Adaptation of gas-dynamic characteristic arrays to automated ballistics support of spacecraft flight

The current level of the design and use of new-generation spacecraft calls for a maximally automated ballistics support of engineering developments. An integral part of the solution of this problem is the development of an effective tool to adapt discrete functions of gas-dynamic characteristics to the solution of various problems that arise in the development and use of space complexes. Simplifying the use of bulky information arrays together with improving the accuracy of approximation of key coefficients will significantly improve the ballistics support quality. The aim of this work is to choose an optimum method for the approximation of a discrete function of two variable spacecraft aerodynamic characteristics. Based on the analysis of the advantages and drawbacks of basic methods of approximation by two fitting criteria: the maximum error and the root-mean-square deviation, recommendations on this choice were made. The methods were assessed by the example of the aerodynamic coefficients of the Sich-2M spacecraft’s simplified geometrical model tabulated as a function of the spacecraft orientation angles relative to the incident flow velocity. Multiparameter numerical studies were conducted for different approximation methods with varying the parameters of the approximation types under consideration and the approximation grid density. It was found that increasing the number of nodes of an input array does not always improve the accuracy of approximation. The node arrangement exerts a greater effect on the approximation quality. It was established that the most easily implementable method among those considered is a step interpolation, whose advantages are simplicity, quickness, and limitless possibilities in accuracy improvement, while its significant drawbacks are the lack of an analytical description and the dependence of the accuracy on the grid density. It was shown that spline functions feature the best approximating properties in comparison with other mathematical models. A polynomial approximation or any approximation by a general form function provide an analytical description with a single approximating function, but their accuracy of approximation is not so high as that provided by splines. It was found that there exists no approximation method that would be best by all criteria taken together: each method has some advantages, but at the same time, it has significant drawbacks too. An optimum approximation method is chosen according to the features of the problem, the priorities in approximation requirements, the required degree of approximation, and the initial data organization method.

Application of Bullseye Diagram in Teaching of “Variance”

Variance and standard deviation, as a kind of discrete function, are an important content in high school teaching, and they are widely used in data statistics. For high school students, they have difficulty understanding abstract concepts such as stability, volatility, and degree of dispersion when they are first exposed to discrete functions. In actual teaching, you can apply “bullseye chart” to “variance” teaching, and use abstract concept diagrams. Demonstration helps high school students to intuitively understand the concept of “variance” and the nature of dispersion.

On the seven-diagonals splitting for the cubic spline wavelets with six vanishing moments on an interval

This study uses a zeroing property of the first six moments for constructing a splitting algorithm for the cubic spline wavelets. First, we construct a system of cubic basic spline-wavelets, realizing orthogonal conditions to all polynomials up to any degree. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the orthogonality to all polynomials up to the fifth degree on the closed interval. The originality of the study consists of obtaining implicit finite relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, the resulting transformation matrix has seven diagonals, instead of five as in the previous case with four zero moments. A modification of the system is performed, which ensures a strict diagonal dominance, and, consequently, the stability of the calculations. The comparative results of numerical experiments on approximating and calculating the derivatives of a discrete function are presented.

On ways to increase protection of special structures from impact action

The technique of research of dynamic processes of elements of engineering constructions of special purpose from explosive action of projectiles is developed. Elastically reinforced beams with hinged ends were chosen for the physical model of elements of engineering structures. It is assumed that the elastic properties of the latter satisfy the nonlinear technical law of elasticity. A mathematical model of the process of a series of impact actions of projectiles at different points of the element of the protective structure is constructed. The latter is a boundary value problem for a partial differential equation. Its peculiarity is that the external dynamic action is a discrete function of linear and time variables. To determine the dynamic effect of a series of impacts on the object under study, and thus the level of protection of the structure, the basic ideas of perturbation theory methods are extended to new classes of systems. This allowed to obtain an analytical dependence of the deformation of the elastically reinforced element on the basic physical and mechanical characteristics of the material of the protective element, its reinforcement and the characteristics of the external action of the projectiles. It is shown that the most dangerous cases, given the security of the structure, are those when the impact is repeated at equal intervals, in addition, the point of impact is closer to the middle of the protective element. The obtained theoretical results can be the basis for selection at the stage of designing the main physical and mechanical characteristics of the elements of engineering structures and their reinforcement in order to reliably protect personnel and equipment from the maximum possible impact on it of the shock series of projectiles. The reliability of the obtained results is confirmed by: a) generalization of widely tested methods to new classes of dynamical systems; b) obtaining in the limit case the consequences known in scientific sources concerning the linearly elastic characteristics of the elements of protective structures; c) their consistency with the essence of the physical process itself, which is considered in the work.

On the use of the scattering amplitude in coherent X-ray Bragg diffraction imaging

Lens-less imaging of crystals with coherent X-ray diffraction offers some unique possibilities for strain-field characterization. It relies on numerically retrieving the phase of the scattering amplitude from a crystal illuminated with coherent X-rays. In practice, the algorithms encode this amplitude as a discrete Fourier transform of an effective or Bragg electron density. This short article suggests a detailed route from the classical expression of the (continuous) scattering amplitude to this discrete function. The case of a heterogeneous incident field is specifically detailed. Six assumptions are listed and quantitatively discussed when no such analysis was found in the literature. Details are provided for two of them: the fact that the structure factor varies in the vicinity of the probed reciprocal lattice vector, and the polarization factor, which is heterogeneous along the measured diffraction patterns. With progress in X-ray sources, data acquisition and analysis, it is believed that some approximations will prove inappropriate in the near future.

On Five-Diagonal Splitting for Cubic Spline Wavelets with Six Vanishing Moments on a Segment

In this study, we use the vanishing property of the first six moments for constructing a splitting algorithm for cubic spline wavelets. First, we construct the corresponding wavelet space that satisfies the orthogonality conditions for all fifth-degree polynomials. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the closed interval. The originality of the study consists in obtaining implicit relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, in contrast to the case with two zero moments, the resulting transformation matrix has five (instead of three) diagonals. The results of numerical experiments on calculating the derivatives of a discrete function are presented.

METHODS OF SYNTHESIS OF REGULATORS FOR HYDRODYNAMIC CONTROL SYSTEMS WITH DISTRIBUTED PARAMETERS

An important problem of choosing a method for synthesizing regulators for constructing a system for controlling the parameters of hydrolithospheric processes is considered. The article deals with the issues of system analysis and synthesis of control systems with distributed parameters. The existing ways of solving mathematical equations describing models with spatial coordinates are shown. The main methods when there are solutions to the mathematical model are identified: analytical construction of optimal regulators and the structural method of analysis. Methods of approximation in cases where there is no solution to the mathematical model are described: methods of finite-dimensional approximation; decomposition of bilinear control systems; frequency method. The main provisions of the methods for solving the problem of discretization in partial derivatives are given. The preference of the frequency method of regulator synthesis in the creation of control systems for hydrolithospheric processes is shown. A hodograph is obtained in the form of logarithmic amplitude and phase frequency surfaces, which can be used to interpret the Nyquist stability criterion from graphs. A frequency method for the synthesis of multidimensional systems is considered, when the input effects to a distributed controller are implemented as a discrete function in space. The condition under which the object belongs to the class of spatially invariant objects is shown. It is concluded that the frequency method of regulator synthesis is the most convenient tool for creating systems with distributed parameters.

Discrete Hardy Spaces for Bounded Domains in $${\mathbb {R}}^{n}$$

Discrete function theory in higher-dimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as half-space, while the case of bounded domains generally remained unconsidered. Therefore, this paper presents the extension of the higher-dimensional function theory to the case of arbitrary bounded domains in $${\mathbb {R}}^{n}$$ R n . On this way, discrete Stokes’ formula, discrete Borel–Pompeiu formula, as well as discrete Hardy spaces for general bounded domains are constructed. Finally, several discrete Hilbert problems are considered.

JUSTIFICATION OF PARAMETERS OF THE 2D CDP FIELD SYSTEM

When performing seismic observations, 2D seismograms of a common shot point are represented as a discrete function of two variables, i.e. time and receiver – source offset. When recording a wave field using single seismic receivers placed small distance apart (UniQ technology), two goals are pursued : maintaining high frequencies of reflected signals by eliminating the effect of microstatics and fulfilling the Kotelnikov sampling theorem when discretizing a continuous field with respect to a spatial variable, thereby eliminating the effect of spatial aliasing of regular interference waves. At the stage of digital processing, this allows to solve the problem of extracting useful signals and suppressing noise more effectively. Taking the idea of a close array of receivers as a whole, it is proposed to optimize the profile observation system by source – receiver spacing combining analog and digital grouping of seismic receivers. In this case, the spatial sampling of the field is determined by the distance between the centers of receiver groups, and the parameters of the analog – digital grouping are calculated from the condition of suppressing spatial aliasing frequencies. Based on the analysis of static corrections obtained during processing of previous seismic studies, a method is proposed for assessing the effect of lateral microstatics variations on the results of analog grouping.