COMPUTER SIMULATION OF SUPERSONIC FLOW ABOUT BLUNT BODIES GIVEN ENERGY RELEASE SOURCES IN THE FLOW

N.E. Afonina, V.G. Gromov, P.Yu. Georgievsky, V.A. Levin
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 38.

      A technique and software are developed for complete Navier-Stokes equation system base computation of flow about blunt bodies given energy release sources in the flow. The technique is implemented for models of constant adiabatic exponent perfect gas and air of constant chemical composition at thermal equilibrium. For numerical integration of the Navier-Stokes equations a finite-difference Riemann problem type UNO-scheme constructed with the finite volume method is used. Required methodical computations are conducted in order to evaluate and verify the computational technique and software system. These computations were used as a basis for selection of optimal values of the computational scheme parameters.
      To find the basic mechanisms for the class of flows under consideration, a series of parametric computations for flow about bodies of a simple geometric shape were conducted. For illustration some computed results for supersonic flow about a spherical blunt entity R_s, in radius at the Mach number M∞ = 3, the Reynolds number Re= 8300, the temperature factor T_w/T_∞ = 1,2 and the adiabatic exponent γ = 1,4 are presented. The obtained results correspond to two modes of flow in the region of energy release whose intensity was given in the cylindric coordinate system (x, r) with the formula of the form
      .
      In the first case, ; x_q/R_s = -1; R_g/R_s = 0,1 - the flow in the energy release region remains supersonic. At = 2000 and the same values of x_q and R_q the second mode of flow is realized when ahead of the energy release region a jump arises where the flow is decelerated down to the subsonic speed, then accelerated up to the supersonic speed and again decelerated in the leading shock wave ahead of the blunt entity. In both the cases a stagnation zone filled with low-mobility gas is formed ahead of the body. A similar result was obtained for a sphere by the invicid flow-about model in terms of Eulerian equations. The computed results were used to construct pressure, temperature, Mach number isolines, as well as the flow line picture using the code ISOLIN of the graphic program package Grafor. The analysis of distribution of pressure and friction factors, as well as the Stanton number over the sphere surface showed a considerable increase in heat exchange in the region of flow attachment which takes place in both the modes of flow.

A.V. Alekseev, I.D. Sofronov, L.P. Fedotova, R.M. Shagaliev
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 38-39.

      This work considers the parallelization algorithms for 3-D group neutron diffusion and transport calculations on distributed memory multiprocessors. These algorithms are implemented in SATURN complex intended for 3-D stationary and nonstationary neutron-nuclei interaction calculations and for neutron transport in group diffusion kinetic approximation.
      The parallelization algorithm for diffusion equation rests on splitting the system into subdomains using the iterative process for special type internal boundary conditions ensuring unconditional process convergence.
      The parallelization algorithm for transport equation is iteration-free and uses the pipeline type scheme with sequential loading of processors.
      The testing results for parallelization algorithms are given obtained on the domestic 8-processor MP-3 system and on western distributed-memory multiprocessors (up to 256 PEs) Cray T3D and IBM SP2.

A.N. Andrianov, K.N. Efimkin, S.V. Zybin
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 39.

      The issues are considered for the use of declarative (nonprocedural) NORMA language to solve 2-D Poisson: equation in cylindrical coordinates on nonuniform grids in the problem of streamer propagation through the cathode layer.
      NORMA language is a tool designed for the specification of numerical methods for the calculations in computational physics on parallel computer systems. It actually allows to automate the programming phase needed to convert from the computational formulas specified by the applications specialists to the computer-specific code.
      There is no considerable difference between the computational formulas and NORMA, representation (of the algorithm; these formulas are the source information for the translation system. This description retains the natural parallelism of the problem without containing any restrictions related with the, wish to adapt the program to a parallel architecture or programming language features. NORMA representation does not require any information about the computational procedure, organization techniques for the computational (cyclic) processes. The order of language clauses can be arbitrary: the informational relations are identified and taken into account by the translator during the computational process organization. This resulted in the following high-level automatization of applications program development (the programmer operates primarily in terms of computational formulas from the application domain):
      — development of reliable applications programs (if the computational formulas are written, correctly, then the correct target program is guaranteed);
      — portability of NORMA programs (the synthesis NORMA translator takes into account the architecture features).
      In this paper NORMA language is used for the representation of the parallel.-algorithm for the calculation of 2-D Poisson equation in cylindrical coordinates using SOR method. The use of NQRMA allowed to obtain automatically the target Fortran program for IBM PG for a single (shared memory) node of the parallel Convex SPP-1000, and Fortran GNS program for. the distributed- memory i860XP parallel system with message passing. The source NORMA program was not actually changed; in the last case it is sufficient to add the specification of the number of processors to the, program that would be desirable for the execution., NORMA programming does not require the knowledge of complicated message passing mechanism; NORMA translator automatically organizes the computations and the communications between parallel tasks.
      The results obtained allow to suggest that the use of NORMA language for the calculations in computational, physics would considerably reduce the development cost of high-performance algorithms especially on parallel architectures.

A.M. Anikin, A.Yu. Bisyarin, L.A. Gorbatova, V.M. Gribov, A.V. Kim
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 40.

      At present computers of small and medium performance are usually integrated into computer local area networks (LAN). Each network can be considered as a single computing resource, that is similar in something to multiprocessor parallel computer. Nodes of a network are analogs of processors. This problem deserves attention because of widely spread LAN and permanently increasing both capacity of its relatively cheap components, and channel capacity of communication hardware.
      Base means for organization of parallel calculations in LAN on various types of computers under OS Unix supervision was created by authors of this report. The set of such means makes possible both calculations in a parallel mode, and debugging new parallel programs, intended for the use on real multiprocessor computers with distributed memory. The application program interface on the basis of Berkley Sockets is implemented in the language C and is made out in the form of library of procedures.
      The program for calculations of two-dimensional problems of gas dynamics with heat conduction in the complex geometry domains is created with the use-of this package. For parallelizing computations the method of geometrical decomposition with introduction of internal boundary conditions is applied.
      Factors of speedup and performance of the distributed computer system in calculations on LAN with different number and types of units were represented in the report.

E.M. Antonenko, G.V. Dolgoleva, V.F. Ermolovich
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 40-41.

      The studies of physical processes occurring in inertial fusion applications commonly use numerical simulation methods. This motivates the development of codes allowing to simulate the characteristics of thermonuclear burning and compression. One of these codes is represented by ALF.
      ALF is a 1-D code for the calculation of two temperature emitting gas including the nonstationary energy, momentum and mass transfer by neutrons and fast charged particles, kinetics of thermonuclear and neutron-nuclear reactions as well as local variations of the ion distribution function.
      The ALF code allows to calculate the following processes:
      - two-temperature gas dynamics in terms of physical viscosity, momentum transfer to material by neutrons and
      - charged particles, energy, release from thermonuclear and neutron-nuclear reactions, variations of mass related to the transport of neutrons and fast charged particles;
      - radiation transfer in the approximation of nonequilibrium spectral quasi-diffusion in terms of Compton scattering;
      - heat transfer by electrons and ions;
      - material heating under the exposure to fast ion flux;
      - kinetics of thermonuclear and neutron-nuclear reactions;
      - transport of fast charged particles (these include both the products of thermonuclear and neutron and recoil nuclei resulting from the elastic scattering on H, D, T, 3He, 4He).
      - neutron transport (the calculations involve the thermal dispersion of TD neutrons and energy distribution of neutrons resulting from nonthermal TD reaction);
      - local variation of the distribution function of thermal ions (H, D, T, 3He, 4He).
      The report contains:
      - a system of differential equations serving the base for the code;
      - results of numerical operation studies for the target proposed for the use within the laser NIF project (USA).

L.V. Antonova, O.V. Verbitskaya, N.S. Dyadina, V.I. Legon´kov, V.I. Murashkina, V.R. Sokolov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 41.

      The specialized code system EOS Package (equations of state of substances) is a part of a shared software, necessary for “physical support” of numerical experiments. It is intended to meet the requirements of code complexes concerning the description of thermal properties of substances. By now the EOS Package serves more than 10 complexes.
      The package satisfies the needs in calculations of thermodynamic functions of substances (TDF) and track length (TLF) functions for a wide class of problems, solving one- and two-dimensional hydrodynamics equations with heat conduction and other dissipative processes in one- and three-temperature approximation. The EOS Package can be used for solution of problems accounting for phase transitions and flow.
      The list of TDFs computed in the EOS Package by one or another EOS includes dozens of quantities and can be extended. Currently the Package contains 17 EOSs beginning with the ideal gas equation up to a wide-range equation of state.
      The TLF codes calculate the photon range lengths (Rosseland-averaged), heat conduction factor, their derivatives, as well as the ion and electron heat conduction for problems in three-temperature approximation. Currently the EOS Package contains 9 TLFs.
      The EOS Package is a specially organized set of multi-purpose EOS and TLF modules, data sets for them and service software, ensuring service and use of these objects in application codes. The Package is intended for service of codes, written in Fortran. It runs on a distributed computer network including BESM6, EC, Elbrus-2, Sun in different operating systems. In particular they are MS DOS, Windows, UNIX for IBM PC.
      In order to facilitate the interaction with the Package and to speed up the operation, window interfaces are implemented for its separate components in the form of interactive environments.
      There is an opportunity to add new functional objects to the Package. Practically every EOS or TLF code written as a Fortran subroutine can be included in the Package structure after small modification related to standard batch interface.

A.Yu. Artemiev, Yu.D. Chernyshev
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 41-42.

      To suppress the short-wave velocity perturbations, the “D” codes [1] implement artificial viscosities of various types. For this purpose, the most efficient are contour and angular artificial viscosities. Furthermore, for highly 2-D flows an algorithm was developed and implemented on “D” codes for smoothing the normal velocity components at the interface. To some extent, the velocity smoothing algorithm is an analog of contour and angular viscosities. This algorithm is generalized to the inner points of domain where each family of the grid lines is considered as an interface during smoothing.
      For a uniform grid, Јhe smoothing operator has the first order approximation in time and third order approximation in space. To conserve the total energy, the kinetic energy transferred due to the velocity variations at the cell nodes is transformed to the internal energy. The pulse received by the grid node is transferred to the neighbors with inverted sign at the distance inversely proportional to the distance between the nodes.
      The smoothing algorithm proposed differs from the commonly used algorithms in that:
      — the timestep is explicitly included into the smoothing operator which makes the smoothing procedure relatively flexible (when the timestep size is reduced the complement to the velocity vanishes),
      — the smoothing is accomplished only for a given mode of angle variations in one of the families of the grid lines.
      This algorithm is generalized to the 3-D case and implemented in DF complex [2]. The complement to the velocity during the smoothing at the node represents a sum of complements obtained from the velocity smoothing on each of three surfaces passing through this node and determined by two families of grid lines.
      1. Dmitriev N.A., Dmiirieva L. V., Malinovskaya E. V., Sofronov I.D. A method for 2-D gas-dynamic nonstationary calculations in Lagrangian variables // Theoretical Fundamentals and Construction .of Numerical Algorithms for Computational Physics / Ed. by Babenko K.I. ?.: Nauka, 1979. P. 175-200.
      2. Artemiev A. Yu., Delov V.L, Dmiirieva L. V. A method for the 3-D nonstationary gas-dynamic calculationsin Lagrangian variables // Voprosy Atomnoy Nauki i Tekhniki. Ser. Matematicheskoye Modelirovaniye Fizicheskikh Protsessov. 1989. N 1. P. 30-39.

S.M. Bakhrakh, G.P. Simonov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 42.

      Using the Lagrangian representation for equations of dynamics of an accelerated thin liquid layer the analytic solutions are found for the problem of Raleigh-Taylor instability at the process stage non-linear in the observer’s space. Evolution of various perturbation types in layer shape and component velocities is considered. It is shown that there are both exponentially growing and limited, oscillating solutions.
      This analysis is also important at consideration of Raleigh-Taylor instability regarding a relatively thick layer. This is substantiated with the results of numerical studies of compressible ideal fluid semi-space interface perturbation evolution. It is noted that there are qualitative differences between the cases when perturbations are given in semi-space interface shape and initial velocity form.
      The work was carried out under the auspices of International Science and Technology Center (grant NM-4000) and Russian Fundamental Research Foundation (project N 96-01-00043).

S.M. Bakhrakh, G.P. Simonov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 42-43.

      Analytical solutions of the problem of Raleigh-Taylor instability for a thin liquid layer in 3-D formulation at the stage non-linear in the space of the observer are obtained. Lagrangian representation is used for the equations of accelerated layer dynamics. Evolution of perturbations both in the layer shape and in velocities of layer elements is studied, Solutions depending on initial perturbation shape and amplitude are found. Existence of both exponentially growing and limited solutions is shown.

V.N. Bakulin, V.A. Potopakhin
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 43.

      Novel multi-layer structures composed of reflecting, dispersing, absorbing, thermoinsulating and other layers of arbitrary structure have been recently proposed to protect people against injurious effect of penetrating radiations (highly intense fluxes of electrons; ions, neutral atoms), enhance strength, reduce mass and cost of vessels of NPP reactors, particle accelerators and other entities at action of intense thermoforce loads.
      When studying the thermoforce state (temperature fields and stressed-strained state parameters) and strength of such structures it is necessary to take into account features of poorly studied dynamic thjermoforce loading type which are concentrated energy fluxes (GEF); features of GEF interaction with structural material which results in essentially nonuniform energy release over the structure Volume leading to occurrence in it of highly intense local temperature, pressure fields rapidly varying with time, to variation in physical, mechanical, geometrical characteristics during the effect. The computation methods should therewith account the above layer properties and features, structural and other features of multi-layered vessels made of composite and conventional materials (inhomogeneity, anisotropy, transversal pliabilities, property variation with coordinates and time, etc.), multiple effects of CEF various in nature, variations in structure parameters from one effect to another:
      The approach developed by the authors is discussed which is based on derivation of non-linear dynamic 3-D and 2-D equations, including equations of bound thermal viscous elasticity, time-dependent heat conduction, etc. for layers imparted with all necessary properties at dynamic thermoforce loads, CEF and their solution using a combination of modified methods: the direct method, the discrete orthogonalization method, the Striklin method, the Khubolt finite-difference scheme. In this case it is possible to take into consideration the specificity of the CEF effects, multi-layered structure features, advantages of every method and obtain fairly simple computer algorithms.
      The advantages of the developed computational models, algorithms and computer programs for solving 3-D and 2-D non-linear dynamics problems are as follows:
      — no time step limitation is imposed;
      — the time step may be changed during problem solution and both static and dynamic problems can be solved using a single algorithm;
      — preliminary static loading and the possibility of repeated introduction of temperature fields, pressures, variations in structure parameters are taken into consideration.
      Employment of the developed software models considerably extends the range of solvable dynamic non-linear problems, including problems of coupled and uncoupled thermoelasticity, allows to refine the computed results arid formulate recommendations for designing, rhanufacture; operation of the structures under consideration.
      The developed experimental methods, developed and improved rigs and devices were used to conduct studies of the thermoforce state and strength of the above multi-layered structural elements under action of concentrated energy fluxes which confirmed the computed results.
      The work was carried out under the auspices of Russian Fundamental Research Foundation (project N-94-01- 01797a).

S.P. Belyaev, L.I. Degtyarenko, I.Yu. Turutina
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 43-44.

      The codes for distributed-memory parallel systems must:
      - ensure the interprocessor communications via message passing;
      - provide the uniform loading of processors;
      - use the hardware-based compatibility of the menage passing and computations on the processor.
      For continuum mechanics problems with time-dependent loading, it is possible to implement the dynamic balance of the loading by transmitting the computational points from the most loaded to less loaded processors.
      The possible re-allocation of an arbitrary point from one processor to another necessitates the pointwise parallelization.
      The parallel gas-dynamic code can be written as a sequence of steps each of them first calculating the boundary- processor points which require interprocessor transfers and the waiting time is used for the calculation of internal processor points. The message passing is accomplished on the initiative of the transmitter: as soon as the point is calculated necessary transfers to neighbors are initiated and the calculations for the next point start in parallel with transfers. This organization is simpler and allows to reduce the global waits as compared to other approaches.
      The parallel heat-conduction code can be written based on parallel-pipeline approach where each processor sequence executes a group of 1-D runs in the pipeline mode and all processor sequences execute in parallel. The parallel-pipeline algorithm is also implemented for an arbitrary set of processor points.
      The computational codes are written as event processing routines. A special order of event processing allows the maximum balance between the computations and communications.
      For the maximum computations-communications balance, nonblocking transfers are used and the prohibition is introduced to use global communications.
      The communications buffering is provided to increase the efficiency.
      The dynamic loading balance does not depend on the computational method, can be accomplished after the given number of steps and is implemented based on the following:
      - minimization of distant transfers;
      - local decisions about the load redistribution.

S.P. Belyaev, I.Yu. Turutina
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 44.

      Parallel codes intended for the calculations on distributed memory systems must take into account the specific of these systems. First, it is necessary:
      - to provide the communications between the processes running on different processors through the message passing;
      - to achieve the uniform loading of processors;
      - to use the hardware-based balance between the message passing and computations.
      The multidimensional continuum mechanics problems with variable loading allow to implement the dynamic load balance by transmitting the computational points from the most to less loaded processors.
      The algorithms for automatic redistribution of computational points among the processors must not necessarily have high accuracy. If the automatic redistribution algorithms provide parallel computations efficiency of 70-80% on a ten-processor system and about 50% on 100 processors for a considerable part of production problems they can be said to do well. The preliminary estimates indicate that this needs only the load unbalance of 10-20% as the computations run.
      However the redistribution algorithms should necessarily have the following properties: simplicity of the computational procedure to determine the number of points with minimum information and local decisions about the redistribution.
      It is assumed that prior to the algorithm execution each processor has received from the neighbor the information about the step termination including the computational time and number of points that can be received by each neighbor/rom the given processor.
      The general algorithm consists of the following sequential steps:
      1. Evaluation of the number of points to be transinitted to each neighbor.
      2. Generation of transmission lists.
      3. Transmission of points.
      4. Generation of new control lists.
      5. Evaluation of the number of points received for each subsequent balance step.
      The processors communicate only at step 3. All remaining steps are executed by each processor independently.

S.A. Bel´kov, P.D. Gasparyan, Yu.K. Kochubey, E.I. Mitrofanov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 45.

      Ionization kinetics of multi-charge, time-dependent, nonequilibrium plasma in the medium ion approximation is studied. Approximations hydrogen-like ion whose energy levels depend-both only on the principal quantum – number and on the principal and orbital quantum numbers are considered as the atom models. Computed results were presented for various plasrna characteristics found within this model, comparison with earlier published data, as well, as with computations in the chemical bond approximation was made. The medium ion model was shown to describe spectral characteristics of nonideal plasma satisfactorily.

D.N. Bokov, N.N. Bokov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 45-46.

       The work major equations of characteristic directions method and algorithms for separation of strong and weak discontinuities. Peculiarities are discussed of solving hyperbolic components of system of equations describing invicid motion of heat conducting gas.
      A main dissimilarity from the wide known mesh-characteristic methods described in monograph [1] involves the fact that the solution is constructed on four independent mesh sets:
      - Eulerian observation mesh;
      - mesh of characteristics corresponding to eigennumber (u - α);
      - mesh of characteristics corresponding to eigennumber (u);
      - mesh of characteristics corresponding to eigennumber (u + α);
      The first set can be absend.
      The determining system of difference equations is written in the form of finite increments of conservative dependent variables along the characteristic directions. Such form of presenting solution doesn’t lead to appearance of limitations on time step of Currant condition type, and enables to track formation and evolution of strong discontinuities.
      Dissimilarity from the classical method of characteristics [2] is that arbitrary equations of state are considered and the mesh solution is constructed for one time. This enables to apply implicit methods, of solving parabolic components of the initial system of equations.
      1. Magomedov K.M., Kholodov A.S. Meshicharacteristic Numerical-Methods. M.: Nauka Publishers,; 1988.
      2. Zhukov A.I. Application of Method of Characteristics to Numerical Solution of One-Dimensional Problems of Gas Dynamics // Proceedings of MIAN USSR. I960. Vol. 58.

Yu.A. Bondarenko
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 46-47.

      For a sequence of processing elements (PE), we consider an explicit local algorithm similar to the explicit difference scheme with monotonic distribution of grid points m(x,t) on a PE where m is the point number, x is the PE number t is the timestep number. The algorithm is considered for the PE dynamic load balance accomplished by sending the grid points from one PE to the neighbors at each timestep. It is assumed that the computation time of the cell number m, σ(m,t), at the time t is a positive function continuously differentiated over m and α(m), the time of transfer of the, m cell from one PE to the neighbor, is also a positive function continuously differentiated over m. The functional
      
      is the computational time for a single timestep where m₀(x) = m(x, t - Δt) is the distribution of points over the sequence of PEs at the previous timestep. The time (1) should be minimized in terms of boundary conditions and monotonicity condition
      
      and monotonicity condition
      For this variational problem of step-by-step optimization of the computational time, the following theorems are proved.
      Theorem 1. If the functional S[m] reaches its minimum on the function m(x) satisfying the boundary conditions (2) and monotonicity condition (3) then this function satisfies equation
      
      Inversely, each continuous solution of equation (4) satisfying the boundary conditions (2) and monotonicity conditions (3) satisfies also the necessary condition of the minimum value of σS[m] > 0, Vσm(x).
      Theorem 2. Let the function m₀(x) be limited and continuously differentiated. Then equation (4) always has the solution {Φ, m(x)}, satisfying the boundary conditions (2), this solution is unique and the function m(x) is continuously differentiated.
      Theorem 3. Let the function m₀(x) be limited and continuously differentiated. If the function mo (x) satisfies the monotonicity condition and the boundary conditions
      m₀(x), _x=0 = 0, m₀(x), _x=X1 = M1, then the solution m(x) of equation (4) satisfying the boundary conditions (2) is also a monotonically growing continuously differentiated function.

Yu.A. Bondarenko, O.A. Vinokurov, V.V. Zmushko, F.A. Pletenev, P.V. Rybachenko, V.A. Saraev, I.D. Sofronov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 47.

      The report presented the results obtained with MIMGZA codes for the parallelization of 2-D gas-dynamics and heat conduction equations on the eight-processor MP-3 system. The computational domain is divided into subdomains in accordance with the number of processors. When the gas-dynamic equations are solved, the neighboring subdomains have the overlapping portions through which the processors communicate. The parallelization of 2-D heat conduction equation uses the pipeline algorithm arid the transposition algorithm. The results of two computations are given.

V.I. Delov, L.V. Dmitrieva, V.V. Sadchikov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 47.

      The report is devoted to the description of the method developed for the calculation of spatial problem of the intersection of two arbitrarily located in space; hexahedra and evaluation of their total volume of intersection, if any. The need in such algorithms and codes occurs particularly in the calculation of gas-dynamic problems in Lagrangian coordinates. It is known that in this case the numerical calculation of complicated problems necessitates one-shot grid reconfiguration and rescaling the values to the new grid because further computations are impossible. These situations are encountered, for example, in the event of strong grid distortions, pinch of physical domains, formation of small-scale jet flows.
      The calculation of the problems presented here is also addressed when creating Lagrangian-Eulerian methods for continuum mechanics problems with strong deformations. In addition, such algorithms are extremely needed to create the graphics packages and codes, oriented to the operation with 3-D geometrical objects.
      The problem solution reduces to finding the exact intersection volume of two tetrahedra arbitrarily located in space.
      The methods are considered to split the polyhedra into elementary tetrahedra together with the main program implementation features for the algorithms proposed. The most computationally cost efficient ways are reported to divide the polyhedra without additional computational errors.
      The report presented the results of test computations for the intersection of hexahedra with the faces representing second order surfaces-hyperbolic paraboloids that are used in 3-D “D” code.

V.I. Delov, O.V. Senilova, I.D. Sofronov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 47-48.

      The report proposed an approach to the construction of conservative differential-difference representations of equations describing nonstationary elastic-plastic flows in Lagrangian variables. The method is the further development of 2-D method for the generation of spatial approximations to the equations of motion in gas dynamics [1,2] for elastic-plastic media.
      In this work the matrix of kinetic energy determining the approximation technique for the pressure gradient is taken in the canonical form that is traditionally used in gas-dynamic codes.
      The report presented the difference formulas for the components of deformation rate tensor and the resulting difference approximations for the evaluation of derivatives with respect to the components of the stress deviator.
      The computational results were reported obtained with difference schemes where the grid distribution of quantities in time is taken like in “D” code [3] and the time derivative is approximated with the second order accuracy. The problem describing the elastic oscillations of membrane is taken to show unquestionable advantages of the resulting difference schemes as compared to the classical Wilkins scheme.
      1. Isaev V.N., Sofronov I.D. Construction of discrete models for gas-dynamic equations based on the interconversion law of kinetic and internal energies of continuum // Voprosy Atomnoy Nauki i Tekhniki. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. N 1(15). P. 3-7.
      2. Delov V.I., Isaev V.N., Sofronov I.D. Conservative and invariant differential-difference representations of gas- dynamic equations in cixisymmetric case // Ibid. 1987. N 1. P. 3-10.
      3. Dmitriev N.A., Dmitrieva L.V., Malinovskaya E.V., Sofronov I.D. A method for the calculation of 2-D gas-dynamic problems in Lagrangian variables // Theoretical Fundamentals and Construction of Numerical Algorithms in Computational Physics / Ed. by Babenko K.I. M.: Nauka, 1979. P. 175-200.

V.V. Dryomov, D.G. Modestov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 48.

      Model equation of state for methane chlorine derivatives is constructed on the basis of variation theory of perturbations [1]. Directly for the calculations we use proposed by Ross decomposition of free energy with base potential of solid spheres and with correction for agreement with results of computer modeling of a system of particles interacting through potential R-12 [2]. To describe interaction between molecules potential exp-6 was used in this work [3] which describes this class of substances well. To calculate melting curves analog of Lindeman’s law of melting is used, namely, constancy of package parameter along the melting curve ( for liquid we must speak about solidification curve) [4].
      This work provides comparison with experimental data on melting of methane chlorine derivatives under low pressures. It is shown that dichloromethane and chloroform are liquids under shock compression. As to carbon tetrachloride, even under low pressures it transforms into solid state and melts only under pressures on the order of 25-30 GPa.
      To describe experimental data on shock compression [5] under high pressures where Hugoniots of methane chlorine derivatives have a break, dissociation was introduced into the model. To describe
      a multi-component mixture we used one-liquid Van der Waals’s model (see, e.g., [3]). It is
      shown that in the region of experimental Hugoniots breaks a considerable role is played by disbalance. Comparison with experimental values of shock front temperature [6] besides that shows influence of oscillatory relaxation upon measurement results under low temperatures. Comparison is also given for calculated and experimental sound velocity [6] behind the shock front.
      1. Barker J.A., Henderson G. // Rev. Mod. Phys. 1976. Vol. 48. P. 587.
      2. Ross M. // J. Chem. Phys. 1979. Vol. 71. P. 1567.
      3. Ree F.N. // J. Chem. Phys. 1984. Vol. 81, N 3. P. 1251.
      4. Ross M.//Phys. Rev. 1973. Vol.-8, N 3. P. 1466.
      5. Dick R.D. // J. Chem. Phys. 1981. Vol. 74, N 7. P. 4053.
      6. Gogulya M.F., Dolgoborodov A.Yu. // Chemical Physics. 1994. Vol. 13, N 12. P. 118.

A.D. Gadzhiev, S.Yu. Kuz´min, S.N. Lebedev, V.N. Pisarev, A.A. Shestakov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 49.

      When coupling simulation of the Lagrangian hydrodynamics with other physical processes, the difference grid becomes significantly nonorthogonal. The efficiency of numerical techniques solving such problems depends on the accuracy of the nonorthogonal difference algorithms applied and on their ability to ensure adequate accuracy on the grids with strong deformations.
      The limitations of traditional nine-dot difference scheme for solving the diffusion-type equations are rather well known at present. The accuracy rapidly decreases when nonorthogonality of a grid increases. This follows from the fact that from among two operators divergence and gradient - the gradient operator is poorly approximated.
      By now some approaches have been proposed for designing difference schemes ensuring adequate accuracy when solving difference equations on nonorthogonal grids. Our approach based on the diamond-type approximation is applicable to a wide class of equations of mathematical physics. In these techniques both divergence and gradient operators are approximated within a preset grid cell, the face values of velocity and pressure being used. Based on the above approach the ROMB-type techniques are easily designed, are efficient, and ensure adequate accuracy on nonorthogonal grids.
      The review considers applications of the ROME technique to:
      — heat conduction equation;
      — system of equations for energies in the three temperature model;
      — gas dynamics equations;
      — hyperbolic systems of general equations;
      — transport equation in both diffusion and P1-approximations;
      — self-conjugate transport equation.
      Results of numerical experiments were presented.

V.L. Gasilov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 49-50.

      Issues of development of 3-D discrete models of Earth’s crust with a great number of miscellaneous deformable elements are considered. Known 2-D disk and single-layer block models of the lithosphere require much memory and computer time at computations on sequential computers and do not allow to conduct modeling with a great number of elements. The suggested computer crust dynamics models are intended for numerical experiments on a parallel super-computer and provide for complexity build-up both quantitatively, i.e. increase in the number of the elements up to several hundred thousands, and qualitatively, i.e. employment of actual geophysical and seismic data, earthquake catalogs and results of seismoactive region morphostructural analysis, complex deformation and destruction mechanisms, taking into consideration geological structure evolution processes, etc.
      Using dynamic destruction models requires computation of stress fields, displacements, injuries, temperature and other fields describing the process thermomechanics. To do this, one often has to set up non-classic boundary problems of strained body mechanics. In addition to algorithmic difficulties, solution of such problems involves quite a great amount of computations and may be efficient only provided high-performance computers are used. RAS Urals Branch IMM is developing algorithmic tools and software for destruction mechanics problem modeling and study which are oriented to employment of parallel multi-processor systems and algorithms.
      In particular, a program system is being developed for a powerful computer with many processors operating in parallel and large-capacity main memory. The software package is based on the finite element method in the destruction mechanics problems and direct minimization of an appropriate dynamic system state functional. It is designed for modeling and studying elastic-plastic (including shock) interaction of spatial bodies with fatigue and destruction effects and allows to compute the processes which occur in complex mechanical systems in detail.
      Corresponding algorithmic support and software developed for the RAS Urals Branch IMM computer is discussed. Results of the computations made to identify model and external effect parameters are given.
      The work was carried out under the auspices of Russian Fundamental Research Foundation (project N 94-01- 00361a).

P.D. Gasparyan, N.V. Ivanov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 50.

      In target laser irradiation experiments of great interest is the process of collision „of plasma counter-flows. In the region of flow interaction mean ion ranges are known to be comparable with or higher than characteristics flow sizes. Therefore, to study this phenomenon, numerical techniques based on kinetic plasma models are needed. At the same time in unperturbed flow regions ion ranges are considerably less than the characteristic sizes and consideration of these regions at the kinetic level involves unacceptable computer costs. This, work proposes a technique for numerical computations of planar plasma flows where ion processes are considered at the kinetic level. It is based on the Monte Carlo method and involves about the same computer costs in the regions of low and high ion ranges.

V.N. Glukhov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 50-51.

      The prediction of weather and climate variations requires high computational performance. One of the ways to resolve this situation.is to use multiprocessors.
      This report is devoted to the problems relating to the parallelization of the model for global atmospheric circulation developed by the Institute of Computational Mathematics on the multiprocessor MVS-100 (Keldysh Institute) and to the results obtained. The development of this model took many years and now it is extensively used in various international experiments (AMIP, FANGIO).
      The model rests on the system of complete nonlinear equations for atmosphere hydrothermodynamics in Lamb form on a sphere using the vertical s coordinate. The difference approximation: of the spatial operator over the horizontal is accomplished on shifted Arakawa C grid regular with respect to latitude and longitude. The grid step along the latitude circles is Δα = 5° and Δφ= 4° along the meridians; the uniform splitting into 7 levels is used, vertically. The integration over time uses the semi-implicit scheme with the step of 20”. At each integration step, Helmholtz equations on sphere are solved with the reduction relative to the variable λ.
      Two parallelization techniques were used: in the first case the data were distributed among the processors in terms of latitude, in the second case this was done in terms of latitude and longitude. To solve the Helmholtz equations a pipeline was organized, and the summation of data over the processors used the coupling scheme.
      It was assumed that if one processor needs the time t for execution, p processors can execute in time t/p. However this ideal acceleration is achieved only in very special cases. The experiments showed that the performance increases nonlinearly with the number of processors and there exists a certain number of processors where the maximum performance is obtained. For example, for the first parallelization technique the computational time was reduced only about by a factor of 4 while the minimum was observed on 9 processors.
      By summing the execution time of individual procedures over the processors one can identify those for which the total execution time increases and those for which it remains actually unchanged. It could be noted that those procedures for which the total time increases contain the dependencies on data residing on different processors.
      For the first parallelization method the maximum performance was estimated to be 10.3Mfiops (without optimization or compilation).
      The computer system resources are not completely used because; the interprocessor transfers are needed. The second approach must be more efficient since in this case the amount of data transferred decreases is the number of processors grows while it remains fixed in the case of the first approach.

E.V. Groshev, V.A. Zhmailo
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 51.

      Some engineering and science applications use plasma radiation. Furthermore, it is frequently used for the diagnostics of plasma parameters.
      If the emitting plasma is in nonequilibrium, the calculation of its properties and radiation characteristics requires combined computations of kinetics, energy and radiation transport equations. -
      This work presents one of the approaches to the solution of such problem. Its specific feature is the kinetics model considered: it is oriented to the description of plasma in a wide range of temperature and density variations. The gas is assumed to be composed of one mixture of particles (atoms, molecules and ions). The kinetics includes the following processes: electron shock ionization, molecules dissociation by the electron shock, dissociation recombination, photo-ionization, associative ionization, molecules dissociation by heavy particles, recharging, photorecombination and recombination in triple collisions. Energy equations are written for ions and electrons; these equations account for the energy transfer between electrons and heavy particles resulting from elastic and nonelastic collisions as well as from the interaction with radiation.
      The absorption coefficients are calculated from Kramers-Unsold approximation, the radiation transport in the lines is ignored. The gas is assumed to be fixed, the problem is one-dimensional and spherically symmetric.
      The process is described by the system of photon transport, kinetics and energy equations for ions and electrons. The system is approximated with the first order implicit scheme in time. The transport equation is approximated with actually monotonic ST_n scheme in space [1]. The system of difference equations at the timestep is solved with the method of “simple” iterations involving the convergence acceleration algorithm [2]. The system of kinetics and energy equations is solved with Newton method.
      The illustration is given based on the problem describing the radiation from rarefied air heated by the external source. The description is given for the energy redistribution over space due to the radiant transport and gas composition changes.
      1. Groshev E.V., Pastushenko A.M., Yudinisev V.F. One three-point difference scheme with the weight factor for the transport equation // Voprosy Atomnoy Nauki i Tekhniki. Ser. Metodiki i Programmy Chislennogq Resheniya Zadach Matematicheskoy Fiziki. 1985. N 2. P. 87-96.
      2. Groshev E.V. One iteration convergence acceleration method for numerical calculation of 1-D nonstationary radiation transport in multigroup kinetic approximation // Ibid. 1982. N 1. P. 67-72.

V. Yu. Gusev, M. Yu. Kozmanov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 52.

      This work is devoted to constructing monotone difference schemes for the transport equation. The developed methods may be applied, for example, for solving problems of contaminant transport in the atmosphere, radiation transport with account of absorption and scattering.
      The computed results of model problems were presented in comparison with exact solutions.

V.Yu. Gusev, M.Yu. Kozmanov, N.Ya. Moiseev
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 52.

      One of versions of a new maximum principle base monotonizer is considered. This is used to monotonize solution to difference equations generated by explicit schemes of a high (second and higher) order of accuracy which approximate the equation of transport with a constant coefficient.
      In [1] by monotone difference schemes for the equation of heat conduction were called the schemes satisfying the maximum principle. A similar definition was used in [2,3] for the system of the equation of radiation transport and the equation of energy. This work is further development of the approach from [4] to solution monotonization.
      The essence of the approach proposed consists in the fact that the flows computed at the stage of predictor for the scheme of a high accuracy order secure meeting the maximum principle, namely: u_min ≤ u^j ≤ u_max. Here u_min and u_max are local minimum and maximum of the numerical solution at the time t the vicinity of the point x_j, u_j the numerical solution at the time t_n + τ at the point x_j. When this inequality is violated then the flows are computed from the equation written on the basis of this inequality for the left or right boundary by substitution of u_j with, the expression from the initial difference equation.
      The monotonizer basing on the schemes from [5,6] was used to obtain monotone schemes of the second, third and fourth order of accuracy which are higher than the first-order on the whole and produce numerical solutions of rather a low diffusion as compared to the explicit monotone scheme of the first order of accuracy which is substantiated with the results of numerical solution of various model problems.
      1. Samarsky A.A. Theory of difference schemes. Moscow, Nauka Publishers, 1977.
      2. Andreyev E.S., Kozmanov M. Yu., Rachilov E.B. The maximum principle for the energy equation system and the non-stationary equation of radiation transport // Zhurn. Vych. Mat. i Mat Fiz. 1983. Vol. 23, N 1. P. 151-159.
      3. Kozmanov M.Yu. Monotone schemes for the system of equations of radiation transport // Voprosy Atomnoy Nauki i Tekhniki. Ser. Matematicheskoye Modelirovaniye Fizicheskikh Protsessov. 1989. N 2. P. 51-54.
      4. Gusev V. Yu., Kozmanov M. Yu. Conservative schemes using characteristics and anti-diffusion velocities for solving the equation of transport // Ibid. 199d. N 1-2. P. 24-32.
      5. Moiseev N. Ya. On one modification to the Godunov difference scheme // Voprosy Atomnoy Nauki i Tekhniki. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1983. N 3. P. 35-43.
      6. Moiseyev N.Ya. On one approach to construction of hybrid scheme of an enhanced order of approximation // Ibid.1988. N2. P. 11-17.

A.O. Ignatiev, A.A. Kalinin, V.K. Koryakin, A.I. Mel´nikov, L.S. Talantova, N.I. Shulepov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 53.

      The report reviews problems and techniques of parallel program-debugging and monitoring for massively parallel computers with MIMD-architecture and distributed memory. The following components are considered:
      - system of dynamic debugging during program actual run;
      - post-processing system for analyzing data accumulated in the course of program run after the latter is completed.
      Some issues are discussed related to implementation of dynamic debugging and post-processing systems, proposals on functional structure of debugging system are formulated.

Ivanov A.G.
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 53.

      Since the mid 20-th century the need to account for reasons on unexpected brittle fracture has led to development of the linear fracture mechanics. A painful process of fracture criteria replacement began. The new criteria based on energetic relations required knowledge of presence of defects for an entity under consideration which considerably impeded their use and sometimes even made this impossible.
      The report presented critical consideration of some papers on space body failure description at interaction with the planet atmosphere. The use of conventional failure, criteria, such as critical shearing or breaking stresses, is shown to lead to inadequate description of a phenomenon as a whole. When using the integral approach based on meeting a needed energetic failure condition it is possible to adequately describe the qualitative phenomenon picture and also obtain quantitative results under certain conditions.
      Some requirements were formulated which the failure criteria should meet.

M.G. Keidzhyan, M.F. Ivanov, A.V. Ivlev
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 53-54.

      This work studies interactions of powerful laser radiation with supercritical density plasma. At radiation rates of 10¹⁶ W/cm² the plasma dynamics under irradiation is severely non-linear, while at rates of 10¹⁸ W/cm² and higher relativistic effects become essential which motivates employment of computer experiment methods.
      To model processes (rates of 10¹⁸W/cm² and higher, the collisionless case, this work uses the developed electromagnetic codes implementing the particle method in the 2-D and 2^1/2D-gepmetry with account of electron relativism and ion mobility. The following was considered within a wide initial data range: normal and oblique radiation incidence; cases of planar and focused waves at various polarizations; evolution of processes on the density gradient. Some new effects were considered, such as generation of high-energy (MeV) electron fluxes on the boundary, generation of long-lived eddy structures in the plasma volume, excitation of higher harmonics on the boundary and, hence, laser radiation travel into dense plasma, essential dependence of dynamic processes on radiation polarization. Intense evolution of instabilities occurring on the density gradient was found to lead to generation of spatial structures in plasma. For oblique radiation incidence non-linear processes of plasma wave excitation and energy transfer to plasma are considered.
      To model kinetic processes (at rates up to 10¹⁶ W/cm²) with account of Coulomb collisions, developed computer codes are used which are based on Langevin equations stochastically equivalent to the original Focker-Plank equation. The collision effect on radiation absorption processes was analyzed.

K.V. Khishchenko, I.V. Lomonosov, V.E. Fortov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 54.

      The equation of state within a wide density and pressure range is a necessary element for mathematical simulation of time-dependent hydrodynamic processes of intense energy flux pulsed effect on material [1]. Owing to serious problems involved in computation of a complex collective inter-particle interaction in a heated multi-component medium [2], semi-empirical models are traditionally used to uniquely describe thermodynamic material properties within a wide parameter range on the phase diagram where the general form of the functional dependencies of the thermodynamical potential is determined without theoretic representations being involved, while the experimentally measured data at high energy densities are used to estimate numerical values of free coefficients in these dependencies. This work presents the caloric form of wide-range equations of state which allows to effectively describe properties of various materials (both elements and compounds) in the condensed and quasi-gaseous phases. The developed semi-empirical model which in the analytical form expresses the relation of internal energy, pressure and volume extends the Mie-Grueneisen equation to the region of rarefied states and arbitrary energies. Various options to estimate the volume dependencies for the cold curve and Grueneisen factor are discussed. The computed results are given which were obtained on the base of the developed caloric model of thermodynamical characteristics of molybdenum, iron, lead and plexiglas under shock loading and isentropic unloading. For each material studied the computed dependencies were compared with the set of experimental data, within a high energy density range.
      1. Bushman A.V., Kanel G.I., Ni A.L., Foriov V.E. Thermal physics and dynamics of intense pulsed effects. Ghernbgolovka: USSR Academy of Sciences OIKhF, 1988.
      2. Bushman A.V., Foriov V.E. // UFN. 1983. Vol. 140. P. 177.

Yu. I. Kolosova
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 54-55.

      The asynchronous medium for the messages in distributed computations may lead to the deviations from the real reception sequence. An algorithm is considered for post-program analysis of parallel process histories to identify the discrepancy between the reception of messages by a process and the transmission sequence.
      A common model is used where the distributed computation is represented by the set n of processes p_i (i = 1, 2,..., n), where the parallel history of process p_i is described by the set {e_i1, e_i2,…} of events in the execution order. Here only the message transmission and reception are recorded.
      The algorithm rests on the variant of logical clock characterizing the relation “occurred before” usually denoted as “→”and defined.as: iff: 1) (i = j)∧(q < u) or 2) there is a message m such that e_iq is its transmission and e_ju is its reception; or 3) there is an event e_ks, such that e_ig → e_ks → e_ju.
      Each event e,-j is related to the time mark V(e_iq) which is n-dimension vector of integers and represents the current logical clock. This clock is updated such that the component V(e_iq) [i] is equal to the number of events in message pacing executed in till (including) e_iq, and the remaining components are equal to the number of similar events “occurred before” e_iq in other processes.
      For eawh pair of reception events e_ks→e_ku in the process pt for which the transmission events are e_iq and e_ju, respectively, the algorithm will identify the deviation if e_ju → e_iq. To do this the known relation is used:.
      The proof for the algorithm correctness is given, the examples of its usefullness are presented.

A.V. Kondrashenko, N.V. Prudov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 55.

      The report presented the physical and mathematical model of a many-region electrochemical cell with account of electrochemical kinetics, interphase transitions, diffusion, convection, self-consistent rnbtion of charged particles in electric field of the cell. Material balance equations are used as the equations for time variations in cell component concentrations. The equation of electric field potential with account of outside electromotive force sources is used for consistent description of the electric held of the cell. The proposed model allows to compute volt-ampere characteristics of various cell types.

V.E. Kondrashov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 55.

      The report proposed purely algebraic method for convolution inversion in contrast to methods such as regularization, quasisolutions, etc., which are based on some variational principles. We give qualitative arid quantitative comparisons with variational type methods. The basic report statement is that for convolution inversion computational difficulties are of algebraic nature and, indeed, in the variational approach they are analyzed quite superficially.

V.B. Kostousov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 55-56.

      The report discussed the problem of terrain stereo model construction using a couple of overlapping aerophotographs (a stereo couple) which is well known in photograph metering. The problem is solved with the algorithm reproducing the human visual system capability to recover the volume image of a scene by two planar images. For the initial data the. algorithm uses several couples of respective points indicated in the images.
      The problem is solved in several steps. First parameters of mutual image orientation are computed, then the images are transformed and reduced to a single plane. Next, the algorithm determines correspondence of all (or most) photograph points through analysis of the images of these photographs. Finally, for the corresponding points found the volume model of a scene is recovered. The algorithm suggested in this work possesses good capabilities of paralleling most labor intensive operations: image transformation and correspondence verification for all points. Quality of the recovered volume scene is assessed using the digital model scene. The report presented actual aerophotograph processing results.
      The work was carried out under the auspices of Russian Fundamental Research Foundation (project N 94-01- 00361a).

V.M. Kryukov , D.V. Mogilenskikh, V.V. Fyodorov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 56.

      Visualization promotes the labor productivity increase and enables the effective and fast analysis of obtained results, as well as duly detection of features during solution of complex two- and three-dimensional problems.
      The existing 3-D-visualization systems are as a rule oriented to specific applications much different from scientific visualization.
      Presentation of the created three- and two-dimensional scientific visualization systems is offered.
      There are practically no scientific visualization systems which are well oriented and adapted for specific mathematical modeling applications. At present, the transition to calculations by three-dimensional techniques is taking place resulting in huge quantity of data and conventional methods are not very effective and involve much time for analysis. Therefore, problems of development and creation of scientific visualization systems become important.
      Not only geometrical properties of models are of interest, but also three-dimensional interpretation of various physical magnitudes through isosurfaces, isolines, applying of color, shades and isolines to geometrical forms.
      Scientific visualization is the combination of systematized tools, methods, operations on geometrical and, first of all, on physical data, as well as on functions of the image, that permits to reflect the behavior and development of physical or any other processes on a monitor screen using computer graphics.
      The following issues were stated in the report:
      1. Main mathematical objects for 3-D-visualization, the data classified according to their structure and physical sense:
      — the data described by structured and unstructured three-dimensional grids;
      — the functions of two variables, set on regular, irregular, structured, unstructured grids, as well as set by lines of constant functions value or by random discrete point set;
      — three-parameter data, that is data set in each volume point;
      — data, domain of definition of which do not tear-apart to a plane;
      — three-dimensional tracks of some processes presented in the form of three-dimensional bent tube;
      — results of two-dimensional calculations, which can be presented as three-dimensional objects with the help of creation of figures of rotation about axes of symmertry or time can serve as the third dimension;
      — analytical functions.
      2. Methods and tools of scientific 3-D-visualization.
      3. Methods of three-dimensional and two-dimensional presentation of physical system evolution dynamics.
      4. Formalization of some concepts of 3-D-visualization.
      5. Principles of reception of three-dimensional objects images in the form of a framework and its application to scientific visualization.
      6. Main functions, interface of developed scientific 3-D and 2-D visualization systems.
      7. Potentials for development of similar scientific visualization systems.
      Illustrations are available.

V.M. Ktitorov, V. Yu. Meltsas
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 57.

      The problem of finding the cases of unstable evolution of the, point blast in perfect gas was actually formulated since 1980’s when the self-similar, approach applied to 2-D linearized hydrodynamic equations showed [1,2] that for the gas adiabatic ratio close to unity (less than 1,20) Raleigh-Taylor instability occurs at the front of such blast when the gas compressed by the shock wave to a thin dense shell is decelerated by the arriving flow relative to low density gas.
      The difficulty in experimental validation of this instability and on the other hand the significance of this result both conceptually and for astrophysical applications motivated the wish to verify it in a numerical experiment.
      This work examines the evolution of 2-D perturbations in the blast with a gas adiabatic ratio close to unity (1,15). The initial values were represented by various sets of 2-D perturbations:
      - perturbations with harmonic number n = 2,4,8,16;
      - linear combinations;
      - bubble perturbation on the axis.
      The computations allowed:
      1. To show that irrespectively of initial conditions the evolution mode of perturbations in a short time (the increase in the shock wave radius by 3-4 times) becomes self-similar further behaving in accordance with [1,2].
      2. To study quantitatively further decay (with the perturbation amplitude increase) of the linear self- similar mode [1,2] followed by the transition to turbulization. The critical amplitudes are found after which the evolution pattern becomes nonlinear.
      3. Ktitorov V.M. // Voprosy Atomnoy Nauki i Tekhniki. Ser. Teoreticheskaya i Prikladnaya Fizika. 1984. N 2(2). P. 28.
      4. Ryu D. and Vishniac E.T. // Astrophys. J. .1987. Vol. 313. P. 820. Vishniac E.T. and Ryu D. // Ibid. 1989. Vol. 337. P. 917.

E.V. Kuznetsova, V.N. Bakulin
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 57-58.

      The problem of porting application software developed for the third and fourth generation computers to a vector- pipeline super-computer was considered using the “Kompozit” application software package implementing the finite element method for strength and building mechanics problem solution.
      The objective of the application software porting is primarily to achieve a considerably, higher speed of computations as maximum possible performance of modern super-computers is considerably higher than that of previous generations. However, when executing most Fortran programs developed for serial computers, as a rule, it is impossible to achieve high performance on a super-computer due to incompatibility of the vector-pipeline architecture algorithms implemented in them which does not allow to use the advantages of the architecture in full measure.
      Many Fortran programs can be adapted to the vector-pipeline architecture with using the “vectorizing compiler” which effectively vectorizes simple and nested loops, as well as inner product type macroinstructions. Other Fortran programs can be adapted to the vector-pipeline architecture using loop rearrangement type local changes. Finally, there are Fortran programs whose adaptation to the vector-pipeline architecture requires more extensive transformations up to complete algorithm reconstruction.
      The window-oriented frontal method for solution of linear algebraic equation systems was discussed which combines the advantages of the frontal and band methods and allows to effectively vectorize the algebraic phases of the “Kompozit” package. Fortran-77 texts of the programs were given which implement the window-oriented frontal method. For vector computations vector subprograms in the “Elektronika SSBIS” super-computer system assembler language are developed which allow to enhance performance and reach super-vector performance at making the Kholesky frontal matrix factorization.

V.D. Liseikin
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 58.

      The variational method was considered for the generation of adaptive grids based on the generalization of the smoothness functional. The studies were carried out for the method properties and its potential use in general- purpose packages. Numerical examples were given.

I.N. Lomov, V.I. Kondaurov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 58.

      The Godunov method and higher order methods constructed on its base are widely used in aerodynamics to solve hyperbolic systems of the laws of conservation. The results found show that the method provides considerable opportunities for solving these problems. Therefore, it is interesting to use these methods to solve condensed media mechanics problems as it is necessary to take into consideration both smooth and discontinuous solutions. The Godunov type methods should be based on a closed divergence form equation system. The divergence equations of continuity, conservation of momentum and energy are well known, but the equation for the symmetric strain tensor can not basically be written in the divergent form and is replaced with the law of conservation of the velocity and strain consistency which uses the asymmetric distortion (strain gradient) tensor [1]. The laws of conservation are closed with the governing relations: the wide-range equation of state and kinetic equations of damageability increase and plastic strains. This approach corresponds to the elastic-viscous-plastic model and removes the problems relating to the impossibility of the full divergent form for the scleronomous elastic-plastic model.
      For that equation system the Riemann problem was solved with method [2] using the parabolic approximation. When solving the Riemann problem between boundary and imaginary cells various boundary condition types may be set. The method is implemented on an arbitrary movable Lagrangian-Eulerian grid. To determine the cells used in the finite-volume formulation of the laws of conservation, the Delaunay triangulation of the computational domain is used.
      The program system developed was used to compute various problems, for example, of high- speed collision, space body motion in the planet atmosphere, high-energy effects, on condensed targets. These computations showed the capability of the method to model shock waves, phase transitions, inelastic finite strains and condensed medium damageability accumulation.
      1. Kondaurov V.I., Nikitin L.V. Theoretic fundamentals of geomaterial reology. ?.: Nauka Publishers, 1990.
      2. Dukowicz J. //J. Comput. Phys. 1985. Vol. 61. P. 119.

V.V. Lunev, D.M. Obuvalin, I.N. Orlov, S.V. Sivolgin
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 59.

      The BLAS library (Basic Linear Algebra Subprograms) is an integral part of the well known LAPACK package intended for numerical analysis applications. The BLAS library includes frequently used linear algebra subprograms. LAPACK is based on block algorithms; the BLAS programs are used as much as possible as the lower level blocks. The library optimization (adaptation) to the specific architecture substantially increases the performance of the LAPACK package as whole.
      Shared Memory Parallelism is oriented to the configuration consisting of several processors with shared main memory. In such systems the memory is global for all processors, the basic workstation was represented by the Intel platform Altera that can contain from one to four Pentium Pro processors interconnected through the shared memory bus. This configuration ensures a good performance of UNIX and Windows NT.
      For the implementation of the shared memory parallelism, the SMP package was created representing a simple and efficient tool for . the development of parallel applications. The basic package advantages include; the dynamic adjustment to the number of processors in the cluster and single-shot use of extensive mechanisms for systems management of subtasks. The package designed for the parallelization of BLAS codes can be also used in any other SMP application.
      BLAS subprograms are divided into three levels depending on operations to be executed. The first level subroutines execute vector operations, the second level subroutines execute, matrix-vector operations, the third level subroutines implement matrix-matrix operations. The development of parallel algorithms was meaningful only for the third level because the routines from the second and especially from the first levels have a much lower amount of computations. The main difficulty in the implementation of parallel algorithms is the need of uniform load of all system processors. The unbalance results in, lower parallelization coefficient. The specific feature and at the same time the bottleneck of the shared-memory systems is the bus shared by all processors that cannot process several results at a time. This factor is the basic restriction prohibiting the growth of the number of processors. To design the parallel algorithms, we had to choose the version that smooth the construction in memory access
      The third level routines operate various matrix types: general, triangular, symmetric. For the review generality the algorithms for general and symmetric matrices were described.
      The report presented the parallelization results. The parallelization coefficient varies from 80 to 99 percent for different routines.

V.G. Morozov, L.I. Karpenko, L.V. Dmitrieva, N.V. Korepova, T.L. Grebennikova, S.S. Sokolov, A.D. Kovtun, V.A. Komrachkov, L.A. Tolstikova, Yu.M. Makarov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 59.

      The report presented the results of numerical simulation of X-ray recording detonation initiation and evolution in a plane HE layer from a linear initiator located on the surface of a plane layer confined with a steel plate from the opposite side. When the diverging wave is reflected from a steel barrier a shock or detonation wave is generated (depending on HE layer thickness) which propagates along the barrier surface. With decreasing HE thickness a delay in detonation propagation along the layer is observed. This effect is a significant characteristic of shock-wave HE sensitivity and accuracy of its simulation which was made by 2-D gas-dynamical programs using the detonation kinetics model. The computed result reproduces the experimental picture fairly well.

V.A. Novichikhin, A.V. Pastukhov, V.A. Gusev, A.M. Lyakishev, G.A. Popovidchenko, A.A. Runich, I.D. Sofronov, S.A. Stepanenko, V.N. Timchenko, A.A. Uzentsov, A.A. Kholostov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 60.

      The MIMD multiprocessor MP-X (various instruction threads over various data threads) consisting of 2^x- processing elements is a functional module designed for parallel computations.
      The MP-X rests on the following concepts;
      — combination of the MP-X multiprocessor with the general purpose host computer Y allowing to combine the advantages of parallel, vector and scalar, processing when executing a program that contains non-parallelizable fragments;
      — adaptibility of connectivity structure to the algorithms of the applications to be run (program emulation of various interprocessor connections);
      — design modularity which allows to upgrade the performance according to the user demands a;nd simplifies the transition to new microprocessors and new message passing technologies;
      — hardware support of the routing algorithms to increase the efficiency of the interprocessor transfers;
      — higher failure tolerance level of MP-X due to redundancy;
      — orientation to the common programming languages with the support of parallel program development tools standard in this field.
      The most efficient applications for MP-X-Y include: multidimensional problems in computational physics, design computerization, image processing, artificial intellect and others that require a maximum possible number of operations for this class of Y computers.
      The concepts presented are used in MP-0, MP-3 models and serve the base for MP-7 project.
      The description is given for the component base of existing and future models of some MP-X, as well as the software used for the development of parallel codes. The results are given for some test problems on various types of similar computers including MP-0 and MP-3.
      The report was accompanied by the illustrations arid demonstration of operational models.

V.V. Partsevsky, A.E. Efimov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 60.

      A range of problems relating to residual static and dynamic strength of stratified orthotropic plates with exfoliation type defects is studied. Such defects are characteristic of composite elements due to relatively weak layer interfaces. Exfoliations occur during manufacture, operation, especially at shock loads, storage.
      The Reissner type theory was used as a basis to construct approximate solutions in series for the plate regions including exfoliations. Solution convergence in energy was studied. Comparison with
      The results for transversally isotropic plates which allow solutions accurate within the theory is given.
      The energetic approach was used to study the conditions and character of exfoliation fracture growth in arbitrary orthotropic plates. The dependence of fracture growth direction on the anisotropy degree, exfoliation value and location was, in particular, found. The conditions of exfoliated part clicking-out at static and pulsed plate loading with pressure were also studied.

L.A. Pozdniakov, M.Yu. Khramtsov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 61.

      For the development of the portable parallel programming system for distributed memory multiprocessors, a new approach was chosen providing a much higher stability level of parallel programming environment as compared to the library approach.
      The following basic Fortran GNS properties served the base for choosing it for the development of such systems:
      - dynamic generation of subtasks;
      - addressing the subtasks using the dynamically generated identifiers;
      - coupling between each subtask and the remaining ones;
      - three interaction ways for subtasks based on message passing: synchronous, asynchronous, wait-free;
      - data specification for subtasks communications using a standard Fortran I/O list.
      The programming system based on Fortran GNS is composed of the following elements:
      1. A converter transforming Fortran GNS to Fortran-77 codes using the procedures of library support for Fortran GNS.
      2. Systems support libraries for Fortran GNS providing parallel execution. These are available to users as library files coming to the linker input together with the object user modules.
      3. Configurator forming the logic task configuration and mapping onto the physical configuration of the computing system.
      4. Systems tools supporting the subtask triggering mechanisms and message passing.
      These are available to the users as final modules-coming to the configurator input. The implementation of Fortran GNS system on a specific platform would rest on the existing Fortran-77 version for the given platform and standard parallel execution tools.
      The work was carried out under the auspices of Russian Fundamental Research Foundation (project N 96-01- 00493).

R.Z. Samigulina
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 61-62.

      The evolution of multiprocessors allowed to develop new- methods for the calculation of scientific applications; and to adapt the existing codes to parallel computations.
      The report presented a parallelization algorithm for 2-D Lagrangian method (on a regular grid) within “D” complex [1,2] in a single-domain geometry on the 8-processor distributed-memory MP-3 system [3]. The gas- dynamic code of “D” complex combined with RND code represents a relatively complicated product whose significant modification may require much labor and time. Therefore we use the parallelization algorithm allowing to port these codes to a parallel computer with minor modifications or, without any modification.
      The test computations were run to evaluate the algorithm efficiency. The report presented the computational results illustrating the speedup and efficiency of the parallel algorithm.
      The work results indicate that the parallel computations allow to reduce considerably the computational time for gas-dynamics with the increase in computational cells. The approach used can be also applied to the multidomain case.
      1. Dmitriev N.A., Dmiirieva L.V., Malinovskaya E.V., Sofronov I.D. The method for the calculation of 2-D nonstationary gas-dynamic problems in Lagrangian variables // Theoretical Fundamentals and Construction of Numerical Algorithms in Computational Physics / Ed. by Babenko K.I., M.: Nauka, 1979. P. 175-200.
      2. Artemiev A.Yv., Bashurova M.S., Delov V.I., Dmiirieva L.V., Samigulina R.Z., Senihva O.V., Chernyshev Fu.Z. Applications package “D” for the calculation gas-dynamic problems in Lagrangian variables and deforming solid problems on regular grids // Zababakhin Scientific Conference (report summary). 1992. P. 41- 42.
      3. Voronin B.L., Skrypnik S.I., Kozub A.G. et al. RAMZES package for the calculation of gas-dynamic heat conduction problems // Computer Simulation Technology / Ed. by V.P. Il’in Novosibirsk: Computing Center of Siberian Division of the Soviet Academy of Sciences, 1989.

V.V. Samofalov, A.V. Konovalov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 62.

      One of the ways to solve the parallel «program debugging problems is clear-cut understanding of the role of models in debugging and distribution of debugging actions over several levels. The following debugging levels may be distinguished:
      — program model debugging;
      — real program debugging in the model (pseudo-parallel) execution mode;
      — debugging of a real program run on a real machine with using a route.
      Extremely important is control over data flows between the levels and provision of their feedback.
      The model debugging allows to develop a program structure, assess its effect on the performance and stability without any details which are not needed at that stage.
      Debugging in the pseudo-parallel execution mode secures fixing the bugs introduced at the transition from a program model to a real program. At this stage the developed debugging tools existing on PC are extensively used.
      The route operation tools may be broken into two groups. The first group systems secure program operation visualization using various multimedia types, in so doing the burden of decision making regarding the bug reasons and the ways to fix the bugs is entirely placed on the programmer. The second group systems are oriented to development of advices and recommendations for the programrner. Presently the first group tools are most extensively applied. In our opinion, the main way for their development is the route analysis based on the program model notion.
      To support the discussed technology, the authors are developing the T-model system. One of its structure-forming ideas is that of integration of diversiform parallel and sequential programming tools.

V.V. Shumilin, V.M. Vikharev, S.A. Gvozdeva, S.I. Sapronov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 62-63.

      Following issues were considered:
      1. MPI represents in attempt to standardize the message passing on distributed memory with the implementation approaches are as follows.
      2. Low level functions of the communications system. Their adaptation to meet the MPI requirements. The set of the lower level primitives, data transfer from the continuous buffer.
      3. The implementation of point-to-point transfers using the lower level primitives. The reasons for the selection of this implementation.
      4. The selection of a model for the implementation of communications spaces based, on the hardware specifics. Recording node and its functions.
      5. Generation of MPI service functions.
      6. Approaches to the development of this implementation.

A.F. Sidorov, O.B. Khairullina
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 63.

      The results of study series on one- and multi-dimensional shock-free gas compression processes leading to unlimited increase in density and energy cumulation are summarized in paper. The problem of construction of laws for control over such compression leads to unconventional formulations of boundary problems for gas dynamics equations. In these problems laws of motion of movable compressing pistons and pressure distributions in them have to be found by given flow properties (entropy constancy, unlimited density growth).
      The report presented results of research into new options of control over unlimited compression, as well as new estimations of physical parameters for some earlier discussed compression processes.
      A combination of analytical and numerical approaches was used to construct laws of control over compression of cone-shaped gas volumes at an arbitrary taper angle a and adiabatic exponent γ (the inconsistent case). Estimates for the constants of energy spent for compression are found in asymptotic representations.
      Non-self-similar processes are studied to obtain unlimited growth in gas density at prism and tetrahedron compression. A powerful cumulative jet is shown to generate at coupling of arbitrary non- self-similar onedimensional unlimited compression flows. In this the degrees of cumulation of the basic gas-dynamical values are the same as at unlimited self-similar prism compression (in two dimensions) and do not depend on the onedimensional compression control laws, Thereby it was found that the flow field in the severe compression region is of independent character, while the multi-dimensional compression process is stable with respect to perturbations of one-dimensional flow fields.
      Dynamics computations of gas involvement in the super-strong compression process were made, asymptotic estimations of compressed gas optical thicknesses and their dependence on the spent energy values in 2-D planar and axisymmetric cases were constructed.
      The work was carried out under the auspices of Russian Fundamental Reseach Foundation (project N 96-01— 00115).

I.D. Sofronov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 63-64.

      Following issues were considered:
      1. A brief characterization of the codes developed in 1985-1995:
      - 1-D codes;
      - 3-D codes;
      - models for the processes considered.
      2. Some features of the developed computational technology:
      - inadequate performance of computers and requirements for the methods, computers and mathematicians;
      - a low number of cells, computational credibility, various computational techniques;
      - full-scale tests — a necessary component at the computational technology.
      3. Some requirements for the new technology:
      - more accurate models for physical processes;
      - user-independent computations;
      - convergence computations, computational validity;
      - mathematical testing;
      - development of new algorithms allowing a sufficient parallelization.
      4. Technical base:
      - multi-user computer network;
      - up-to-date state of computing base;
      - various classes of machines within the new ensemble of computers;
      - parameters of the desired supercomputer;
      - various ways for computerization.

I.D. Sofronov, B.L.Voronin, O.L. Butnev, A.N. Bykov, S.L. Skrypnik, D. Nielsen, Jr. D. Nowak, N. Medsen, R. Evans
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 64.

      Numerical calculation of 3-D problems requires the threshold computing resources provided by massively parallel distributed memory systems. A successful use of a high potential performance of such systems for the calculation of a single problem is only possible after the development of application programs supporting the parallel processing.
      The paper presents the results of the parallelization efforts for 3-D heat conduction equation. The basic method for numerical calculation of 3-D implicit finite-difference equations is the directional splitting. This method allows to reduce a complicated multidimensional problems to a collection of simpler problems that can be run on parallel processors.
      Two conceptually different approaches were developed for the organization of massively parallel computations. The first uses the decomposition of 3-D data matrix reconfigured at a timestep and is the development of the parallelization algorithms for the shared-memory multiprocessors. The second approach rests on nonreconfigurable decomposition of the 3-D data matrix.
      The resulting algorithms were implemented as a parallel code for massively parallel distributed-memory systems.
      The series of computations allowed the numerical studies of the parallelization efficiency for various techniques of the geometrical decomposition, for two modes of processor loading and depending on arithmetic execution/communications ratio.
      The quantitative estimates are given for the parallelization of the resulting algorithms obtained on MP-3 and Meiko systems.

VI.V. Voevodin
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 65.

      It is commonly adopted opinion that parallel computers are very hard to be used efficiently. The announced values of performance are high but-in practice obtained results look less attractive.-Many scientists have already experienced serious difficulties in programming for a such kind of computers.. These, difficulties are easy to understand since a lot of new and unusual problems occur almost everywhere: what is parallelism, how to; detect it; how one. should express parallelism in a high level language, how.to make a performance closer to a peak (or theoretical) performance of a computer, can we use efficiently a large existing resource of algorithms and programs, and how to guarantee that the efforts which have been spent to develop parallel software will not have to be applied again and again with the emergence of computers.
      This report presented different aspects of program analysis and optimization for up-to-date parallel computers. The first part describes the theoretical basis of so called V-Ray technology directed to comprehensive analysis of programs. This technology is a set of mathematical methods and algorithmic approaches designed to facilitate investigation and transformation of the structure of serial algorithms and programs. The V-Ray technology is developed on the basis of the strict theory and provides a basis for resolving the whole scope of problems related to mapping of real world applications to parallel computers starting from the visual analysis of program structure and detection of data dependencies, description of the total resource of parallelism, search for potential bottlenecks in programs, up to analysis of possible data distributions and analysis of data locality.
      The second part outlines the functionality of the V-Ray system designed on the principles of the V-Ray technology. The main goal of the system is to provide powerful tools for performing comprehensive analysis of program structure. Since the reasons for low performance are different, the system supports investigation and evaluation of different properties of programs on all levels: from an interprocedure level up to individual loop iterations.
      The final part of the report describes our experience in porting several programs to massively parallel (CRAY T3D, IBM SP2) and vector-parallel (CRAY Y-MP C90) computers. Results and the main stages of this process are shown. For instance, , the detection of the total resource of parallelism hidden in the program for modeling large-scale magnetic fields in galaxies enabled us to attain 7-fold speedup on the CRAY Y-MP C90 computer.

B.L.Voronin, A.M. Erofeev
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 65.

      Currently the highest performance is demonstrated by massively parallel computer systems with a large number of processors. The efficient use of high potential performance of these machines requires the development of parallel methods for the calculation of complicated multidimensional applications. An example is represented by 3-D heat conduction equation. One of the ways to calculate the multidimensional heat conduction equation is the directional splitting combined method with the implicit scheme. The resulting equations are solved by funning.
      The most natural approach to the parallelization of numerical solution in this case is a geometrical technique where each computational domain is split into subdomains each being computed by a designated processor. Multiple papers are devoted to the organization of region-by-region computations. At this point, there are many schemes for sequential region-by-region computations. The first one was that of Zaguskin-Kondrashoy. Later some authors developed the schemes with re redundant stability. This work considers some parallel schemes for region-by-region; calculations including the scheme that uses the running parallelization algorithm proposed in a paper by Yanenko. The studies were performed for the computational stability and accuracy, the conclusions are made relative to .the potential use of the schemes for the organization of massively parallel computations.

A.I. Voropinov
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 66.

      The report proposed the modification of the rigid-plastic model for the calculation of plastic flows. The model is based on the introduction of the regularization addend according to Tikhonov method into the formula for the calculation of the stress (expressed via the deformation rate). The proposed model was used for the computations. The .comparison with the elastic plastic model results showed that .the method is well suited for strong deformation domain.
      It is interesting that the model demonstrates the relaxation property, so that the computations can run without artificial viscosity.
      The regularization addend can be chosen from some physical characteristics of the application or by relating it to the computational quantities.

V.A. Vshivkoy, G.I. Dudnikova
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 66.

      The problems of formation and interaction between collisionless shock waves in plasma are important for the outer space processes such as Sun flashes, Supernova fragments dragging by inter-Galaxy medium sun wind flow-over Earth magnetic sphere etc.
      Numerical simulation method was used to study the interaction dynamics bf shock waves resulting from the dragging of dense plasma clouds expanding from one space point with a time delay. The 2-D axisymmetric hybrid plasma model is based on the kinetic description of plasma component and hydrodynamic approximation for electrons. The numerical implementation was accomplished using the particle-in-cell method for the calculation of kinetic Vlasov equation and finite-difference splitting schemes for Maxwell equations. The shock amplitude dependence on initial energy of plasma clouds, parameters of magnetized background delay time is obtained. The conditions are found when the second plasma cloud can expand without perturbation generation in plasma.
      The work was carried out under the auspice of Russian Fundamental Research Foundation (project N 94-01- 00112).

V.A. Vshivkov, G.I. Dudnikova
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 66-67.

      The propagation of strong laser pulses through plasma is the source of multiple problems in nonlinear optics that represent a great interest in the general physics sense and for various applications. Currently extensive studies are underway for the interaction of super short pulses with plasma in connection with new methods for charged particle acceleration.
      The paper presents the numerical simulation results for the wake acceleration of particles based oh 3-D relativistic kinetic code. The source computational model includes Vlasov equations for ion and electron plasma components and the complete system of Maxwell equations. The kinetic equations are solved with the particle-in-cell method the finite-difference method is used for electromagnetic field equations.
      The results presented show that the pulses shorter than the plasma wavelength excite a regular wake wave in plasma with the electric field accelerated by electrons. In addition, the interaction of relativistically strong laser pulses with rarefied plasma demonstrates various nonlinear process types: fast absorption of the pulse energy, energy transformation and other wave types, variations in electromagnetic radiation frequency, shock wave formation.

V.A. Vshivkov, G.L. Dudnikova, M.A. Kraeva, V.E. Malyshkin
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 67.

      The particle-in-cell method (PIC) is commonly used for the calculations in collisionless plasma physics where a great importance is given to the interaction through electromagnetic fields.
      The PIC method represents plasma as a set of model particles with the motion trajectories being the characteristics of Vlasov equation.
      Since the evaluation of new velocities and coordinates of the particle does not directly depend on the other particles the problem is well suited to parallelization. However depending on the technique for data distribution (particles, and field values at the grid nodes) among the processors we deal either with nonuniform loading of processing elements (PEs) or high communications overhead. These problems appear in the case of nonuniform distribution of particles in the modeling space. Thus the parallel code textually depends on the law of particles distribution in the modeling space. In addition, the decision about the data and computation distribution among the processing elements depends on the pattern of particle expansion, their amount, grid size, the difference between sequential algorithms and of course on the computer system architecture, particularly on PE memory size.
      The work presents the development results, for the programming system intended for the implementation of various PIC method versions. The work is performed within the ASSY project (ASsembly SYstems). The project is oriented to the development of the metasystem supporting the development of problem-oriented programming systems. PIC method is one of the problems used to verify actual capabilities of ASSY technology.
      The work was carried out under the auspice of Russian Fundamental Research Foundation (project N 95-01- 01358) and the Commission of the European Communities (grant ITDC-203-822165).

V.A. Yeroshenko, A.V. Kondrashenko
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 67-68.

      Computations for short pulse amplification in active medium with degenerate laser levels (such medium is atomic iodine) often use quasi-classical model which is an extension by analogy of equations for a nondegenerate two-level system. For each permitted transition its own polarization is therewith introduced, while the inverse population is written with account of statistic, weights of the degenerate levels. This model is not quite consistent and computations on its base sometimes yield physically senseless results (negative level populations). A consistent model should be based on equations for a complete density matrix. For the iodine atom this matrix is of 36 x 36 dimension.
      This work studies effects relating to transition from the complete model to the two-level level-degenerate model using a model three-level system with the degenerate lower level as an example. These studies showed that for a pronounced coherent mode only the complete model provides physically proper results. At the same time, when intrusion in the coherent region is not very deep, the two-level approximation yields very good results.
      In the experiments on powerful iodine laser facilities “Iskra-4” and “Iskra-5” in some, cases pulse duration is on the order of the phase relaxation time. Thus, at least to control computation accuracy, as well as in order to be able to compute laser pulse amplification with constant magnetic field present, it is desirable to have the laser pulse amplification model based on the complete iodine atom density matrix of 36 x 36 dimension. This model was implemented in the ZSK program.

S.V. Zybin
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 68.

      Parallel algorithms were considered for the integration of Cauchy problem for the system of ordinary differential equations implementing the large-scale time parallelism by using the iterative processes, particularly, the algorithms based on the discrete convolution.
      The parallelism of these algorithms rests on the transformation of the original ordinary differential equation to the equivalent integral equation and subsequent calculation through the iterations in time subintervals called blocks. The length of these subintervals is usually greater than the step of the grid used to calculate the integral which allows to achieve the large-scale parallelism where simultaneous calculations run in each iteration for the integrand at the grid nodes. These methods were not previously used on a single processor since they require O(n2) operations where n is the number of grid nodes. The emergence of parallel computers changed the situation, since only O(logn) operations are required on n processors for the iteration methods.
      The majority of iterative parallel algorithms also called “waveform relaxation methods” focus on block parallelization or when the integrand is calculated. This paper mainly considers the algorithms allowing additionally to use the potential parallelism of the discrete convolution. These algorithms rest on the approximate iterative Newton- Kantorovich method and are similar to the algorithms such as shifted Picard splitting. The basic idea is to solve linear ordinary differential equations at each iteration by evaluating the convolution integral. Upon the discretization with Gregory formulas the convolution integral can be calculated with parallel algorithms of the discrete convolution or using special processors (signal or pipeline).
      The theoretical results obtained include the studies of the convergence properties, approximation and stability of the algorithms used the estimates of the convergence rates of iteration processes used. The iterative processes were studied with the piecewise-constant Jacobi approximation (basic and modified) and the process with internal iterations. The structure was also developed for their parallel implementation and the family of parametrized algorithms is constructed with the hierarchy of the parallelism levels.
      These results allow to develop efficient parallel algorithms for the integration of ordinary differential equations based on Picard iterations (without convolution) and approximate Newton-Kantorovich iterations (with potential convolution use). The studies were performed for their computational properties on a set of different test ordinary differential equations the estimates are obtained for the acceleration and parallelization efficiency. The algorithms were also implemented as NORMA program to be translated to parallel Fortran GNS. The preliminary testing was accomplished for the generated program with the allocation of tasks over processors and support of communications on a parallel computer with i860XP processors.

F. Beatke
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 68-69.

      The computer industry is experiencing a major paradigm shift which is clearly visible in the trend towards microprocessor based system even at very high end. The niches still ???u?ied by proprietary CPU designs like vector-processors ?re under attack from a new class of RISC designs which are capable to access main memory with long sequences of vector-like load sequences.
      Different evaluation criteria and performance characteristics have been defined to compare and analyze parallel systems from the level of the processors over the interconnect technology and the memory subsystems up to the features of the operating system and its capabilities.
      Hewlett-Packards new Exemplar SPP series is being evaluated within the contexts defined above with a special emphasis on:
      - PA-RISC 7200/8000 memory access capabilities compared to competing architectures and utilization in the SPP-series;
      - interconnect technology at node and system level and performance metrics like bandwidth and latency;
      - features of the micro-kernel based SPP-UX operating system;
      - features of the parallelizing compilers for C, C + + and Fortran.
      The future development of the Exemplar SPP-series which has established with the SPPlxxx products and is now being continued with SPP2000 will finally be discussed. Additional information on measured and expected performance for test-examples and real industrial applications were provided.

D. Barrett
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 69.

      The presentation describes a parallel code for solving neutron transport problems in 3-D geometry: domain decomposition message passing.
      Measured time characteristics are given. A prediction is made for further development of these works.

Christensen Randy B.
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 69.

      The report presented a method for obtaining high-order non-oscillatory results for staggered mesh (velocities at cell boundaries and thermodynamic variables at cell centers internal energy-based hydrodynamics codes using a simple modification of Artificial Viscosity technique. This modification is derived by applying the Godunov method directly to a staggered mesh and replacing the full Riemann solution by a particular approximate Riemann solver. It was shown to deliver results comparable to cell-centered total energy based FTC, TVD and high order Godunov methods.

Dale E. Nielsen
VANT. Ser.: Mat. Mod. Fiz. Proc. 1997. No 1. P. 68-69.

      The high-performance computing community has seen a steady procession of different parallel computing architectures. For large application codes the time required to change from sequential algorithms to parallel algorithms, or even change between parallel algorithms for different architectures is longer that the lifetime of individual parallel computers. Smooth transition strategies were described which overcome this challenge, and the parallel machine characteristics defined which make these transition strategies possible.