Formation of Characteristic Polynomials on the Basis of Fractional Powers j of Dynamic Systems and Stability Problems of Such Systems

The article presents the creation of characteristic polynomials on the basis of fractional powers j of dynamic systems and problems related to the determination of the stability intervals of such systems.

Stabilised explicit Adams-type methods

In this work we present explicit Adams-type multi-step methods with extended stability intervals, which are analogous to the stabilised Chebyshev Runge – Kutta methods. It is proved that for any k ≥ 1 there exists an explicit k-step Adams-type method of order one with stability interval of length 2k. The first order methods have remarkably simple expressions for their coefficients and error constant. A damped modification of these methods is derived. In the general case, to construct a k-step method of order p it is necessary to solve a constrained optimisation problem in which the objective function and p constraints are second degree polynomials in k variables. We calculate higher-order methods up to order six numerically and perform some numerical experiments to confirm the accuracy and stability of the methods.

INVESTIGATION OF THE GEOMETRY OF THE D-PARTITION OF ONE-DIMENSIONAL PLANE OF PARAMETER OF THE CHARACTERISTIC EQUATION OF A CONTINUOUS SYSTEM

The paper considers two types of boundaries of the D-partition in the plane of one parameter of linear continuous systems given by the characteristic equation with real coefficients. The number of segments and intervals of stability of the X-partition curve is estimated. The maximum number of stability intervals is determined for different orders of polynomials of the equation of the boundary of the D-partition of the first kind (even order, odd order, one of even order, and the other of odd order). It is proved that the maximum number of stability intervals of a one-parameter family is different for all cases and depends on the ratio of the degrees of the polynomials of the equation of the D-partition curve. The derivative of the imaginary part of the expression of the investigated parameter at the initial point of the D-partition curve is obtained in an analytical form, the sign of which depends on the ratio of the coefficients of the characteristic equation and establishes the stability of the first interval of the real axis of the parameter plane. It is shown that for another type of the boundary of the D-partition in the plane of one parameter, there is only one interval of stability, the location of which, as for the previous type of the boundary of the stability region (BSR), is determined by the sign of the first derivative of the imaginary part of the expression of the parameter under study. Consider an example that illustrates the effectiveness of the proposed approach for constructing a BSR in a space of two parameters without using «Neimark hatching» and constructing special lines. In this case, a machine implementation of the construction of the stability region is provided. Considering that the problem of constructing the boundary of the stability region in the plane of two parameters is reduced to the problem of determining the BSR in the plane of one parameter, then the given estimates of the maximum number of stability regions in the plane of one parameter allow us to conclude about the number of maximum stability regions in the plane of two parameters, which are of practical interest. In this case, one of the parameters can enter nonlinearly into the coefficients of the characteristic equation.

The Triads Geometric Consistency Index in AHP-Pairwise Comparison Matrices

The paper presents the Triads Geometric Consistency Index ( T - G C I ), a measure for evaluating the inconsistency of the pairwise comparison matrices employed in the Analytic Hierarchy Process (AHP). Based on the Saaty’s definition of consistency for AHP, the new measure works directly with triads of the initial judgements, without having to previously calculate the priority vector, and therefore is valid for any prioritisation procedure used in AHP. The T - G C I is an intuitive indicator defined as the average of the log quadratic deviations from the unit of the intensities of all the cycles of length three. Its value coincides with that of the Geometric Consistency Index ( G C I ) and this allows the utilisation of the inconsistency thresholds as well as the properties of the G C I when using the T - G C I . In addition, the decision tools developed for the G C I can be used when working with triads ( T - G C I ), especially the procedure for improving the inconsistency and the consistency stability intervals of the judgements used in group decision making. The paper further includes a study of the computational complexity of both measures ( T - G C I and G C I ) which allows selecting the most appropriate expression, depending on the size of the matrix. Finally, it is proved that the generalisation of the proposed measure to cycles of any length coincides with the T - G C I . It is not therefore necessary to consider cycles of length greater than three, as they are more complex to obtain and the calculation of their associated measure is more difficult.

An Accurate SPH Scheme for Hypervelocity Impact Modeling

We focus in this paper on the use of a meshless numerical method called Smooth Particle Hydrodynamics (SPH), to solve fragmentation issues as Hyper Velocity Impact (HVI). Contrary to classical grid-based methods, SPH does not need any opening criteria which makes it naturally well suited to handle material failure. Nevertheless, SPH schemes suffer from well-known instabilities questioning their accuracy and activating nonphysical processes as numerical fragmentation. Many stabilizing tools are available in the literature based for instance on dissipative terms, artificial repulsive forces, stress points or Particle Shifting Techniques (PST). However, they either raise conservation and consistency issues, or drastically increase the computation times. It limits then their effectiveness as well as their industrial application. To achieve robust and consistent stabilization, we propose an alternative scheme called γ -SPH-ALE. Firstly implemented to solve Monophasic Barotropic flows, it is secondly extended to the solid dynamics. Particularly, based on the ALE framework, its governing equations include advective terms allowing an arbitrary description of motion. Thus, in addition of accounting for a stabilizing low-Mach scheme, a PST is implemented through the arbitrary transport velocity field, the asset of ALE formulations. Through a nonlinear stability analysis, CFL-like conditions are formulated ensuring the scheme conservativity, robustness, stability and consistency. Besides, stability intervals are defined for the scheme parameters determining entirely the stability field. Its implementation on several test cases reveals particularly that the proposed scheme faithfully reproduces the strain localization in adiabatic shear bands, a precursor to failure. By preventing spurious oscillations in elastic waves and correcting the so-called tensile instability, it increases both stability and accuracy with respect to classical approaches.

AHP-Group Decision Making Based on Consistency

The Precise consistency consensus matrix (PCCM) is a consensus matrix for AHP-group decision making in which the value of each entry belongs, simultaneously, to all the individual consistency stability intervals. This new consensus matrix has shown significantly better behaviour with regards to consistency than other group consensus matrices, but it is slightly worse in terms of compatibility, understood as the discrepancy between the individual positions and the collective position that synthesises them. This paper includes an iterative algorithm for improving the compatibility of the PCCM. The sequence followed to modify the judgments of the PCCM is given by the entries that most contribute to the overall compatibility of the group. The procedure is illustrated by means of its application to a real-life situation (a local context) with three decision makers and four alternatives. The paper also offers, for the first time in the scientific literature, a detailed explanation of the process followed to solve the optimisation problem proposed for the consideration of different weights for the decision makers in the calculation of the PCCM.

Фазовые переходы в жесткой доменной структуре феррит-гранатовой пленки

Spontaneous and magnetic field-induced phase transitions in a rigid domain structure of a uniaxial ferrite-garnet film are studied. It is shown that the temperature and field stability intervals of the lattice of cylindrical magnetic domains depend on the structure of the domain boundaries.

Cotton fibre selection and grading – a PROMETHEE-GAIA-based approach

Purpose In spinning industries, selection of the most appropriate fibre for yarn manufacturing plays an important role for achieving an optimal mix of several yarn characteristics, like maximum tenacity, elasticity and spinning ability; and minimum unevenness and hairiness. Identification of the best suited cotton fibre from a set of available alternatives in presence of different conflicting physical properties is often treated as a multi-criteria decision-making (MCDM) problem. The paper aims to discuss this issue. Design/methodology/approach In this paper, the preference ranking organisation method for enrichment of evaluations (PROMETHEE) and geometrical analysis for interactive aid (GAIA) methods are integrated to solve a cotton fibre selection problem. The PROMETHEE II method ranks the alternative cotton fibres based on their net outranking flows, whereas GAIA acts as a visual aid to strongly support the derived selection decision. The weight stability intervals for all the considered fibre properties (criteria) over which the position of the top-ranked cotton fibre remains unchanged are also determined. Findings The clusters of cotton fibres formed in the developed GAIA plane act as a yard stick for their appropriate grading to aid the blending process. The ranking of 17 cotton fibres as achieved applying the combined PROMETHEE-GAIA approach highly corroborates with the observations of the past researchers which proves its immense potentiality and applicability in solving fibre selection problems. Originality/value Two MCDM methods in the form of PROMETHEE II and GAIA are integrated to provide a holistic approach for cotton fibre grading and selection while taking into consideration all the available cotton fibre properties.