Optimal with respect to accuracy methods for evaluating hypersingular integrals

In this paper we constructed optimal with respect to order quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions Ωur,γ(Ω,M), Ω¯ur,γ(Ω,M), Ω=[−1,1]l, l=1,2,…,M=Const, and γ is a real positive number. The functions that belong to classes Ωur,γ(Ω,M) and Ω¯ur,γ(Ω,M) have bounded derivatives up to the rth order in domain Ω and derivatives up to the sth order (s=r+⌈γ⌉) in domain Ω∖Γ, Γ=∂Ω. Moduli of derivatives of the vth order (r<v≤s) are power functions of d(x,Γ)−1(1+|lnd(x,Γ)|), where d(x,Γ) is a distance between point x and Γ. The interest in these classes of functions is due to the fact that solutions of singular and hypersingular integral equations are their members. Moreover various physical fields, in particular gravitational and electromagnetic fields belong to these classes as well. We give definitions of optimal with respect to accuracy methods for solving hypersingular integrals. We constructed optimal with respect to order of accuracy quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions Ωur,γ(Ω,M) and Ω¯ur,γ(Ω,M).

Numerical analysis of heating by a current pulse of a niobium nitride membrane in its longitudinal section

A numerical calculation of the evolution of the temperature distribution in the longitudinal section of a niobium nitride membrane when it is heated by an electric current pulse is performed. Mathematical modeling was carried out on the basis of a two-dimensional initial-boundary value problem for an inhomogeneous heat equation. In the initial boundary value problem, it was taken into account that current and potential contacts to the membrane serve simultaneously as contacts for heat removal. The case was considered for the third from the left and the first from the right initial-boundary value problem. Analysis of the numerical solution showed that effective heat removal from the membrane can be provided by current-carrying and potential clamping contacts made, for example, of beryllium bronze. This makes it possible to study the current-voltage characteristics of superconducting membranes near the critical temperature of the transition to the superconducting state by currents close to the critical density without significant heating.

On modelling of thermodynamic interaction of particles suspended in two-dimensional medium

Analytical description of temperature distribution in a medium with foreign inclusions is difficult due to the complicated geometry of the problem, so asymptotic and numerical methods are usually used to model thermodynamic processes in heterogeneous media. To be convinced in convergence of these methods the authors consider model problem about two identical round particles in infinite planar medium with temperature gradient which is constant at infinity. Authors refine multipole expansion of the solution obtained earlier by continuing it up to higher powers of small parameter, that is nondimensional radius of thermodynamically interacting particles. Numerical approach to the problem using ANSYS software is described; in particular, appropriate choice of approximate boundary conditions is discussed. Authors ascertain that replacement of infinite medium by finite-sized domain is important source of error in FEM. To find domain boundaries in multiple inclusions’ problem the authors develop “fictituous particle” method; according to it the cloud of particles far from the center of the cloud acts approximately as a single equivalent particle of greater size and so may be replaced by it. Basing on particular quantitative data the dependence of domain size that provides acceptable accuracy on thermal conductivities of medium and of particles is explored. Authors establish series of numerical experiments confirming convergence of multipole expansions method and FEM as well; proximity of their results is illustrated, too.

Numerical modeling of the formation of D-T mixture spherical layer in micro-targets of LTS

The article presents a two-dimensional economical computational model of the formation of D-T mixture cryogenic layer in a spherical shell. The model is based on the description of the motion of the gas phase in the Boussinesq approximation. The thermal problem is a Stefan problem with a gas-solid phase transition. The technique is based on the finite volume method, the use of a structured mobile grid, whose movement is associated with the separation of the phase front, implicit approximations and the method of splitting two-dimensional equations in directions into one-dimensional equations. It is numerically shown that, due to natural radioactivity, the target is symmetrized. A calculated estimation of the symmetrization time for one geometry of the target with different filling coefficients is carried out.

On topology of manifolds admitting gradient-like calscades with surface dynamics and on growth of the number of non-compact heteroclinic curves

We consider a class GSD(M3) of gradient-like diffeomorphisms with surface dynamics given on closed oriented manifold M3 of dimension three. Earlier it was proved that manifolds admitting such diffeomorphisms are mapping tori under closed orientable surface of genus g, and the number of non-compact heteroclinic curves of such diffeomorphisms is not less than 12g. In this paper, we determine a class of diffeomorphisms GSDR(M3)⊂GSD(M3) that have the minimum number of heteroclinic curves for a given number of periodic points, and prove that the supporting manifold of such diffeomorphisms is a Seifert manifold. The separatrices of periodic points of diffeomorphisms from the class GSDR(M3) have regular asymptotic behavior, in particular, their closures are locally flat. We provide sufficient conditions (independent on dynamics) for mapping torus to be Seifert. At the same time, the paper establishes that for any fixed g geq1, fixed number of periodic points, and any integer n≥12g, there exists a manifold M3 and a diffeomorphism f∈GSD(M3) having exactly n non-compact heteroclinic curves.

Theoretical Analysis of Fully Conservative Difference Schemes with Adaptive Viscosity

For the equations of gas dynamics in Eulerian variables, a family of two-layer in time completely conservative difference schemes with space-profiled time weights is constructed. Considerable attention is paid to the methods of constructing regularized flows of mass, momentum, and internal energy that do not violate the properties of complete conservatism of difference schemes of this class, to the analysis of their amplitudes and the possibility of their use on non-uniform grids. Effective preservation of the balance of internal energy in this type of divergent difference schemes is ensured by the absence of constantly operating sources of difference origin that produce "computational"entropy (including those based on singular features of the solution). The developed schemes can be easily generalized in order to calculate high-temperature flows in media that are nonequilibrium in temperature (for example, in a plasma with a difference in the temperatures of the electronic and ionic components), when, with the set of variables necessary for describing the flow, it is not enough to equalize the total energy balance.

Investigation of dynamic processes in pressure measurement systems for gas-liquid media

Initial-boundary value problems for systems of differential equations are considered, which are mathematical models of the mechanical system "pipeline - pressure sensor". In such a system, to mitigate the effects of vibration accelerations and high temperatures, the sensor is located at a certain distance from the engine and is connected to it via a pipeline. The "pipeline - pressure sensor" system is designed to measure pressure in gas-liquid media, for example, to control the pressure of the working medium in the combustion chambers of engines. On the basis of the proposed models, the joint dynamics of the sensitive element of the pressure sensor and the working medium in the pipeline is studied. To describe the motion of the working medium, linear models of fluid and gas mechanics are used, to describe the dynamics of a sensitive element, linear models of the mechanics of a deformable solid are applied. Analytical and numerical methods for solving initial-boundary value problems under study are presented. The numerical study of the initial-boundary value problem was carried out on the basis of the Galerkin method. In analytical study using the introduction of averaged characteristics, the solution of the original two-dimensional problem is reduced to the study of a one-dimensional model, whose further study made it possible to reduce the solution of the problem to the study of a differential equation with a deviating argument. Also, a numerical experiment is carried out and an example of calculating the deflection of the sensor’s moving element is presented.

Simulation of switchers for CNOT-gates based on optical waveguide interaction with coupled mode theory

The paper is devoted to simulation of interactions in the system of two symmetrical slab optical waveguides, that guide exactly two guided modes with the aim to use the directional coupler as a switcher for CNOT gate in the waveguide model of quantum-like computations. The coupling mode theory is used to solve the system of Maxwell equations. The asymptotic analysis is applied to simplify the system of differential equations, so an approximate analytic solution can be found. The solution obtained is used for the quick directional coupler parameters adjusting algorithm, so the power exchange in the system occurs as that of correctly working CNOT-gate switcher. Moreover, the finite difference method is used to solve the stricter system of equations, that additionally takes into account the process of power exchange between different order guided modes, so the computational error of the device can be estimated. It was obtained, that the possible size of the device may not exceed 1 mm in the largest dimension, while the computational error does not exceed 3%.