All issues

This paper covers deployable systems assembled from trapezium plates. When the plate package is unwrapped, a net shell with six loop cells is formed. It is proved that additional degrees of freedom appear in case of certain correlation between the sizes of the six loop faces. When thin plates were used, the continuum approximation of the deployed net could be interpreted as a shell with a wide variety of local curvatures. Kinematics of the continuum model is analyzed by the method of Cartan moving hedron. Mechanical behavior of continuum nets is studied when cylindrical hinges between the plates are completed of shape memory plastic materials. The paper researches into shell transformations from one stable form to the other. Various practical applications of the continuum nets are demonstrated.

In the current paper, we study a Galerkin–Petrov method for a parabolic equations of higher order in domain with a moving boundary. Asymptotic estimates for the convergence rate of approximate solutions are obtained.

Semiclassical asymptotic solutions with accuracy $O(D^{N/2})$, $N\geqslant3$ are constructed for the multidimensional Fisher–Kolmogorov–Petrovskii–Piskunov equation in the class of trajectory-concentrated functions. Using the symmetry operators a countable set of asymptotic solutions with accuracy $O(D^{3/2})$ is obtained. Asymptotic solutions of two-dimensional Fisher–Kolmogorov–Petrovskii–Piskunov equation are found in explicit
form.

This article is dedicated to the use of higher degree $S$-splines for solving equations of the elasticity theory. As an example we consider the solution to the equation of bending of a plate on a circle. $S$-spline is a piecewise-polynomial function. Its coefficients are defined by two conditions. The first part of the coefficients are defined by the smoothness of the spline. The rest are determined using the least-squares method. We consider class $C^4$ 7th degree $S$-splines.

We consider a model of stacked long Josephson junctions (LJJ), which consists of alternating superconducting and dielectric layers. The model takes into account the inductive and capacitive coupling between the neighbor junctions. The model is described by a system of nonlinear partial differential equations with respect to the phase differences and the voltage of LJJ, with appropriate initial and boundary conditions. The numerical solution of this system of equations is based on the use of standard three-point finite-difference formulae for discrete approximations in the space coordinate, and the applying the four-step Runge-Kutta method for solving the Cauchy problem obtained. Designed parallel algorithm is implemented by means of the MPI technology (Message Passing Interface). In the paper, the mathematical formulation of the problem is given, numerical scheme and a method of calculation of the current-voltage characteristics of the LJJ system are described. Two variants of parallel implementation are presented. The influence of inductive and capacitive coupling between junctions on the structure of the current-voltage characteristics is demonstrated. The results of methodical calculations with various parameters of length and number of Josephson junctions in the LJJ stack depending on the number of parallel computing nodes, are presented. The calculations have been performed on multiprocessor clusters HybriLIT and CICC of Multi-Functional Information and Computing Complex (Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna). The numerical results are discussed from the viewpoint of the effectiveness of presented approaches of the LJJ system numerical simulation in parallel. It has been shown that one of parallel algorithms provides the 9 times speedup of calculations.

We consider the approaches to the construction of methods for solving four-dimensional programming problems for calculating directions for multiple minimizations of smooth functions on a set of a given set of linear equalities. The approach consists of two stages.

At the first stage, the problem of quadratic programming is transformed by a numerically stable direct multiplicative algorithm into an equivalent problem of designing the origin of coordinates on a linear manifold, which defines a new mathematical formulation of the dual quadratic problem. For this, a numerically stable direct multiplicative method for solving systems of linear equations is proposed, taking into account the sparsity of matrices presented in packaged form. The advantage of this approach is to calculate the modified Cholesky factors to construct a substantially positive definite matrix of the system of equations and its solution in the framework of one procedure. And also in the possibility of minimizing the filling of the main rows of multipliers without losing the accuracy of the results, and no changes are made in the position of the next processed row of the matrix, which allows the use of static data storage formats.

At the second stage, the necessary and sufficient optimality conditions in the form of Kuhn–Tucker determine the calculation of the direction of descent — the solution of the dual quadratic problem is reduced to solving a system of linear equations with symmetric positive definite matrix for calculating of Lagrange's coefficients multipliers and to substituting the solution into the formula for calculating the direction of descent.

It is proved that the proposed approach to the calculation of the direction of descent by numerically stable direct multiplicative methods at one iteration requires a cubic law less computation than one iteration compared to the well-known dual method of Gill and Murray. Besides, the proposed method allows the organization of the computational process from any starting point that the user chooses as the initial approximation of the solution.

Variants of the problem of designing the origin of coordinates on a linear manifold, a convex polyhedron and a vertex of a convex polyhedron are presented. Also the relationship and implementation of methods for solving these problems are described.

Non-collapsing soliton-like wave functions are shown to exist in semiclassical approximation for the Bose-Einstein condensate model based on the Gross–Pitaevskii equation with attractive nonlinearity and external field of magnetic trap of special form.

The a priori analysis of approximation of magnetohydrodynamic equations on irregular quadrangular analysis was performed. The values of coefficients wich determine the misalignment norm for difference analogs of operators gradient and divergence were calculated. Was studied the influence of properties of grid cells on misalignment. For the numerical confirmation of obtained estimations were cited the examples of calculations with specifying identical initial data on different grids.

The modified two-sweep iteration procedure was proposed, built according to the least-squares method scheme, which determines progressive approximations to the periodic solution of Mathieu’s equation and his own function, considerably superior according to the accuracy earlier well-known results.

The approach to increase efficiency of Gill and Murray's algorithm of Newtonian methods of unconstrained optimization with step adjustment creation is offered, rests on Cholesky’s factorization. It is proved that the strategy of choice of the descent direction also determines the solution of the problem of scaling of steps at descent, and approximation by non-quadratic functions, and integration with a method of a confidential vicinity.