Vol 38, No 5 (2017)

We consider a rectangular plate connected along the outer contour with an absolutely rigid and fixed support through a low tough elastic support elements included in the class of transversely soft foundations. It is assumed that the contact interaction of plate at its points of faces of the connection to the support elements there is no relative slip and separation, and the opposite boundary surfaces of the support elements are fixed. Deformation of plate’s mid-surface is described by geometrically nonlinear relationships of the classical plate theory based on the hypothesis of the Kirchhoff–Love (the first option), and refined Timoshenko model taking into account the transverse shear compression (second version). The mechanics of support elements is described by the linearized equations of three-dimensional theory of elasticity, which have been simplified in the framework of transversely soft layer model. By integrating the latter equations along the transverse coordinate and satisfying to the conditions of the kinematic coupling of plate to the support elements at their initial compression in the thickness direction, two-dimensional geometrical nonlinear equations and their corresponding boundary conditions have been introduced, which describe the contact interaction of elements of the concerned deformable system. Simplification of derived relationships for the case when foundations have a symmetrical layer structure is carried out.

A 2 D nonstationary problem of an orthotropic one-component elastic layer, accounting for diffusion, is considered. A locally balanced model of elastic diffusion is used, which includes a coupled set of equations of elastic body motion and a mass-transfer equation. On the layer boundaries normal displacement, tangential stress and diffusive flow are assumed. At an initial time, the layer is in an undisturbed state.

The solution is sought in an integral form, as a double convolution in time and along the space coordinate of Green functions and the right-hand sides of the boundary equations. To construct a solution of the initial problem, Laplace time transform, exponential Fourier transform along the space coordinate in the direction of the layer surface, reduction to zero boundary conditions, and expansion into incomplete Fourier series along a coordinate directed through the depth of the layer are successively used. Thus, the initial problem is reduced to a set of linear algebraic equations, from which forms of the sought functions are found. It is shown that the necessary condition of the applicability of the present algorithm is orthotropy of the medium in question. Inversion of Laplace transform of the sought functions is reduced to computing the originals of the rational functions, which is done analytically, using deductions. It is found that all Green functions are even (odd) relative to Fourier transform parameters, which makes it possible, when the right-hand sides of the boundary conditions are even (odd), to reduce the inversion of exponential Fourier transform to the inversion of sinus-, cosine-transform. In that case, quadrature formulas of medium rectangles are used to find the originals of Fourier transformants.

An example is considered, where surface perturbation is modeled with Heaviside time function and a Gaussian along the space coordinate. A homogenized medium of interlaced layers of aluminum and copper is used as a model of an orthotropic medium. The computational results are presented both analytically and in the form of 3D diagrams of the diffusant concentration increment and displacement vector components.

The problem of one-dimensional non-stationary wave propagation in viscoelastic half space is considered. Relaxation kernel is considered exponential. Initial conditions are set to zero, displacement is determined on the half space boundary. The solution is represented in the form of perturbation and surface Green function resultant. For determination of Green function time Laplace transform is used. Its inverse transform is carried out both analytically expanding Green function in series and numerically. A good coincidence of analytical and numerical Green function calculation results is shown. The final solution is determined analytically. Examples of calculations are represented.

The paper presents a method of instantaneous construction of relative permeability pseudo functions in analytical form upscaled to a coarser computational grid using a system of artificial neural networks. The coefficients of these functions can be forecasted by the neural network. The learning dataset is based on a preliminary series of calculations at the reference values of the system parameters the exponents of the initial functions, the liquid phases viscosity ratio, the statistical parameters of distribution laws of the reservoir’s properties. The latter may be obtained according to the primary well logging data with no need for building a detailed geological model.

A positive semi-definite eigenvalue problem for second-order self-adjoint elliptic differential operator definedon a bounded domain in the planewith smooth boundary and Dirichlet boundary condition is considered. This problem has a nondecreasing sequence of positive eigenvalues of finite multiplicity with a limit point at infinity. To the sequence of eigenvalues, there corresponds an orthonormal system of eigenfunctions. The original differential eigenvalue problem is approximated by the finite element method with numerical integration and Lagrange curved triangular finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions are established.

The paper studies approaches to numerical solving huge-scale quasiseparable optimization problems. The main idea is based on using gradient methods with simple iteration structure instead more intelligent techniques, which is widely used for solving traditional, small-sized problems. The results of numerical experiments for a number of test quasiseparable optimization problems with dimensions up to 10¹⁰ variables are presented.

The paper is presented some classes of MDS matrices of size 4 × 4 with the maximum number of units and minimal number of non unit elements. This class of matrices is widely used as diffuse maps when building block type algorithms and hash functions that provide protection against certain methods of analysis.

Presented are the results of experiments on calculation of Probability Distribution Functions for elevations of waters waves in numerical tank. Statistics of waves of anomalous amplitude, or freak-waves were compared both for nonlinear and linear models. Obviously, linear model demonstrates the exact Rayleigh distribution of surface elevations while PDFs for nonlinear equation have tails (for large elevations) similar to Rayleigh distribution, but with larger σ.

In the paper we consider the problem of searching for coexisting modes in a nonlinear boundary value problem with a delay from population dynamics. For this we construct the asymptotic of spatially homogeneous cycle using the normal forms method and research the dependence of its stability on the diffusion parameter. Then we find coexisting attractors of the problem using numerical methods. Numerical experiment required an application of massively parallel computing systems and adaptation of solutions search algorithms to them. Based on the numerical analysis we come to the conclusion of the existence in the boundary value problem of solutions of two types. The first type has a simple spatial distribution and inherits the properties of a homogeneous solution. The second called the mode of self-organization is more complex distributed in space and is much more preferred in terms of population dynamics.

An approach to monitoring temporal evolution of thematic clusters with evaluating their relations on base of probability and entropy methods is presented. It allows to get a temporary map of nested topics with their short annotations, concerning a predetermined main theme. The methods of semantic analysis of texts to generate topics and to find the most emotive of them to reflect a social significance are used. The technology word2vec was implemented to determine the relation of topics and evaluate their proximity to the main theme.

To increase the usability the visualization of nested topics is realized on base of a WEB interface. The proposed approach complements well the popular software for analyzing big volumes of data such as Elasticsearch (search for thematically similar documents). Results of case study of analyzing the theme “AEROFLOT” on base of news corpus which consists of 3 million messages is presented.

Let us consider a mathematical model of dynamic system, which is presented as a chain of three connected, singularly perturbed nonlinear differential equations. In the further text there were researched the questions of existence and stability of periodic solutions of this system due to a bifurcational analysis of special two-dimensional map. Also the special attention is paid to the number of coexisting stable regimes.

The results of analytical and numerical investigation of quantum dissipative dynamics a particle moving in a potential well with two equilibrium states is presented. A classical analogue of this problem is the capture into the resonance a particle in a double-well potential, when two states of equilibrium in the phase space are separated by a separatrix. The probability of finding the system in one of states of equilibrium for the quantum particle, depending on initial conditions and noise, is discussed.

Paper considers magnetic properties of the transition metals at nanoscale level, the total energies of the several collinear spin-configurations are calculated by using VASP software package. Energy of the film formation and the magnetic moment values were obtained for trilayer structures Ni/Cu(001)/Ni, Ni/Cu(111)/Ni and Co/Cu(001)/Co as a function of the convergence parameters and taking into account the relaxation effects, and magnetic film thickness. An influence of various Ni atom positions (ontop, bridge, fcc) and additional Cu cap layers on Ni/Cu/Ni system total energy was investigated. The exchange interaction parameters for Co/Cu(001)/Co and Fe/Cr(001)/Fe structures were calculated within a classical Heisenberg model. The finite temperature magnetism of these structures was studied by Monte-Carlo simulations with the use of the exchange interaction parameters obtained from the ab initio results.

We introduce a class of stochastic networks in which synchronization between nodes is modelled by a message passing mechanism with heterogeneous Markovian routing. We present a series of results about probability distributions related to steady states of such models.

This report focuses on technology of supercomputer simulation of nonlinear processes in the cores, extracted from oil and gas production wells in order to study the properties of hydrocarbon reservoirs. One of modern approaches to solving these kind problems is to create multiphysical mathematical model of core for its study by computer methods. This approach minimizes the number of natural experiments and predicts the evolution of layers properties. Also it allows to predict oil and gas recovery of layers for a long time period. However, implementation of this technology called “virtual core” requires the following: 1) to create multiparametrical model of core as close as possible to the reality; 2) to include the multicomponent and multiphase composition and complex real geometry of core in consideration; 3) to develop a computational framework for modeling the seepage of multicomponent liquid and gas mixtures through the core; 4) to carry out large-scale calibration calculations. In this paper, an attempt to create such a multifactor mathematical model and computational foundations for its computing and supercomputing analysis is made.

We simulate model for evolution of local virtual time profile in conservative parallel discrete event the simulation (PDES) algorithm with long-range communication links. The main findings of simulation are that i) growth exponent depends logarithmically on the concentration p of long-range links; ii) utilisation of processing elements time decreases slowly with p. Thismeans that the conservative PDES with long-range communication links is fully scalable.

The density functional theory (DFT) is a research tool of the highest importance for electronic structure calculations. It is often the only affordable method for ab initio calculations of complex materials. The pseudopotential approach allows reducing the total number of electrons in the model that speeds up calculations. However, there is a lack of pseudopotentials for heavy elements suitable for condensed matter DFT models. In this work, we present a pseudopotential for uranium developed in the Goedecker–Teter–Hutter form. Its accuracy is illustrated using several molecular and solid-state calculations.