V. Yanilkin, O. O. Toporova, V. Yu. Kolobyanin, G. S. Firsova VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 4557.
The paper describes investigations of the impact of various von Neumann—Richtmyer viscosity forms on simulation results for a number of typical flows, for which exact solutions are available. A new 2D viscosity form developed within EGAK technique is presented. As test computations show, the viscosity proposed in the paper improves the simulation accuracy for a number of flows in comparison with the classic von Neumann—Richtmyer viscosity (both scalar and oriented in the moving shock wave direction). The specific feature of this new viscosity consists in; that it is simple and easy to implement in codes. Key words: EGAK technique, von Neumann—Richtmyer viscosity, characteristic mesh size, shock wave.
 METHODS OF MANUPULATING ENTITIES IN 3D VISUAL ENVIRONMENT V. L. Averbukh, A. V. Zyryanov VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 5869.
The paper describes approaches to the development of entity manipulation techniques for modern visual data representation systems, including those constructed on the base of virtual reality. The notion of actionoriented language is considered. The paper overviews some capabilities of threedimensional input, including 3D manipulators of various kinds and means for capture of motions. A prototype implementation of a simple and easytouse interface for entering gestures on the base of a webcamera and a simple torch is described. Such interface can be useful to arrange interactions both within the viewing systems of computers and management systems using large displays. Key words: visualization, entity manipulation, virtual reality.
 MODELING OF WATER EXCHANGE PROCESSES IN AGGREGATED NATURAL POROUS MEDIA G. S. Goncharova, M. G. Khramchenkov VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 7073.
The paper considers a mathematical model of water exchange in an aggregated natural porous environment (clays, soils and rocks) with regard to nonuniform contraction of its porous structure. The model allows predicting unfavorable environmental consequences of storing hazardous wastes of various industry fields in storage ponds, or longterm storing wastes in special storage facilities due to areas of nonuniform contraction that could result in evolving fissures in the isolating material layer (bentonitic clays, as a rule). The novelty of problem is in the use of a new equation describing the rate of contraction. Key words: unsaturated flow, aggregated porous medium, water exchange, deformation, nonuniform contraction.
 MODEL OF OPTOELECTRONIC SERVO SYSTEM IN A SPACESHIP S. V. Artamonov, B. P. Mukhachev, I. I. Shanin VANT. Ser.: Mat. Mod. Fiz. Proc. 2009. No 3. P. 7480.
The paper describes a mathematical model of a spaceship’s optoelectronic servo system, its purpose, interactions between the model components and some ways of modeling these components. During the mathematical model development, its authors used a common structural diagram of the system. Key words: spacecraft, mathematical model, servo system, photodetector, photosensitive cell, optical system, intraframe processing, calibration, contouring, threshold formation, thresholding.
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