50 years of the development of grid-characteristic method

MIPT & ICAD RAS || March 31 - April 3, 2018

**Petrov I.B.**

The scientific way of the academician Kholodov A. S. Development of а grid-characteristic method

AbstractIn the report, there is a speech about the scientific activity of the academician of the Russian Academy of Sciences, professor of Moscow Institute of Physics and Technology Alexander Sergeyevich Kholodov. Also in the report, it is told about development and applications of a grid-characteristic method

**Chetverushkin B.N, Golovizin V.M.**

New generation algorithms for computational fluid dynamics

AbstractA new approach for creation of CFD algorithms, linking characteristic methods and conservation laws is discussed.

**Gulyaev Yu.V.**

Radiophysical digital methods in biomedicine: achievements and challenges

AbstractAt present in connection with the active development of radiophysical methods and tools, we get the absolutely new great possibilities for application of these approaches in biomedicine. The devices based on new physical and technological principles for biomedicine have been developed. Human body is a dynamic self-governing system whose stability (homeostasis) provided by simultaneous operation of distributed physiological systems for neuroregulation, blood circulation, metabolism, etc. Continuous operation of all life support systems is reflected in real time on one side in the complex picture of the physical fields and radiation emitted by the human body, and on the other side in the parametric changes of natural background fields and radiation, that normally surround a person. Application of the actual radiophysical methods for precise measurements and dynamic mapping of these fields, radiation and changes of the background field gives us the possibilities for development of new methods for non-invasive medical diagnostics that is a very important component in preventive medicine. One of the clear example is the development of a quasi-static electric impedance, electric field and magnetic field computer tomography for a human body. The results of the application this type tomography allow to allocate the changes in tissue electric impedance that take place in pathological tissue. Mentioned changes come much earlier than the changes in tissue density that could be registered by regular X-ray Computer tomography. So, mentioned new technology could be very useful for early medical diagnosis. The other clear example is the development of infrared thermography for early differential medical diagnosis. It is well known, that the aggregation of cancer cells has a little bit higher temperature in comparison with the neighbor healthy cells. So, it is necessary to make precise measurements of the temperature to allocate the dangerous area. The special devices already developed and applied successfully. This approach provides early medical diagnosis. Actual radiophysical methods could be successfully applied not only for medical diagnosis but for treatment also. Description of the experimental equipment for precise measurements and medical diagnosis examples are presented in the report.

**Tyrtyshnikov E.E.**

Development and application of methods of the low-rank approximation of tensors

AbstractThe methods of the low-rank approximation of matrices and tensors have been developing rapidly for several decades, while in the most successful methods the tensors are transformed into matrices associated with them and the methods of low-rank approximation for matrices are actively used. The report will present the current state of the science field, in particular, new results [2,3], recently obtained and substantially developing methods proposed in [1]. Among the most interesting new applications, we note the methods of global optimization in docking problems [4] and methods of increased accuracy for solving the type of Smoluchowski type [5]. [1] S. Goreinov, E. Tyrtyshnikov, N. Zamarashkin, A theory of pseudo-skeleton approximations, Linear Algebra Appl. 261 (1997) 1-21. [2] A. Osinsky, N. Zamarashkin, Pseudo-skeleton approximations with better accuracy estimates, Linear Algebra Appl. 537 (2018) 221-249. [3] A. Osinsky, Probabilistic estimation of the rank 1 cross approximation accuracy, arXiv:1706.10285 (2017). [4] A.V.Sulimov, D.A.Zheltkov, I.V.Oferkin, D.C.Kutov, E.V.Katkova, E.E.Tyrtyshnikov, V.B.Sulimov, Tensor Train Global Optimization: Application to Docking in the Configuration Space with a Large Number of Dimensions. -Supercomputing. Third Russian Supercomputing Days, RuSCDays 2017, Moscow, Russia, September 25–26, 2017, Revised Selected Papers, Ser. Communications in Computer and Information Science (CCIS), Springer Cham, vol. 793, p. 1-532 (2017). [5] S.A.Matveev, P.L.Krapivsky, A.P.Smirnov, E.E.Tyrtyshnikov, N.V.Brilliantov, Oscillations in Aggregation-Shattering Processes, Physical Review Letters, 119 (26), p. 260601 (2017).

**Dymnikov V.P.**

Modeling of the Earth system dynamics

AbstractThe central trend in the development of climate models is currently the transition from climate models to models of the Earth system through the inclusion of biological and chemical processes, upper layers of the atmosphere and ionosphere in climate models. At the same time, the development of computer technology makes it possible to complicate climate models within the framework of describing regional processes that require high spatial resolution. The report examines the current state and perspective directions of the development of the national model of the Earth system, as well as some approaches to solving one of the most important tasks-the sensitivity of the climate system to small external influences.

**Nikitin I. S., Burago N. G.**

Explicit-implicit schemes for solving the problems of the dynamical isotropic and anisotropic elastoviscoplasticity

AbstractA method for the numerical solution of the dynamic equations of elastoviscoplasticity is developed for isotropic and anisotropic cases. Anisotropic elastic-visco- plastic systems of equations often deduced from models of deformable solid media with a discrete set of slip planes layered, block media) and with a nonlinear slip condition at contact boundaries. In all these cases the defining relations of the system include equations with a nonlinear free term and a short relaxation time of the stresses. For a stable numerical solution of the system, an explicit-implicit method is proposed with an explicit approximation of the equations of motion and an implicit approximation of the defining relations containing a small parameter in the denominator of the free term. Various effective formulas for the stress components correction on after the "elastic" step of the calculation were obtained using the method of expansion with respect to a small parameter. Examples of numerical solutions of dynamic problems are given.

**Son E.E., Aksenov A.A., Zhluktov S.V., Gavrilov A.D., Son K.E.**

Physical and mathematical models and computational simulation of hypersonic flight vehicles

AbstractThe analysis of the current state of development of hypersonic aircraft (HSA) in countries that have national development programs for HSA, experimental data, physical and mathematical models and software packages used for modeling HSA has been carried out. Models that take into account the aerodynamics and thermal protection of HSA are proposed, numerical models for implementing the simulation of HSA both in laboratory conditions and in modes of free flight in the upper atmosphere are proposed. Optimization algorithms for the design of intake, hull and power plant have been created. Models and algorithms are implemented in the package "Flow vision" and presented in the form of ready-to-use modules for calculating HSA. Models have implemented equilibrium and nonequilibrium models of chemical reactions in the bulk and on the surface of the HSA. Models of partial local chemical equilibrium (PLCE) are based on minimizing the free energy and finding the Lagrange multipliers corresponding to PLCE. In contrast to most works on the modeling of HSA, exact methods of accounting for multicomponent diffusion and chemical reactions are used for different degrees of wall catalytic activity. The destruction and ablation of the HSA surface and the change in the aerodynamic shape of the body, associated with ablation, were calculated using the method of phase fields. The results of direct numerical statistical modeling based on the Boltzmann equation on the "SMILE" package (Ivanov M.S., Bondar Ye.) were used for the arcraft descent into the atmosphere from ballistic trajectories. The results of numerical simulation were used to design the HSA. Degtyar V.G., Son E.E. Hypersonic flying machines, M., "Janus K", 2017, vol.1, 980 p.

**Krasilnikov A. V.**

In memory of A.S. Kholodov. Influence of catalytical properties of the surfaces on heat flow in diccosiated gases**Konovalov A.N.**

Effective explicitly solvable mathematical models for linear parabolic problems

AbstractEffective explicitly solvable mathematical models for linear parabolic problems are constructed and substantiated. An initial-boundary parabolic problem with which, for example, a real physical process of heat propagation in a bounded phase volume is connected is considered. The specified parameters of the physical process under study are the following scalar functions: density, heat capacity, thermal conductivity and specific density of internal sources (sinks) of heat. The resulting parameters are a scalar temperature function, a flow vector, and also a flux density vector. The relationship between the resulting parameters of the process under study is determined by the law of the amount of heat conservation, the equation of state, and the governing equation. An approach is proposed for modelling a particular material that has different aggregate states. An approach is proposed for constructing a uniform spatial grid that is consistent with the boundary. The fulfilment of the conjugate-consistent approximation condition is justified. The constructed models allow us to compare the standard classical approach, based on the conservation laws, with various approximate models of heat propagation in thin films.

**Lazareva, G. G.**

Calculation of heat sink under pulsed heat loads using economical explicitly solvable mathematical model

AbstractOptimal explicitly-solvable mathematical model for linear parabolic problems [1,2] used for calculating the heat sink under pulsed heat loads near the surface tungsten plates of the diverter. The experimental and numerical simulations of the conditions causing the intensive erosion and expected to be realised infusion reactor were carried out. The influence of relevant pulsed heat loads to tungsten was simulated using a powerful electron beam source in BINP [3]. The laboratory experiments are accompanied by computational ones. The computational experiment allowed to quantitatively describe the overheating near the cracks, caused by parallel to surface cracks. The approach used to simulate a specific material, which has a different state of aggregation [2]. The mechanical destruction, melting and splashing of the material were observed. The calculations confirmed the hypothesis that heterogeneity of the surface temperature of tungsten associated with the presence of subsurface cracks. The schedule of the boundaries of the melt-solid tungsten was obtained in the axially symmetric formulation. If the evaporation process is calculated, the graph exactly corresponds to the results of laboratory experiments. 1. Konovalov A.N., Popov Y.P. Explicitly Solvable Optimal Discrete Models with Controlled Disbalance of the Total Mechanical Energy for Dynamical Problems of Linear Elasticity // Siberian Mathematical Journal 56, N 5, 2015, P. 872-878. DOI: 10.1134/S0037446615050092 2. Konovalov A.N. Optimal, an explicitly solvable mathematical model for linear parabolic task // Abstracts. International conference “Mathematics in the modern world”, 14-19 August, Novosibirsk (in Russian). 3. Arakcheev A S, Burdakov A V, Trunev Y A, Kandaurov I V, Kasatov A A, Kurkuchekov V V, Mekler K I, Popov V A, Shoshin A A, Skovorodin D I, Vasilyev A A, Vyacheslavov L N 2016 AIP Conf. Proc. Heating of tungsten target by intense pulse electron beam 1771 060016

**Shishlenin M. A., Kabanikhin S. I.**

Linear regularisation of data of the area-based systems of seismic measurements

AbstractCurrently, due to the area systems of observations, it is possible to create a fundamentally new method of solving three-dimensional inverse problems, which are used: three-dimensional analogues of the equations of Gelfand-Levitan-Krein, parallel computing on high-performance clusters, methods Monte-Carlo, fast algorithms for the inversion of block-Toeplitz matrices of large dimensions. The main problem of the investigations of three-dimensional elastic media is the large size of the area in which high-precision calculations are produced. Even for a relatively small area of 2 km * 2 km * 2 km the solution of the direct problem of seismic prospecting is a very difficult. Let us note that the most of the modern methods for solving inverse problems are based on iterative procedures and even the number of operations required for carrying out multiple iterations, may lead to uncontrolled errors. The work was supported by Russian Foundation for Basic Research (projects NNo. 16-29-15120 and 15-01-09230).

**Rakesh Khare, Nikunj Binnani**

Probabilistic Seismic Hazard Analysis of Punasa Dam Site in India

AbstractSeismic hazard assessment is a procedure to assist effective designing of structures located in seismically active regions. Traditionally, in any seismically active region as the seismic hazard evaluation was based primarily on the historical seismicity, and to a lesser extent based on the consideration of the geological information. The importance of the geological information in seismic hazard assessment is significant, for the reason that earthquakes occur on faults. In the present work, we have examined the Probabilistic seismic hazard for the specific site of Punasa Dam, considering the seismically active faults. The fault considered in this investigation consists of 35 active faults out of total 126 faults, for which a minimum amount of information was available (i.e. length, maximum magnitude etc.). For this purpose, an area of radius 500-550 km is selected. There are 154 earthquakes (moment magnitude 3.5 to 6.6) occurred in this region during last 210 years from 1808 to 2017. Completeness of this data is checked by Stepp’s method. For this data Gutenberg-Richter (G-R) parameters (a and b values) are estimated by regression analysis a=2.363 and b=0.605. The seismic hazard for the Punasa Dam is estimated by the deductive approach of probabilistic seismic hazard analysis. In the deductive approach, it is assumed that seismic sources will cause seismic hazard at the site. Seismic hazard in the form of PGA having 2 % probability of exceedance in 50 years (return period 2500 years) and 10% probability of exceedance in 50 years (return period 500 years) at Punasa Dam, Khandwa is estimated by deductive approach considering various active faults within the radius of 550 km. The seismic hazard was quantified regarding seismic hazard curves for the region of interest. Finally, the results of this thesis were compared with the Indian Seismic Code (IS 1893:2002 Part I)."

**Shuhua Zhang**

Modeling and computation of energy efficiency management with emission permits trading

AbstractIn this paper, we present an optimal feedback control model to deal with the problem of energy efficiency management. Especially, an emission permits trading scheme is considered in our model, in which the decision maker can trade the emission permits flexibly. We make use of the optimal control theory to derive a Hamilton-Jacobi-Bellman (HJB) equation satisfied by the value function, and then propose an upwind finite difference method to solve it. The stability of this method is demonstrated and the accuracy, as well as the usefulness, is shown by the numerical examples. The optimal management strategies, which maximise the discounted stream of the net revenue, together with the value functions, are obtained. The effects of the emission permit price and other parameters in the established model on the results have also been examined. We find that the influences of emission permit price on net revenue for the economic agents with different initial quotas are quite different. All the results demonstrate that the emission permits trading scheme plays an important role in the energy efficiency management.

**Jain Lakhmi**

Intelligent data processing paradigms and real-world applications

AbstractThe main paradigms for generating intelligence in machines include artificial neural networks, evolutionary computing techniques, chaos theory, fuzzy systems, intelligent agents, conventional techniques as well as the fusion of these paradigms. There is a tremendous interest among researchers in developing techniques to process data and extract knowledge from the data. This talk will summarize the research projects undertaken by me and my research team in recent years. The progress made in solving the practical problems related to engineering and aviation based on intelligent data processing techniques will be presented.

**Gushchin V. A.**

On a family of quasi-monotonic finite-difference schemes of the second order of approximation

AbstractUsing a simple model of a linear transport equation a family of hybrid monotone finite difference schemes has been constructed. By the differential approximation, it was shown that the resulting family has a second-order approximation in the spatial variable, has minimal scheme viscosity and dispersion and being monotonic. It is shown that the area of intersection of the basic schemes (Modified Central Difference Schemes (MCDS) and Modified Upwind Difference Schemes (MUDS)) is non-empty. The local criterion for switching between the basic schemes is based on the velocity sign and the signs of the first and second differences of transferred functions. Within the studied schemes, the optimal pair of basic schemes, possessing the properties mentioned above and being closest to the third-order scheme, is obtained. On the solution of the Cauchy problem provides a graphical comparison of the calculation results obtained using the known schemes of the first, second and third order approximation. Keywords: finite-difference schemes, monotonicity of finite-difference schemes, hybrid finite-difference schemes, criterion for switching

**Babakov A.V.**

Numerical simulation of massive stars evolution based on gasdynamical model

AbstractApplication of the conservative numerical method of finite volumes for modelling massive stars evolution in self-gravitation conditions is considered. The large-scale vortex structures are obtained at evolutionary calculation. The numerical simulation is carried out by means the parallel algorithms which are realised on a cluster architecture supercomputer.

**Fortova S.V., Godunov S.K., Kluchinski D.V., Shepelev V.V.**

Experimental investigation of different models of gas dynamics with shock waves

AbstractA linearised edition of the classical Godunov scheme with non-linear decays of the discontinuities is introduced in the paper. Proved is that this scheme has the property of guaranteed decree of the entropy, which allows simulating entropy growth on the shock waves. Investigated is the structure of the shock waves after disintegrations of the discontinuities. Shown are the dependencies of the width of the shock wavefronts and their formation time lengths on the choice of the Courant number. The results of verification of the accuracy of discontinuous solutions are attached.

**Lobanov A.I.**

The analysis of difference schemes in the space of uncertain coefficients for the transfer equation

AbstractThe family of the differential schemes approximating the linear equation of transfer on an obvious four-dot template is considered. Spaces of uncertain coefficients of differential schemes for the model equation are entered in [1]. Development of technology of research in spaces of uncertain coefficients is described in [2]. For a problem of minimisation of an error of approximation of schemes the solution of a task is found in [2], the offered way allows describing a decision method, excellent from [2]. The method is based on consideration of couples of self-dual problems of linear programming [3]. Along with minimisation of an error of approximation on smooth decisions other functionalities, linear on scheme coefficients, are considered. If they are connected with approximation conditions, it is possible to speak about the generalised approximation of a differential task. It is by way of illustration brought the new families of differential schemes constructed on the basis of the solution of a problem of linear programming. 1. A.S. Kholodov. O postroenii raznostnyh skhem povyshennogo poryadka tochnosti dlya uravnenij giperbolicheskogo tipa // Zh. vychislitel'noy matem. i matem. fiziki, t.20, №26, s.1601-1620. 2. K.M. Magomedov, A.S. Kholodov. Setochno-harakteristicheskie metody. ‒ M.: Nauka, 1988, 288s 3. A.I. Lobanov. Raznostnye shemy v prostranstve neopredelennyh koefficientov I dvoistvennye zadachi lineinogo programmirovaniya / V sb.: Matematika, computer, obrazovanie. Tezisy konferentsii – Izhevsk, Izd-vo RHD. 2017. Str. 176.

**Laitinen E.J., Lapin A.V.**

Using ﬁctitious domain method for a class of parabolic optimal control problems

AbstractConstrained linear-quadratic parabolic optimal control problem in a curvilinear space-time domain is solved numerically using finite difference approximation on a regular orthogonal mesh. To construct the approximation we "embed" the initial problem in a "wider" problem using so-called algebraic fictitious domain method. Mesh problem in the wider domain is loaded by additional constraints for control and state functions, and the restriction of its solution to the initial domain coincides with the solution of the initial problem. Iterative methods are developed for a new problem. Efficient locally one-dimensional solvers based on ADI and fractional steps methods are used for the state and co-state problems. The optimal mesh problem is solved by a preconditioned Uzawa method. Numerical tests demonstrate the efficiency of the proposed approach for solving the formulated problem.

**Vyshinsky V.V., Sizykh G.B.**

The verification of the calculation of stationary subsonic flows and the presentation of the results

AbstractThe principle of pressure maximum is proved for a stationary three-dimensional vortex flow of an ideal gas (without the assumption of barotropicity). Based on the fact that in areas where the solution with high accuracy modeled by the Euler equations must be fulfilled and consequences of these equations, obtained subsonic principle is proposed to be used for verification of numerical solutions of boundary value problems for Euler equations for an ideal gas and the Navier – Stokes equations for viscous gas. Conditions of the maximum principle include the value of the Q-parameter, image surface level of which is currently widely used to visualise the flow pattern. The proposed principle of maximum pressure reveals the meaning of the surface Q = 0. It divides the flow region into the subdomain Q > 0, which cannot has a local pressure maximum points, and subdomain Q < 0 which cannot has a local pressure minimum points. A similar meaning of the parameter Q was known for an incompressible fluid (H. Rowland, 1880; G. Hamel, 1936). The expression for the Q-parameter contains only the first derivatives of the components of the velocity, which allows determining the sign (+/-) of Q even for numerical solutions obtained by the methods of low order. An example of the numerical solution verification using subsonic maximum pressure principle is presented. Analysis of the results of numerical calculation of the flow around the aircraft carrier ship during its movement and the presence of atmospheric winds showed that if the calculation results are used for the simulation of complex flight modes and to analyze the state of the atmosphere from the point of view of safe air traffic, visualization the flow pattern by surfaces is not informative. In particular, these surfaces do not reflect the true picture of the wind shear, which is perceived directly falls into a flying vehicle. To verify the numerical method, it is sufficient to provide only a surface, which has a clear physical meaning.

**Tolstykh A.I.**

Multioperators technique for constructing arbitrary high-order approximations and schemes: main ideas and applications to fluid dynamics equations

AbstractGeneral principles of constructing arbitrary high-order approximations via multioperators as linear combinations of basis operators are presented. Several multioperators-based numerical analysis formulas are described. The main emphasis is placed on multioperators-based schemes for the Euler and the Navier-Stokes equations with up to 32th-order approximations to the convection terms with optimised spectral properties aimed at the high resolution of small scales. Solutions to model problems illustrating the very high accuracy and the ability to preserve small phase errors during large time intervals are presented. Examples of direct numerical simulations with the Navier-Stokes and the Euler equations of flow instabilities and sound radiation as well as of the initial stage of the laminar-turbulent transition are displayed. The ability of the schemes to describe pressure pulsation which are 5-7 orders smaller than mean pressures are shown. Hybrid multioperators-based schemes for hypersonic flows with illustrative examples are presented.

**Chetverushkin B.N., Yakobovskiy M.V.**

Fault-tolerant algorithms for exaflop computing systems

AbstractThe report is devoted to discussion of algorithms aimed at providing the possibility of using advanced calculating systems of exaflop and higher level of performance for mathematical modeling of problems of continuum mechanics. Exaflop computer systems, because of the expected short period of their trouble-free operation, will require qualitatively new solutions to ensure the very possibility of long-term calculations. Algorithms are considered that ensure the independence of the calculation time from the occurrence of computational failures in computing.

**Tishkin V.F., Krasnov M.M., Kuchugov P.A., Ladonkina M.E.**

Discontinuous Galerkin method for solving the Navier-Stokes equations on grids of arbitrary structure.

AbstractDiscontinuous Galerkin method characterised by a high order of accuracy in areas of smooth solutions has well proven itself for numerical modelling of a wide class of problems of mathematical physics. As it is known for solving applied problems with a complex geometry of the object under study, it is necessary to use grids of arbitrary structure [3], consisting of mesh elements of two or three different types. The discontinuous Galerkin method is fairly easily realised in this case too without loss of accuracy of the solution. However, if the solution contains strong discontinuities, to ensure its monotonicity, it is necessary to use limiting functions that can significantly affect the accuracy of the solution. The questions of ensuring the monotonicity of the solution and preserving the order of accuracy of the method on grids of an arbitrary structure are considered in this work. The research has been performed under the financial support of the Russian Foundation for Basic Research, grants Nos. 17-01- 00361_а, 16-01- 00333. References 1. B. Cockburn. An introduction to the discontinuous Galerkin method for convection dominated problems, an advanced numerical approximation of nonlinear hyperbolic equations // Lecture notes in mathematics, 1998, 1697, 151-268. 2. M.E. Ladonkina, O.A. Neklyudova, V.F. Tishkin, Application of the RKDG method for gas dynamics problems, Mathematical Models and Computer Simulations, 2014, 6, 4, 397-407. 3. M.M. Krasnov, P.A. Kuchugov, M.E. Ladonkina, V.F. Tishkin, Discontinuous Galerkin method on three-dimensional tetrahedral grids: Using the operator programming method, Mathematical Models and Computer Simulations, 2017, 9, 5, 529-543.

**Shananin A.A.**

On the Cauchy–Gelfand Problem

AbstractThe report is based on joint work with G. M. Henkin. The problem posed by Gelfand on the asymptotic behaviour (in time) of solutions to the Cauchy problem for a first-order quasilinear equation with Riemann type initial conditions is considered. By applying the vanishing viscosity method with uniform estimates, exact asymptotic expansions in the Cauchy–Gelfand problem are obtained without a priori assuming the monotonicity of the initial data, and the initial data parameters responsible for the localisation of shock waves are described. References 1. G.M.Henkin, A.Shananin Asymptotic behaviour of solutions of the Cauchy problem for Burgers-type equations. Journal de Mathematiques Purées et Appliquees, 2004, v.83, N12, p.1457-1500. 2. G.M.Henkin, A.E.Tumanov, A.Shananin Estimates for the solution of Burgers-type equations and some applications. Journal de Mathematiques Pures et Appliquees, 2005, v.84, N1, p. 717-752. 3. G.Henkin. Asymptotic structure for the solution of the Cauchy problems for Burgers-type equations, J. Fixed Point Theory Appl., 1, (2007), 239-291. 4. G.M.Henkin, A.A.Shananin, Cauchy-Gelfand problem for quasilinear conservation law, Bull. Sci. Math. 138, (2014), p. 783-804. 5. G.M. Henkin, A.A. Shananin, On the Cauchy–Gelfand Problem Doklady Akademii Nauk, 2015, Vol. 465, No. 4, pp. 415–418. 6. Henkin GM, Shananin AA The Cauchy-Gel'fand problem and the inverse problem for a first-order quasilinear equation // Functional analysis and its applications, vol. 50, no.1, p. 61-74.

**Shaidurov V.V., Yakubovich M.V., Schepanovskaja G.I.**

Semi-Lagrangian approximations of the conservative laws in the Navier-Stokes equations

AbstractThe report examines a supersonic flow of a viscous heat-conducting gas around a body. A numerical algorithm for solving the initial-boundary value problem for the Navier-Stokes equations written for the conservation of the laws of mass and total energy (kinetic + internal) is proposed. To approximate the time-transfer operator in each equation, a Lagrangian approach is used on adaptive patterns along the particle trajectories. The spatial discretization of the remaining terms of the equations on each time layer is carried out by the finite element method with piecewise bilinear basis functions and simple quadrature formulas. To solve the resulting systems of algebraic equations, the Jacobi method is used with an improved initial approximation within the outer iterations of nonlinearity. For the constructed discrete problem, the first-order approximation in space and time is verified, the fulfillment of the laws of conservation of mass and total energy at a discrete level is proved, the stability of the discrete problem and its first order accuracy convergence are proved. As a numerical illustration, calculations of flow past a wedge-shaped body by a viscous heat-conducting gas are given for different Mach and Reynolds numbers.

**Kholodov Y. A., Kholodov A. S., Tsybulin I. V.**

Developing of Nonlinear Grid-Characteristic High Order Methods for Hyperbolic Type Equations**Utjuzhnikov S.V.**

Non-overlapping domain decomposition for efficient modelling near-wall turbulent flows

AbstractIt is well recognised that the near-wall turbulence modelling is computationally a very expensive problem. The talk addresses a novel approach to tackle this problem, based on non-overlapping domain decomposition. The approach has proven its efficiency for industrial applications. It is shown that the computational time can be saved as much as one order of magnitude while the error does not exceed a few percent.

**Yegorov I.V., Novikov A.V.**

Direct numerical simulation of a laminar-turbulent transition at high flow rates

AbstractDirect numerical simulations of a three-dimensional wave train propagating over a 5.5 deg compression corner at the freestream Mach number 5.373 are carried out. The Navier–Stokes equations are integrated using an implicit finite volume shock-capturing method with the second-order approximation in space and time. After computing the laminar flowfield, unsteady disturbances are imposed onto the steady solution via local suction blowing on the wall surface. The undisturbed boundary layer separates upstream of the corner and reattaches downstream, forming a shallow separation bubble. The suction-blowing generates a three-dimensional wave train propagating downstream. At sufficiently strong forcing, the nonlinear effects destabilize the wave packet in the separation region and lead to its nonlinear breakdown downstream of the reattachment line. A young turbulent spot is formed in the reattached boundary layer.

**Stupitsky E.L.**

Physical studies and mathematical modeling of large-scale geophysical experiments

AbstractThe report presents the results of physical studies and numerical simulation of ionization-chemical, optical and magnetohydrodynamic perturbed regions formed during large-scale geophysical experiments in near-Earth space. A significant place is given to the physical analysis of the phenomena under consideration and the development of numerical algorithms adapted to them. For the first time, as a result of numerical studies and calculations, three-dimensional MHD patterns of the development of a plasma region of high specific energy in a wide range of heights: 100-1000 km were obtained. A numerical analysis of the interaction of two large-scale plasma regions in a geomagnetic field and an inhomogeneous ionosphere is carried out.

**Succi G., Ivanov V.V.**

Comparative analysis of the mobile operating systems

AbstractOne of the main problems of software engineering is currently the reliability of software. A variety of models for evaluating the reliability of software products and the software development process are proposed. Many of these studies are aimed at improving the measurement and prediction of the reliability of software products using models of reliability growth. However, a relatively small part of the work is devoted to approaches to comparing existing systems in terms of software reliability. In addition, currently there is no complete and proven methodology for comparing the reliability of software products. In this paper, we propose a methodology for comparing the reliability of software products, in which software reliability growth models are widely used, based on the representation of the number of software defects in the form of a non-homogeneous Poisson process. The proposed methodology has the following functions: it provides a certain level of flexibility and abstraction while preserving objectivity, that is, providing measurable comparison criteria. Finally, by comparing the comparison methodology (how to compare reliability) with the evaluation criteria (which is the result of comparison), it becomes possible to compare reliability information in a wide range of software systems. The methodology was evaluated on the example of three mobile operating systems with open source: Sailfish, Tizen, CyanogenMod. It was found that: • Tizen OS is the most efficient mobile operating system among the three main modules and components considered from the point of view of reliability. • CyanogenMod is the worst operating system in terms of speed of failure detection. • In terms of detecting defects, the Sailfish operating system proved to be the best.

**Nikitin I.S., Burago N.G.**

On effective methods for solving of nonlinear unsteady problems of continuum mechanics

AbstractTechniques that improve the accuracy of numerical solutions and reduce their computational costs are discussed as applied to continuum mechanics problems with complex time-varying geometry. The approach combines shock-capturing computations with the following methods: (1) overlapping meshes as well as discrete and continuous markers for specifying complex geometry; (2) elastic arbitrarily moving adaptive meshes for minimizing the approximation errors near shock waves, boundary layers, contact discontinuities, and moving boundaries; (3) matrix-free implementation of efficient iterative and explicit–implicit finite element schemes; (4) balancing viscosity (version of the stabilized Petrov–Galerkin method); (5) exponential adjustment of physical viscosity coefficients; and (6) stepwise correction of solutions for providing their monotonicity and c onservativeness. Example solutions of problems in fluid and solid mechanics are presented.

**Skrzypacz S.P.**

On nonlinear differential equations related to Micro-Electro- Mechanical-Structures (MEMS)

AbstractThe recent results on stability and pull-in analysis of MEMS parallel-plate capacitors made of graphene will be presented. Lumped-parameter modelling with nonlinear restoring forces is used to establish some nonlinear mass-spring systems for various cantilever beams of some nonlinear materials subject to the standard constraints in MEMS applications. Nonlinear differential equations are used for studying statics and dynamics of materials like annealed metals obeying the Hollomon nonlinear stress-strain constitutive equation. The parameters in the corresponding nonlinear restoring forces and the effective masses are presented based on the extension of Euler-Bernoulli's beam assumptions and the nonlinear stress-strain model for the constraints. [1] D. Wei, S. Kadyrov, Z. Kazbek: Periodic Solutions of a Graphene-based Model in Micro-Electro-Mechanical Pull-in Device, Applied and Computational Mechanics, Vol 11, No 1(2017)DOI: 10.24132/acm.2017.322 [2] P. Skrzypacz, S Kadyrov, D. Nurakhmetov, D. Wei: Analysis of Pull-in Dynamics of a nonlinear Material NEMS Model, accepted for publication in Materials Today Proceedings."

**Favorskaya A.V.**

Investigation of wave processes in fractured media by grid-characteristic method

Abstract"The paper considers the problem of revealing the features of wave processes in fractured media using full-wave numerical simulation and their application significance for seismic exploration of hydrocarbons. Various variants of a fractured medium are considered, and their comparative analysis is conducted, aimed at identifying effects that can later help separate one fractured cluster from another one. Based on the analysis of spatial dynamic seismic wave fields arising in the process of seismic exploration, the regularities that can be expected on seismograms are revealed. The correspondence of these regularities to the corresponding synthetic seismograms was checked. The study was carried out with the financial support of the Russian Foundation for Basic Research in the framework of the scientific project No. 16-29-15097 ofi_m."

**Beklemysheva K.A.**

Numerical modelling of hybrid composites under impact load

AbstractPolymer composites that are being used in aviation are often exposed to low-velocity strikes - hail, gravel, hit during maintenance, etc.). These strikes do not leave visible marks or dents but lead to the emergence of barely visible impact damage, which severely reduces the residual strength of the whole part. In this work, we propose the application of an existing set of programs based on the grid-characteristic method for the investigation of various parameters that can influence the final strength of hybrid composites concerning low-speed impacts. The grid-characteristic method proposed in this work was previously used for numerical simulation of polymer composites and helped to obtain a good correspondence to the experiment. Its application for the analysis of failure processes in hybrid composites will allow to modify the structure of the hybrid composite and increase its strength at low-velocity impacts. This work was supported by the Russian Science Foundation (project 17-71-10240).

**Grigorievykh D.P.**

Numerical modelling of dynamic destruction using grid-characteristic methods

AbstractThis work discusses the possibility of modeling cracks and irregularities in solid deformable bodies using the grid-characteristic method on one of Lagrangian hexahedral grids. The problems of elastic waves propagation through fluid or gas-saturated cracks and irregularities are discovered as well as processes of formation of new cracks.

**Kozhemyachenko A.A., Favorskaya A.V.**

Development of the method of coating the rail profile with hexahedral grids

AbstractRails are an object of complex shape. The use of conventionally structured grids does not allow for computations. While the calculation of the spatial dynamic load on the rail and the process of ultrasonic rail non-destructive testing is a long-term task, which requires saving of computing resources. Thus, it is of interest to develop the appropriate design of grids, which will allow modelling of the spatial dynamic wave processes occurring inside the rail during its loading and during the process of ultrasonic flaw detection. The method of covering the rail profile with hexahedral grids is considered in this work. The study was carried out with the financial support of the Russian Foundation for Basic Research in the framework of the scientific project No. 17-20-01096 ofi_m_RZD.

**Kabisov S.V., Favorskaya A.V.**

About the possibility of detecting railroad defects

AbstractThe conditions of the possibility of detection or non-detection of defects in railway are investigated by full-wave computer modelling using the finite-difference grid-characteristic method. The conditions under which devices for non-destructive testing of railway do not allow to detect these defects are studied. The approaches for modifying the methods of nondestructive testing of railway that will allow diagnosing these types of defects are proposed. The study was carried out with the financial support of the Russian Foundation for Basic Research in the framework of the scientific project No. 17-20-01096 ofi_m_RZD.

**Stognii P.V.**

Comparative analysis of wave patterns and seismograms in modelling wave processes in the Arctic with the help of grid-characteristic method

AbstractThe report presents the analysis of the influence of ice formations, such as ice cover, icebergs and roses, on wave patterns and seismograms. All the models were computed with the grid-characteristic method, which confirmed its good application to this kind of direct seismic exploration problems in the Arctic. The study was carried out with the financial support of the Russian Foundation for Basic Research in the framework of the scientific project No. 16-29-02018-of-m.

**Simakov S.S.**

Development of computational physiology and medicine by A.S. Kholodov. Review.**Chupakhin A.P., Cherevko A.A., Khe A.A., Parshin D.V., Orlov K.Yu**

Complex study of cerebral hemodynamic: clinical monitoring and computer modelling

AbstractThe results of a study of cerebral hemodynamics based on clinical monitoring and computer modelling of blood flow in the presence of cerebral aneurysms (CA) are presented. Measurements of blood pressure and velocity in the brain vessels are performed using the ComboMap–ComboWire measuring complex. The data is computer processed and systematised in the form of operation records characterising the anomaly, changes in hemodynamics in the operating field, and the success of the operation. Based on computer tomography and magnetic resonance imaging, 3D configurations of blood vessels in the area of influence of anomalies were constructed. A computer simulation of blood flow during the operation of CA treatment was carried out. The obtained data serve as the basis for constructing a hemodynamic atlas of cerebral vessels: a three-dimensional network of blood vessels equipped with the values of blood flow velocity and pressure. Such an atlas is of interest for both fundamental and clinical medicine. This work was supported by the Russian Science Foundation (project 17-11- 01156).

**Shestopaloff Yu.K.**

Growth equation as a new type of mathematical equation. General growth law and its biological and medical applications.

AbstractLife origin is of a great general, academic and practical interest. Presently, life phenomenon is considered almost exclusively from a biomolecular perspective. On the other hand, other factors, acting at higher scale levels, also affect growth and reproduction of living organisms. Here, we present a heuristic growth equation, a mathematical representation of the discovered general growth law. This law, in cooperation with biomolecular mechanisms, governs the growth of living species from cellular to whole organism levels. We show that the growth equation adequately describes growth and reproduction of unicellular organisms and organs, explains known growth and reproduction phenomena and predicts new ones, and can be used in many applications. It has important mathematical properties. Such, for particular geometrical shapes, it produces analytical solutions, similar to logistics equation, but without ad hoc parameters. This confirms the validity of both equations as adequate means for a mathematical description of real growth phenomena.

**Simakov S.S., Vassilevski Yu.V., Gamilov T.M., Danilov A.A., Golov A.V.**

Mathematical models of blood circulation and respiration: applications in medicine

AbstractIn silico personalised modelling of the patient-specific conditions is the state of the art of the modern computational medicine. Academician A.S. Kholodov has made a substantial contribution to this field. One of his prominent results is the 1D model of the hemodynamics and respiratory flow. In this work, we present the recent success of this approach to personalised computational modelling of some cardiovascular and respiratory pathologies and their treatment.

**Vasyukov A.V.**

Adaptation of grid-characteristic method for unstructured tetrahedral meshes with large topological inhomogeneities

AbstractHigh computational complexity is the major problem that grid-characteristic method encounters when using tetrahedron grids of real engineering structures. The formally grid-characteristic method can be used on any tetrahedral grid. However, for tetrahedral grids, there is a time-step limitation analogous to the Courant step for a uniform rectangular grid. For a complex geometry of the calculated region, the grid always contains some very small or very flat tetrahedra. From a practical point of view, this leads to an unjustified drop in the time step (by 1-3 orders of magnitude for real constructions) and, accordingly, to an unjustified increase in the required amount of calculations. In the classical works of A.S. Kholodov and K.M. Magomedov, an approach was proposed for constructing grid-characteristic methods on unstructured grids using flexible mesh stencils. In this work, this technique is used to construct an efficient numerical method for tetrahedral grids. The method solves the problem of a small time step due to large topological inhomogeneities of the initial grid, and also covers time step degradation due to the movement of the grid in the zone of deformations.

**Danilov A.A., Pryamonosov R.A., Yurova A.S.**

Segmentation algorithms for cardiovascular modelling

AbstractIn this work, we present automatic and semi-automatic user-guided methods and algorithms for patient-specific image segmentation and generation of discrete geometric models for several cardiovascular biomedical applications. A new technique for dynamic heart ventricles segmentation and mesh generation using dynamic contrast-enhanced Computed Tomography images is presented in details.

**Kislukhin V.V., Kislukhina E.V.**

Math model for blood passing cardio-vascular system (CVS) with congenital heart defects. Search for cardio-output and intracardial shunts.

AbstractCVS can be considered as an oriented closed graph, with orientation given by blood flow. Structure of graph follows anatomy of CVS. For math equations one divides edges of graph into three types: (a) heart chambers as pumps, (b) transport vessels that are passed with delay, (c) microvessels, with random blood flow and complex geometry. Math description of congenital heart defects does not change the description of flow of blood (and indicator) through edges however introduce pathological connections among heart chambers (and vessels, connected with heart). Thus having an indicator dilution registration and math model one can estimate flow through and between heart chambers.

**Beklemysheva K.A., Kovalev V.V.**

Numerical modelling of mechanical impact on human body

AbstractSeveral problem statements are considered, including a low-velocity impact on a human torso and the propagation of a diagnostic phased ultrasound pulse in a human head.

**Shishlenin M.A., Kabanikhin S.I.**

Continuation problems of the solutions from the part of the boundary of equations of mathematical physics.

AbstractWe consider continuation problems in Geophysics and numerical methods for their solution. The continuation problems of physical fields with the data on the part of the boundary are ill-posed. Continuation problems are formulated in the form of operator equation Aq = f, for which the minimisation of the objective function and the method of singular value decomposition are applied. We study the properties of the operator A and the algorithm of minimisation of functional J(q) = by the conjugate gradient method. In series of numerical experiments are shown that it allows us to recover the boundary conditions on the inaccessible part of the boundary, as well as to obtain information about inhomogeneities (the number, location, approximate volume) located in the region of inaccessibility. The work was supported by Russian Foundation for Basic Research (projects NNo. 17-51-540004, 16-29-15120, 16-01-00755 and 15-01-09230).

**Marchenko M.A.**

Numerical stochastic simulation of kinetic processes of diffusion, coagulation and charged particles transfer on supercomputers

AbstractIn our talk, the effective algorithms of distributed numerical stochastic simulation for precocious evaluation of functionals determined by rare events on trajectories of diffusion processes are presented. The algorithms are based on the splitting and weighted techniques. We consider such functionals as the probability of non-reaching the boundary of the domain within the given time interval and the full concentration of trajectories in the point within the given time interval. The stochastic model for numerical stochastic simulation of the spatially inhomogeneous coagulation process is presented. The model is based on the spatial regularisation of the coagulation kernel and the method of the majorant frequency. The parallel algorithm for implementation of the stochastic model is also given. The stochastic model for numerical stochastic simulation of the process of electron avalanches evolution in the gas is given. The model is based on the use of branched processes. The parallel algorithm for implementation of the stochastic model is also presented. The parallel long-period pseudorandom numbers generators are introduced. The technique of distributed numerical stochastic simulation on the high-performance computational system is also described. The imitation model of execution of distributed numerical stochastic simulation programs is presented. The model is used for evaluation of scalability of such kind of programs. The universal software libraries for the realization of distributed numerical stochastic simulation programs on high-performance computational systems and the parallel program for numerical analysis of stochastic oscillators are presented.

**Zhdanov M.S.**

Generalized joint inversion of geophysical fields based on Gramian constraints

AbstractWe have introduced a new approach to the joint inversion of multimodal geophysical data using Gramian spaces of model parameters and Gramian constraints, computed as determinants of the corresponding Gram matrices of the different physical parameters. We also show how this method can be incorporated into a standard algorithm of Tikhonov regularisation. By imposing an additional requirement of minimising the Gramian during the regularised inversion, one can recover multiple model parameters with the enhanced correlation between the different physical properties. We demonstrate that this new approach is a generalised technique that can be applied to the simultaneous joint inversion of any number and combination of geophysical datasets. Our approach includes as special cases those extant methods based on correlations and/or structural constraints of the different model parameters. The developed method is illustrated by examples of joint inversion of multiple geophysical datasets.

**Golubev V.I.**

The usage of the grid-characteristic method in seismic migration problems

AbstractIn this report, we consider the problem of elastic wave propagation in a fractured medium. The mathematical formulation of the problem includes a system of equations based on linear elasticity and additional correctors describing the dynamic behaviour of gas- and liquid-saturated media. An algorithm for seismic imaging is proposed that take into account a priori data on fractured objects. Specifically, an explicit description of crack boundaries is included in the background medium model. This algorithm was verified on simplest models with a variation of parameters in a wide range. This work was supported by the Russian Foundation for Basic Research and by the Foundation “National Intellectual Development” for supporting undergraduate and graduate students and young scientists, project no. 17-37-80004_mol_ev_a.

**Yavich N.B., Malovichko M.S., Khokhlov N., Zhdanov M.S.**

Parallel low-frequency electromagnetic modelling based on a special contraction operator transformation

AbstractWe introduce two novel preconditioning methods for 3D finite difference (FD) frequency domain electromagnetic modelling. The first preconditioner is based on the layered conductivity model Green’s function; the second one is based on a special transformation of the original system of FD equations into a system with a contraction operator. This transformation extends to the FD modelling the approach originally developed for the integral equation modelling method. We also examine spectral properties of the two preconditioned methods. For the numerical study of the developed methods, we have designed a complex marine 3D geoelectrical model. The numerical experiments confirm the results of the theoretical analysis: an iterative solver with the contraction preconditioner converges faster or at near the same speed as with the Green’s function preconditioner. We apply these two solvers to controlled source modelling and demonstrate that both approaches are very effective, and memory saving. We have also developed a parallel version of the algorithm and studied the scalability of shared and distributed memory parallelisation.

**Kazakov A.O.**

Numerical modelling of non-destructive testing of composite panel after low-velocity impact

AbstractThe process of ultrasonic (pulse-echo) non-destructive testing of a composite panel after a low-velocity impact is modelled. Typical internal damage patterns are considered - delamination, destruction of the contact between matrix and fibres. The grid-characteristic numerical method is used for calculations. Major elements of an ultrasonic scanning device are modelled explicitly to calculate correct A-scan and obtain complete wave pattern recorded by a receiver. Direct comparison of numerical and experimental results is presented.

**Kurashov V.V., Lunev V.V., Malinin V.P.**

Numerical and experimental research of supersonic flow past blunt bodies of complex shape

AbstractBodies of complex shape in the form of blunt compound cones are interesting for the study. Flow around these bodies have difficult effects interaction of jumps and laminar-turbulent transition The results of experimental studies of these flows are an excellent material for validating programs for the numerical solution of the Navier-Stokes equations. In order to study the flow of such bodies, a series of experiments with supersonic (M=6) flow around the blunt compound cones were carried out (one variant is given in [1]). As a result, an extremely nonmonotonic pattern of the current was detected, with a strong maximum of pressure at the periphery, and the Central zone of the recurrent flow. A more detailed study of such flows is carried out numerically, within the framework of laminar-turbulent systems of Navier-Stokes equations with one of the versions of differential turbulence models [2] and using the OpenFOAM library[3]. The internal system of compaction jumps is drawn and it is shown that the occurrence of peak pressures is caused by interference of internal compaction jumps with the head shock wave. The contours and structure of the internal return region of the flow are described. The results of calculations and experiments are generally satisfactorily consistent with each other, especially in the distribution of pressure, which indicates the opportunity to explore such complex flows with the help of modern computational methods.

**Skubachevskii A.A., Lapshin V.B.**

Simulation of Spiral Electromagnetic Wave Propagation in the Medium Using FDTD Method

AbstractThe second-order of approximation numerical method for the solution of Maxwell Equations in nondispersive medium, called FDTD (Finite-Difference Time-Domain), also known as Yee algorithm, was used to create the software package for modelling of electromagnetic waves propagation in medium (with the Courant stability condition). The Total Field/Scattered Field (TFSF) source is considered in detail. Inhomogeneous medium is also included into the model, and there are some examples, which show, how the program works with it. The different boundary conditions (periodic, reflecting and absorbing) were looked into. Perfectly Matched Layer (PML) absorbing boundary conditions are used in the program, and the corresponding results are discussed. Different types of sources are realised in the program, the results of the waves propagation are shown. One of the applications is spiral electromagnetic waves. The results of modelling of these waves are shown. Some new features of these waves are looked into.

**Utkin P.S., Fortova S.V.**

Mathematical modelling of high-speed flows of two-phase media with shock waves

AbstractThe work presents the mathematical model based on Baer-Nunziato equations and the numerical algorithm for the investigation of high-speed flows of two-phase media with shock waves. Two applications are considered. The first one is the problem of shock wave – dense particles cloud interaction. The second one is the high-speed impact of the metal plates.

**Solomatin R.S.**

Mathematical modelling of turbulent mixture layer in multicomponent gas mixtures

AbstractMathematical modelling of the turbulent mixing process is the most important stage of research of detonation wave propagation in non-premixed gas mixtures. Supersonic mixing of gas components is taking place just in front of detonation wave front. That forms a concentration gradient, which has a crucial influence on DW velocity and propagation character. This work is devoted to the development of the mathematical model and computational algorithms of turbulent mixture layer formation between two components of the gas mixture. In the model problem mixing process between hydrogen and oxygen is explored in 2D and 3D formulations. The Navier-Stocks equations system for viscous compressible gas is integrated over time using hybrid explicit-implicit scheme. GMRES+LU-SGS is used as the main numerical method. For turbulent processes, DES-based on SA turbulence model is applied. Also, the turbulent diffusion process is taken into account. Computational experiments are conducted on high-performance clusters.

**Lishankov S.I.**

Multistep ionisation and radiation in shock-heated argon

AbstractIn this paper, the process of ionisation of argon in the relaxation region behind the shock wave is considered in detail. A one-dimensional steady-state flow of a two-temperature gas mixture consisting of argon in the ground and excited states, its ions and electrons is considered. The transition through the shock wave and the corresponding change in the parameters (such as density, velocity, temperature, pressure) are assumed to be frozen, without changing the composition of the gas, and we study its relaxation behind the front of the shock wave. The considered atomic model includes 30 energy states. The collisional-radiative model takes into account multistage ionisation by atomic and electronic impacts, radiative transitions between excited states of argon, three-body recombination, photorecombination and Bremsstrahlung emission.

**Lopato A.I., Utkin P.S.**

The usage of grid-characteristic method for the simulation of flows with detonation waves

AbstractThe work is devoted to the mathematical modelling of the pulsating detonation wave propagation in the shock-attached frame. The numerical algorithm is based on grid-characteristic method. The pressure pulsations in case of stable, weakly unstable, irregular and highly unstable detonation modes in calculations in the shock-attached frame are compared with those in calculations in the laboratory frame.

**Sidorenko P.A., Utkin P.S.**

Mathematical modelling of the shock wave – system of moving and colliding bodies interaction

AbstractMultidimensional gas dynamics modelling is used for the clarification of the mechanisms of shock wave – dense particles cloud interaction. The work describes the numerical algorithms for the simulation of shock wave propagation in the areas with complexly shaped boundaries and with variable boundaries. The problems of one and several moving and interacting cylinders relaxation behind the transmitted shock wave are considered.

**Nevmerzhitski Ya.V.**

Numerical simulation of steam injection in heavy oil reservoir

AbstractAt the present day, modern oil production is characterised by the growing number of fields with tight reserves to which reservoirs with heavy oil belong. The main feature of these oils in comparison with traditional light oils is both much higher density and viscosity and anomalous rheology. The latter effects regarding threshold pressure gradient, below which oil does not move, and at pressure gradients exceeding the threshold value filtration obeys Darcy’s law. Because of decreasing of oil viscosity as well as threshold pressure gradient with the temperature increasing, thermal methods of development such fields have found wide application. One of these methods is steam flooding. For the correct determination of reservoir pressure and temperature and therefore bottom hole pressure and rates, the program was created that allows simulate steam injection into the reservoir, where oil filtration deviates from Darcy's law. Multiphase two-dimensional filtration of oil, water and steam was considered. To ensure high computational speed, the method of streamlines with splitting by physical processes was applied. It is assumed that the temperature and pressure are equal for all components, therefore, for solving the equations, it is appropriate to use IMPES method, the main assumption of which is the constancy of capillary pressure from time. To take into account the nonlinearity in the filtration equation associated with oil rheology, the method of coefficients smoothing was applied. It is most effective for multidimensional problems. Flash calculation was performed by solving Rachford-Rice equation, where equilibrium constants were determined from Wilson’s correlation. Obtained results were compared with known analytical solutions (rectilinear-parallel filtration of an elastic Bingham fluid, the Buckley-Leverett problem) and also with the results of modelling on a commercial simulator. The result of this work is a tool for reliable estimation of wells and pads depleting reservoirs with heavy oils.

**Favorskaya A.V.**

Numerical modelling of destruction of complex structures as a result of seismic influences

AbstractThe results of numerical modelling of the destruction of complex structures (residential buildings of various types, foundations of various types, underground structures) as a result of intensive seismic influences are considered in this work. The grid-characteristic method was used for calculations, which makes it possible to carry out full-wave modelling in this complex formulation and to calculate the dynamic damages of the structures under consideration by applying the criterion of the principle components of the stress tensor. The influence of various types of waves propagating from the hypocenter of an earthquake is investigated. We used a system of nested hierarchical grids. The possibility to investigate the seismic resistance of various design solutions is shown. This work was supported by the Russian Science Foundation, grant no. 17-71-20088.

**Ermakov A.S.**

On the application of the method of markers in conjunction with grid-characteristic method for modelling solid mechanics with large deformations

AbstractModeling large deformations is a quite complex task when simulating solid mechanics problems related to intense dynamic loads. The method of markers and cells is one of the well-known methods for tracing displacements of the medium in the numerical solution. One of the main advantages of the grid-characteristic method when solving dynamic problems of solid mechanics is the correct consideration of boundary and contact conditions of various kinds. However, a topology of a calculation mesh is extremely important for modelling these conditions. The classical method of markers and cells is designed for the calculation of hydrodynamic problems and involves the use of special points (markers) for the determination of grid cells that are filled with liquid. An attempt to apply the same method for calculating the problems of solid mechanics in conjunction with the grid-characteristic method faces certain difficulties, connected both with the difference of the equations in question and with the design features of both numerical methods. The classical method of markers does not provide an option to restore the boundary accurately, so this work pays special attention to this procedure.

**V.L. Yakushev, Yu. P. Nazarov, E.V. Poznyak**

Development of wave theory methods for seismic resistance of construction structures

AbstractNew techniques are algorithmized and tested in practical applied calculations: modeling of spatiotemporal wavefields of seismic motion of soil with given accelerograms; formation of a vector of seismic action with components of translational and angular motion and compilation of relative motion equations for integral and differentiated models of impact; generalized linear-spectral method, applicable both for integral and for differentiated seismic motion. The types of the wave model of soil motion are investigated, as well as the ultimate states for seismic calculations, analysis of the spectral composition of the seismic effect and obtaining a consistent estimate of its spectral density, the variability of seismic motion of the soil, the effect of angular motion on the dynamics of structures, the filtration of short seismic waves by hard foundations.

**Ruzhankaia A, Khokhlov N.I.**

Usage of hierarchical grids in seismic computations

AbstractThe investigation of wave response in places with of micro- and macro-cracks is one of the mathematical modelling directions in seismic computations. In this paper was introduced an algorithm, which allows combining grids of different space layout, size, relative offset for computing a wave response from cracks of different rotation angle using structured grids. The algorithm is based on Chimera grids technique. Earlier this approach was widely applied in hydrodynamics computations, in proposed work, it was adapted for cracking inhomogeneities computations using the grid-characteristic method. A model of ideal isotropic linear elastic material was used. The algorithm consists of two parts: generation of point and coefficient list for direct and reverse interpolation and updating of the values in points during computation. The time of the computation was reduced dramatically due to the usage of a structured grid for each rotated crack instead of curvilinear grids.

**Muratov M.V., Petrov I.B.**

Mathematical models of geological fractures and their application in practical exploration seismology problems modelling

AbstractThe report contains the description of developed mathematical models of fractures which can be used for numerical solution of exploration seismology problems with the use of grid-characteristic method on unstructured triangular and tetrahedral meshes. The base of developed models is the concept of infinitely thin fracture. This fracture is represented by contact boundary. Such approach significantly reduces the consumption of computer resources by the absence of the mesh definition inside of fracture necessity. By the other side, it lets state the fracture discretely in integration domain. Therefore one can observe qualitative new effects which are not available to observe by use of effective models of fractures, actively used in computational seismic.

**Ivanov A, Khokhlov N.I.**

Application of GPU for elastic wave modelling using grid-characteristic method

AbstractTechnologies of parallel programming on GPU namely CUDA and OpenCL currently are applied to wide variety of problems with the aim of increasing computation speed. We show how these technologies are applied for modelling of elastic waves in rigid, deformable media with the grid-characteristic method. We apply explicit method so that this problem can be parallelised with high efficiency. The task of grid-characteristic method optimisation for execution on GPU is considered. We examined the impact of reducing the amount of conditional branching, placement of grid nodes in memory sequentially, choice of sizes of GPU computational blocks. Also, we measured the efficiency of parallelisation by the number of GPUs and performance improvement due to exchanging grid data directly between GPU bypassing CPU. Each stage of research accompanied with a comparison of the result of applying CUDA and OpenCL technologies.

**Frolov M.E.**

Functional approach and mesh adaptation for deformable solid mechanics

AbstractThis work is devoted to recent developments of functional approach [1] to posteriori error control for various 2D problems in mechanics of deformable solids. The approach provides reliable error majorants (upper bounds) that are valid for any approximate conforming solution. For classical and Cosserat elasticity (plane strain) and Reissner-Mindlin plates, it is shown that conforming finite element approximations in the Hilbert space H(div) yield efficient error control. This conclusion is approved by numerical tests with mesh adaptations using MATLAB tools. Work is supported by the Grant of the President of the Russian Federation MD-1071.2017.1. [1] S. Repin. A posteriori estimates for partial differential equations (de Gruyter, Berlin, 2008).

**Zavyalova N. A., Perepechkin I.M., Bykov A.A.**

Modelling of Hydraulic Fracturing

AbstractTIn modern practice, the problem of correct and rapid calculation of fracture geometry is urgent. Until recently, analytic and one-dimensional numerical models were perfectly capable of dealing with it. The vast majority of hydraulic fracturing is carried out with the help of design, performed on these simulators. However, the number of readily available stocks is constantly decreasing and the development of more complex deposits and those cases where hydraulic fracturing was considered an unacceptable operation becomes necessary. These include a reservoir with thin heterogeneous interlayers, with a gas cap or underlying water, oil rims, etc., in other words, all those cases where it is necessary to determine the geometry of the fracture correctly. In the world practice, there are two main trends for solving this problem. The first is the complication of one-dimensional models and an attempt to add to them the possibility of correct calculation of a crack in height. The second is the reduction of the computational complexity of two-dimensional models. The work is devoted to the creation of a model for the modelling of a crack in the Planar3D approximation, which meets the accuracy and speed requirements for modern fracturing models and the realisation of this setting. The difficulty lies in the simultaneous modelling of geomechanical and hydrodynamic processes, the characteristic times of which are very different. Therefore, the splitting of physical processes was done. The flow in the crack was calculated in the 2D approximation, taking into account the power rheology of the liquid, the geomechanical problem was solved in the approximation of a planar crack. An implicit scheme was used. The stabilised method of bi-conjugate gradients with preconditioning was used to solve the system of linear equations obtained after Newton's linearization.

**Konyukhov I.V., Konyukhov V.M., Chekalin A.N.**

Numerical simulation of heat and mass transfer during the commissioning the system “Producing pumping well – Porous-fractured reservoir” into operating regime

AbstractMathematical, numerical and algorithmic models of transient heat and mass transfer during the commissioning into the operating regime the oil producing well equipped with an electrical centrifugal pump (ECP) in developing of the porous-fractured reservoir with bottom water are proposed. These models are implemented in computer simulator with the use of the modern parallel computing technologies. A special program unit is developed to simulate the work of the surface control station and realise the direct and back data exchange with computer simulator of the thermal - and hydrodynamic processes in the oil-producing system. The features of transient non-stationary processes caused by origination, movement and disappearing of boundaries between gas-liquid mixture and water, and between the water-in-oil emulsion and oil-water-gas flow in the tubes of well and in the channels of ECP are studied. It is shown that characteristics of the oil-producing system at the operating stage are quite agreed with its characteristics obtained in solving the corresponding quasi-stationary problem.

**Gaidaenko V.I., Babakov A.V.**

Numerical simulation of unsteady flows in adjustable nozzles of braking engine

AbstractOn the basis of a conservative numerical method, the simulations of a viscous heat conductive gas flow inside diffusor part of spatial nozzles with partial overlapping of a critical section are carried out. Calculations are performed over a broad range of variations of parameters which affect the flow nature and patterns, in particular, flow separation inside the nozzle. Flow patterns both inside and outside the nozzle were visualised. The computations are performed using parallel algorithms implemented on a multi-processor cluster-architecture supercomputer.

**Lukashenko V.T., Maksimov F.A.**

Simulation of meteoroid parts spacing after fragmentation

AbstractThe algorithm of conjugation of aerodynamic and ballistic calculations for a system of bodies is presented, which is based on the method of modelling on a system of grids. The algorithm has been tested and used to calculate problems of supersonic spacing two meteoroid's fragments. The dependence of the escape velocity of identical fragments on the shape of the cross section is obtained, a tube of possible trajectories is constructed. It is shown that algorithm allows modeling the dynamics of bodies with rotation.

**Krivtsov V.M., Zubov V.I., Koterov V.N.**

Experience of numerical modelling of some aerodynamics problems

AbstractThe gas flows around prospective flight vehicles are considered. To compute the characteristics of these flows is a complicated task combining external and internal aerodynamics problems. The mathematical model used is based on the three-dimensional unsteady Reynolds-averaged compressible Navier-Stokes equations for a two-component equilibrium turbulent medium and a two-parameter semi-empirical turbulence model. The problems of equations approximation, grid generation and other details promoting achievement of the goal are discussed. The numerical investigations of three complex flows in modern devices are described: gas flows around aircraft with allowance for the flow/exhaust jet interaction, three-dimensional turbulent gas flows in complex nozzle systems, simulation of gas flow in cooled axial turbines. The conclusion is made about the ability to use the proposed algorithm for solving a wide range of stationary and non-stationary aerodynamic problems.

**Wolkov A.V., Duong D.T.**

Two approaches to the solution of the problem about water-gas finely dispersed mixture flow around wing element

AbstractIn this paper the system of water droplet equations is written out, the type of equations is investigated, and it is shown that this system is parabolic. It is caused by the absence of a complete system of linearly independent eigenvectors. Two ways of this system regularisation, which permit to transform the problem to a hyperbolic type, are presented. One of these methods is realised in the framework of high-accuracy Galerkin method with discontinuous basis functions (DGM), the other method is used in the finite volume method with second-order accuracy. The testing was carried out, which showed the advantage of the DGM method. As the tests, two examples of practical use of these methods are given: 1) the flow around a circular cylinder; 2) the flow around the NACA 0012 profile. It is shown that the results of numerical calculations are converged with grid step, and they are in satisfactory agreement with the experimental data.

**Aksenov A.**

A multidimensional multi-temperature gas dynamic code and the neutrino transport at the gravitational collapse and the supernova explosion.

AbstractA multitemperature code intended for the numerical solution of the multicomponent gas dynamics equations in problems with a high energy density in the matter is described. The velocities of all components with nonzero masses are assumed to be identical. The dynamic gas part is based on Godunov’s scheme and an efficient Riemann problem solver with an approximate local equation of state. As an example of the code application, the gravitational collapse of the massive star’s core with a neutrino transport is considered. The model includes the tabular equation of state of the matter (nuclei, nucleons, radiation, degenerate electron and positrons) and neutrino transport in the fame of the diffusion approximation with the flux limiters. Calculations show the importance of the large-scale convection instability in the centre of the forming proto-neutron star.

**Babakov A.V.**

Program package FLUX for the simulation of fundamental and applied problems of aerodynamics

AbstractBased on parallel algorithms of a conservative numerical method, a software package for simulating fundamental and applied fluid dynamics problems in a wide range of parameters is developed. The software is implemented on a cluster computer system. Examples of the numerical simulation of three-dimensional problems in various fields of fluid dynamics are discussed, including problems of external flow around bodies, investigation of aerodynamic characteristics of flying vehicles, flow in nozzles and interaction of landing multi-nozzles engine with landing field.

**Demchenko V.V.**

High-gradient method for the solution of first order hyperbolic type partial differential equations

AbstractThe effective difference method, based on characteristic directions isolation and consequent approximation of partial derivatives in pre-assigned finite-dimensional space, are suggested for the numerical simulation of physical processes in mechanics of continua and plasma physics. The method was approved on the model problem of one-dimensional discontinuity decay including the four principal variants: 1) with the formation of two shock waves; 2) a shock wave and a rarefaction wave; 3) the two rarefaction waves; 4) the two central rarefaction waves. Numerical solution convergence is investigated to the analytical solution mapping in pre-assigned finite-dimensional space with uniform valuation. The some examples of actual physical multidimensional problem solutions are considered using high-gradient method: 1) the cumulation of shocks waves; 2) Richtmyer-Meshkov instability development; 3) the instability of a collision surface during high-velocity impact.

**Shifrin E.G.**

On approximation of weakly compressible viscous fluids by incompressible ones.

AbstractWe consider the first initial-boundary value problem for the full Navier-Stokes system in the general case of thermodynamically non-piezometric media. We prove that if a solutions consequence for fluids of uniformly decreased compressibility is converged, then the limit, if exists, does not coincide with the solution for an incompressible fluid. A non-reduced part of the approximation error is proportional to the energy dissipation.

**Maksimov F.A., Ostapenko N.A., Zubin M.A.**

Complex Theoretical and Experimental Investigation of the Flow Structure around V-shaped Wings

AbstractThe results of a complex theoretical and experimental investigation of the flow structure around V-shaped wings, including the conical centre body, at supersonic flow velocities are presented. In the inviscid gas model conditions both the flow regime with shock waves attached to the leading edges and the flow regime with centred expansion wave at the leading edge of the leeward panel have been considered. The calculated and experimental data, obtained using the special optical method for the supersonic conical flows visualising in the conditions of the asymmetric flow, have been used as the object of the analysis. The applicability of the previously established criteria of inviscid vortex structures existence in the cases of the appropriate intensity contact discontinuity formation has been studied.

**Matyushin P.V.**

Mathematical modelling of the different flow regimes of a stratified viscous fluid around a disk

AbstractThe density stratified viscous fluid flows around a disk (with diameter d moving in a horizontal direction along his symmetry axis z with velocity U) have been simulated by the Navier-Stokes equations in the Boussinesq approximation using the numerical method SMIF (second-order accuracy in space and monotonous). The following classification of the flow regimes around a disk at 50 - Re - 500 has been obtained: I) homogeneous fluid (Fr > 10), II) quasi-homogeneous case (1.5 < Fr - 10), III) the abrupt reduction of the recirculation zone (RZ) (0.75 < Fr - 1.5); IV) the absence of RZ (0.45 < Fr - 0.75); V) a new RZ (0.35 < Fr - 0.45); VI) the two vertical vortices in new RZ (bounded by the internal waves) (Fr - 0.35). At (Fr > 1.5, Re > 270) and (Fr - 0.35, Re > 170) a periodical generation of the vortex loops has been observed in RZ (where Fr = U/(N•d), Re = U•d/ν, N and ν are the buoyancy frequency and the kinematical viscosity coefficient of the fluid). For the visualisation of the 3D vortex structures of the fluid flows the isosurfaces of β have been drawing, where β is the imaginary part of the complex-conjugate eigenvalues of the velocity gradient tensor.

**Tirsky G.A., Bragin M.D., Petrov M.N.**

Analytical solution of the equations of the physical theory of meteors for a single (non-fragmenting) body with a mass loss in a non-isothermal (arbitrary) atmosphere with a variable ablation parameter.

AbstractNumerical solution of the system of a coupled nonlinear ODE of the physical theory of meteors for determination of velocity, mass and angle of trajectory of meteoroid was obtained. For a straight trajectory equation for velocity and mass of meteoroid was solved analytically. In this case, due to the elimination of mass from the equation of motion analytical solution for velocity was expressed through integral power function with an account for variable ablation parameter. With the use of this solution velocity head, height and velocity at the moment of fragmentation are defined. Analytical solution for fragmenting meteoroid was obtained based on the progressive fragmentation model. Results of the theoretical study were applied for an explanation of aerodynamic parameters of the Chelyabinsk and Lost-City meteoroids.

**Shestopaloff A.Yu., Shestopaloff Y.K.**

New reconstruction and data processing methods for regression and interpolation analysis of big multidimensional data

AbstractThe problems of computational data processing involving regression, interpolation, extrapolation, reconstruction and imputation for big multidimensional datasets are becoming more important these days because of the quickly increasing availability of such data. The existing methods often have limitations, which either do not allow or make it difficult to accomplish many data processing tasks, because of demands for computational resources, difficulty working with high dimensions, etc. We propose a new concept and introduce two high-performance methods, which use local area predictors for finding outcome data. One method uses a gradient-based approach, while the second one employs the introduced family of nonlinear smooth approximating functions. The methods also provide a multidimensional outcome, when needed. We present numerical examples of up to one hundred dimensions and report in detail performance characteristics and various properties of new methods.

**Alekseev A.K.**

A posteriori error estimation on the ensemble of numerical solutions

AbstractThe ensemble of solutions, generated by solvers of different orders of approximation, is analysed from the viewpoint of the approximation error estimation. The distance between the true and numerical solutions in some metric is considered as the approximation error magnitude. The analysis of distances between the numerical solutions provides an opportunity for the estimation of the error magnitude upper bound. Numerical tests for the steady supersonic flows, governed by the two-dimensional Euler equations, are conducted to demonstrate the estimation of the approximation error magnitude in different metrics.

**Alikhanov A.A.**

Difference schemes with higher order of approximation for fractional diffusion equation

AbstractThe time fractional diffusion equation represents a linear integro - differential equation. Its solution not always can be found analytically; therefore it is necessary to use numerical methods. However, unlike the classical case, we require information about all the previous time layers, when numerically approximating a time fractional differential equation on a certain time layer. For that reason algorithms for solving the time fractional differential equations are rather time-consuming even in one - dimensional case. Upon transition to two - dimensional and three - dimensional problems their complexity considerably increases. In this regard constructing stable difference schemes of higher order approximation is a very important task.In the paper [1] has been found a special point for the interpolation approximation of the Caputo fractional derivative and derived a numerical differentiation $L_2-1_{\sigma}$ formula to approximate the Caputo fractional derivative at this point with the numerical accuracy of order $3-\alpha$ uniformly. The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second order in time for the time fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid $L_2$ - norm with the rate equal to the order of the approximation error are proved.In the paper [2] a new difference analog of the Caputo fractional derivative with the order of approximation $3-\alpha$, called $L1-2$ formula, is constructed. On the basis of this formula calculation of difference schemes for the time-fractional sub-diffusion equations in bounded and unbounded spatial domains and the fractional ODEs are carried out. However, until now stability and convergence of difference schemes constructed by the $L1-2$ formula remain a challenge.In the present paper, insignificantly modified $L2$ formula is proposed. The basic properties of this difference operator are investigated and on its basis difference schemes of a higher order of approximation for the fractional, variable and distributed order diffusion equation are constructed. The stability and convergence of the difference schemes are proved. The obtained results are supported by the numerical calculations carried out for some test problems. References [1] Anatoly A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys. 280, pp. 424-438 (2015). [2] G.H. Gao, Z.Z. Sun and H.W. Zhang A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys. 259, pp. 33-50 (2014)."

**Khokhlov N.I.**

On a class of compact grid-characteristic schemes

AbstractA class of two-point compact difference schemes of the 2–3rd orders of accuracy on a two-point coordinate stencil is considered for a one-dimensional transfer equation. All difference schemes are based on interpolation polynomials constructed on a given stencil. Based on the behaviour of the solution and the character of interpolation polynomials, we propose hybrid compact difference schemes of 2–3rd orders of accuracy on smooth functions producing solutions weakly smoothing the front of discontinuities.

**Oshkhunov M.M., Ulbasheva M.M.**

About modification of least square method for date approximation

AbstractThe new modification of the least square method for date analysis is presented. The idea of the method is minimisation of the sum of squared distances between the hyperplanes and the statistical points. The comparing analysis of new method and classical algorithms are given. In particular, non-uniqueness of solutions and the effectiveness of a new method for dispersion are shown. Numerical examples are presented.

**Shilov N.V., Anureev I.S., Promsky A.V., Kondratyev D.A., Berdyshev M.A.**

On the need of formal specification and verification of the standard mathematical functions

AbstractThe problem of validation of the standard mathematical functions is well-recognised by industrial computer engineering and academic software engineering and computational communities (but still is poorly understood by freshmen and inexperienced developers). In the talk, we try to promote (among professionals in computational mathematics) of ideas, experience, and state of the art of the formal specification and verification of the standard mathematical functions. The main case-study is formal (but still human-oriented) specification and formal (but just pen-and-paper) verification of the square root function. The specification is presented regarding total correctness assertions with use of precise arithmetic and the standard mathematical functions; verification is done in Floyd-Hoare style. A part of the purpose of the presented research is to extract and make explicit properties of the machine arithmetic that are sufficient to make verification sound.

**Prutko K.A.**

A structure of nonequilibrium relaxation region behind strong shock wave in the air with consideration of radiation

AbstractThis paper is devoted to the numerical research of the structure of nonequilibrium relaxation region behind strong wave determined by ionization. The ionization relaxation is of interest in studying the phenomena of space vehicle reentry into the Earth atmosphere at escape velocity and the low temperature plasma research in ground experimental facilities. The ionization process depends on population of high excited atom states which are depleting due to emitting the bound-bound transition photons. The collisional-radiative model was applied. It was shown that behind strong shock wave (V = 11-14 km/s) in the case of optically thin plasma, the air cools down due to radiation cooling and the electron concentration after avalanche ionization is lower than equilibrium one (calculated without radiation cooling). Also good agreement between calculation and experimental data for radiation intensity and excited state concentrations was obtained.

**Tikhonychev P.S.**

Automatic layered grid generation methods for high-speed flight simulation

AbstractTTo simulate the flight of bodies with high speeds, it is convenient to use a layered grid because of the simplicity of adapting the grid to the shock wave and clastering the grid to the surface of the body. But during the calculation, the surface of the body may change as result of ablation. Corners can be formed on the surface. The main problem of constructing layered grids on such surfaces is the self-intersection of mesh generators in the regions of corners. The report is based on papers [1] and [2]. For the construction of the grid in these works it is proposed to use mechanical and electrostatic analogies. The first is to use as forming lines of force of the electric field, which do not intersect. The second is the use of elastic forces to smooth the resulting grid. The work contains solutions of various problems arising in the generating of layered computational grids such as the collapsing of edges in the corners, the alignment of the outer boundaries of the grid, the adaptation of the external border of the mesh to the shock wave, creating a super-sonic exit boundary for the calculation of the external inviscid flow, the construction of layered meshes inside the body to calculate heating. All the methods described in the articles allow to generate and to adapt grids for different bodies with complex and changing shape in automatic mode. [1] P.S. Tikhonychev Method for Volume Layered Grid Generation for Problems of Hypersonic Flows around Vehicles with Changing Shape as a Result of Ablation Based on Electrostatic Analogy// Physical - Chemical Kinetics in Gas Dynamics. 2016. V.17 (3) [2] P.S. Tikhonychev Method of Automatic Construction of Dimensional Layered Grids for Calculating Heat- ing and Entrainment of Heat-Shielding Materials Based on a Mechanical Analogy//Cosmonautics and Rocket Engineering, 2017, V.96 (3)

**Golubev V.I., Favorskaya A.V.**

Elastic migration of seismic data.