linear algebraic equations
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Author(s):  
Mariya Ihorivna Shapovalova ◽  
Oleksii Oleksandrovich Vodka

Modern trends in the development of mechanical engineering and other industries related to the production of materials and structures with a given set of physical, mechanical, and technological properties are aimed at reducing material consumption, energy consumption, increasing accuracy, reliability, and competitiveness of the manufactured product. Therefore, the creation of mathematical methods for assessing the stress state of structural elements based on the analysis of the elastic characteristics of a material, taking into account the peculiarities of its internal microstructure, is an actual task. The considered algorithm includes the following stages: identification of strength parameters using data obtained from images of the material microstructure; study of the stress-strain state of the model based on the variational-difference finite element method; formation of a system of linear algebraic equations for solving the problem of analyzing the elastic properties of a material using the plane problem of the theory of elasticity; construction of the material yield surface for a series of tests based on the strength criteria of composite materials, taking into account the different resistance of the material under tensile and compressive loads. Based on the developed mathematical model, the SSS and the yield surface of the plate with a hole are estimated. Structural analysis is performed at the macro and micro levels. The occurrence of plastic deformations at the micro-level can lead to the development of cracks and structural damage at the macro level. As a result of the study, the probability of plastic deformation in the plate is determined, and the critical zones of the model are established. The practical significance of the results obtained is to create an approach to assessing the mechanical properties of a material, such as elastic modulus, shear modulus, Poisson's ratio, and their probabilistic characteristics following the internal material structure. The proposed approach contributes to the expansion of knowledge about the material and allows to increase the valuable information obtained by modeling. To assess the probability of plastic deformations, the generated method uses the entire set of probabilistic characteristics of the yield surface.


2021 ◽  
Vol 20 ◽  
pp. 717-728
Author(s):  
Boris M. Shumilov

In this study, the method for decomposing splines of degree m and smoothness C^m-1 into a series of wavelets with zero moments is investigated. The system of linear algebraic equations connecting the coefficients of the spline expansion on the initial scale with the spline coefficients and wavelet coefficients on the embedded scale is obtained. The originality consists in the application of some preconditioner that reduces the system to a simpler band system of equations. Examples of applying the method to the cases of first-degree spline wavelets with two first zero moments and cubic spline wavelets with six first zero moments are presented. For the cubic case after splitting the system into even and odd rows, the resulting matrix acquires a seven-diagonals form with strict diagonal dominance, which makes it possible to apply an effective sweep method to its solution


2021 ◽  
Vol 5 (4) ◽  
pp. 70-78
Author(s):  
Lev Raskin ◽  
Larysa Sukhomlyn ◽  
Dmytro Sagaidachny ◽  
Roman Korsun

Known technologies for analyzing Markov systems use a well-operating mathematical apparatus based on the computational implementation of the fundamental Markov property. Herewith the resulting systems of linear algebraic equations are easily solved numerically. Moreover, when solving lots of practical problems, this numerical solution is insufficient. For instance, both in problems of structural and parametric synthesis of systems, as well as in control problems. These problems require to obtain analytical relations describing the dependences of probability values of states of the analyzed system with the numerical values of its parameters. The complexity of the analytical solution of the related systems of linear algebraic equations increases rapidly along with the increase in the system dimensionality. This very phenomenon manifests itself especially demonstratively when analyzing multi-threaded queuing systems.  Accordingly, the objective of this paper is to develop an effective computational method for obtaining analytical relations that allow to analyze high-dimensional Markov systems. To analyze such systems this paper provides for a decomposition method based on the idea of phase enlargement of system states. The proposed and substantiated method allows to obtain analytical relations for calculating the distribution of Markov system states.  The method can be effectively applied to solve problems of analysis and management in high-dimensional Markov systems. An example has been considered


2021 ◽  
Vol 28 (4) ◽  
pp. 434-451
Author(s):  
Gleb D. Stepanov

The article considers a method for solving a linear programming problem (LPP), which requires finding the minimum or maximum of a linear functional on a set of non-negative solutions of a system of linear algebraic equations with the same unknowns. The method is obtained by improving the classical simplex method, which when involving geometric considerations, in fact, generalizes the Gauss complete exclusion method for solving systems of equations. The proposed method, as well as the method of complete exceptions, proceeds from purely algebraic considerations. It consists of converting the entire LPP, including the objective function, into an equivalent problem with an obvious answer. For the convenience of converting the target functional, the equations are written as linear functionals on the left side and zeros on the right one. From the coefficients of the mentioned functionals, a matrix is formed, which is called the LPP matrix. The zero row of the matrix is the coefficients of the target functional, $a_{00}$ is its free member. The algorithms are described and justified in terms of the transformation of this matrix. In calculations the matrix is a calculation table. The method under consideration by analogy with the simplex method consists of three stages. At the first stage the LPP matrix is reduced to a special 1-canonical form. With such matrices one of the basic solutions of the system is obvious, and the target functional on it is $ a_{00}$, which is very convenient. At the second stage the resulting matrix is transformed into a similar matrix with non-positive elements of the zero column (except $a_{00}$), which entails the non-negativity of the basic solution. At the third stage the matrix is transformed into a matrix that provides non-negativity and optimality of the basic solution. For the second stage the analog of which in the simplex method uses an artificial basis and is the most time-consuming, two variants without artificial variables are given. When describing the first of them, along the way, a very easy-to-understand and remember analogue of the famous Farkas lemma is obtained. The other option is quite simple to use, but its full justification is difficult and will be separately published.


2021 ◽  
pp. 106-117
Author(s):  
S. O Papkov

It has been for the first time that an analytical solution to the problem of free vibrations of a cantilevered thick orthotropic plate is presented. This problem is quite cumbersome for using the exact methods of the theory of elasticity; therefore, methods based on the variational approach were developed to solve it. The paper suggests using the superposition method to construct a general solution of the vibration equations of a plate in the series form of particular solutions obtained with the help of a variables separation. The particular solutions of one of the coordinates are built in the form of trigonometric functions of a special type (modified trigonometric system). The constructed solution, in contrast to the solutions known in the literature on the basis of the variational approach, accurately satisfies the equations of vibrations. The use of a modified trigonometric system of functions makes it possible to obtain uniform formulas for even and odd vibration shapes and to reduce the quantity of boundary conditions on the plate sides from twelve to nine ones, while five of the nine boundary conditions are also accurately satisfied. The structure of the presented solution on the plate boundary is such that, each of the kinematic or force characteristics of the plate is represented as a sum of two series, i.e. a trigonometric series and a series in hyperbolic functions. Remaining boundary conditions make it possible to obtain an infinite system of linear algebraic equations with respect to the unknown coefficients of the series representing the solution. The convergence of the solution by the reduction method of the infinite system is investigated numerically. Examples of the numerical implementation are given; numerical studies of the spectrum of natural frequencies of the cantilevered thick plate were carried out based on the obtained solution, both with varying elastic characteristics of the material and with varying geometric parameters.


Metrologiya ◽  
2021 ◽  
pp. 17-39
Author(s):  
A. N. Bazhenov ◽  
A. Yu. Telnova

The possibility of application of the interval analysis for data processing in the field of spectral analysis is considered. It is assumed that the data have interval uncertainty; therefore the problem of finding unknown concentrations is posed as a linear interval tolerance problem. The incompatibility of the interval system of linear algebraic equations is shown for the initial data using the apparatus of the recognizing functional. The relevance of the topic is due to the need for regularization of inconsistent interval systems of linear equations. The idea of S. P. Shary of a combined method for correcting a linear tolerance problem has been implemented. A new method for managing the solution by changing the linear algebraic equations interval system matrix elements radii has been developed. The research results can be used for example, to calculate the substance’s concentrations by measurement of the characteristic X-ray radiation.


InterConf ◽  
2021 ◽  
pp. 485-499
Author(s):  
Daria Lys

Systems of linear algebraic equations are a mathematical apparatus that is widely used in solving a significant number of problems in the practical application of mathematics and engineering. The analysis of errors in solving linear algebraic equations and the sensitivity function of nonlinear systems using the method of equivalent excitations, which, in turn, makes it possible to make informed decisions on the choice and development of methods for studying the accuracy of computing devices. Methods for constructing estimates «from below» of the distribution functions of the fatal error of the numerical solution of systems of linear algebraic equations are also presented, in particular, a posteriori estimates of the effectiveness of the methods under study are analyzed.


Author(s):  
I. N. Vankina ◽  
D. A. Fetisov

Modeling the anthropomorphic robot movement is of great interest to researchers all over the world. At the same time, the movement control of a walking mechanism is always a high dimension challenge. The difficulty with the anthropomorphic robot control is also caused by the fact that such a mechanism has always a hybrid dynamics and represents a sequential change of two phases – the single support phase and the double support phase (phase of changing robot’s leg). At the single support phase and at another phase the behavior of the biped robot is described by a system of ordinary differential equations and by a system of linear algebraic equations, respectively.The task of biped robot movement control has been studied in detail for the case when the robot moves over the horizontal surface. Obstacles make the task significantly complicated. The paper considers the movement control of the biped robot over the surface that is a periodic alternation of horizontal sections and obstacles. The obstacles represent steps of the same height known. It is assumed that the lengths of horizontal sections and steps are known as well. The objective is to create a control that provides robot’s periodic movement over the specified surface according to inherent characteristics of a walking human.For the single support phase, the outputs are proposed, the equality of which to zero corresponds to the robot’s movement with a given set of characteristics. The paper presents the feedback controls that stabilize the proposed outputs for a finite amount of time. By choosing the feedback parameters, it is possible to adjust the stabilization time so that the outputs become equal to zero when reached the end of each step.It is shown that for the chosen control law, the problem of constructing the control of robot’s periodic movement is reduced to the solution of a nonlinear equation. In the paper, we discuss the approaches to solving this equation and present the results of numerical simulation.The results obtained can be used to solve the problem of providing control of the biped robot movement over the surfaces with obstacles of a more complicated shape.Modeling the anthropomorphic robot movement is of great interest to researchers all over the world. At the same time, the movement control of a walking mechanism is always a high dimension challenge. The difficulty with the anthropomorphic robot control is also caused by the fact that such a mechanism has always a hybrid dynamics and represents a sequential change of two phases – the single support phase and the double support phase (phase of changing robot’s leg). At the single support phase and at another phase the behavior of the biped robot is described by a system of ordinary differential equations and by a system of linear algebraic equations, respectively.The task of biped robot movement control has been studied in detail for the case when the robot moves over the horizontal surface. Obstacles make the task significantly complicated. The paper considers the movement control of the biped robot over the surface that is a periodic alternation of horizontal sections and obstacles. The obstacles represent steps of the same height known. It is assumed that the lengths of horizontal sections and steps are known as well. The objective is to create a control that provides robot’s periodic movement over the specified surface according to inherent characteristics of a walking human.For the single support phase, the outputs are proposed, the equality of which to zero corresponds to the robot’s movement with a given set of characteristics. The paper presents the feedback controls that stabilize the proposed outputs for a finite amount of time. By choosing the feedback parameters, it is possible to adjust the stabilization time so that the outputs become equal to zero when reached the end of each step.It is shown that for the chosen control law, the problem of constructing the control of robot’s periodic movement is reduced to the solution of a nonlinear equation. In the paper, we discuss the approaches to solving this equation and present the results of numerical simulation.The results obtained can be used to solve the problem of providing control of the biped robot movement over the surfaces with obstacles of a more complicated shape.


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nugzar Shavlakadze ◽  
Otar Jokhadze

Abstract Exact and approximate solutions of a some type singular integro-differential equation related to problems of adhesive interaction between elastic thin half-infinite or finite homogeneous patch and elastic plate are investigated. For the patch loaded with vertical forces, there holds a standard model in which vertical elastic displacements are assumed to be constant. Using the theory of analytic functions, integral transforms and orthogonal polynomials, the singular integro-differential equation is reduced to a different boundary value problem of the theory of analytic functions or to an infinite system of linear algebraic equations. Exact or approximate solutions of such problems and asymptotic estimates of normal contact stresses are obtained.


Author(s):  
A. Ugol’nikov ◽  
B. Demianchuk ◽  
S. Shelukhin ◽  
O. Malynovskyi ◽  
A. Kosenko

The article discusses a probabilistic model of processes in complex systems of technical support for military vehicles. One of the methods for studying such complex systems is their representation in the form of a set of typical states in which the system can be. Transitions occur between states, the intensities and probabilities of which are assumed to be known. The system is graphically represented using a graph of states and transitions, and the subject of research is the probability of finding the technical support system in these states. The graph of states and transitions is associated with a system of first order linear differential equations with respect to the probabilities of finding the support system in its basic states. To obtain a solution, this system must be supplemented with certain conditions. These are, firstly, the initial conditions that specify the probabilities of all states at the initial moment of time. Second, this is the normalization condition, which states that at any moment in time the sum of the probabilities of all states is equal to unity. An approximate solution to the problem is described in the literature. Such approximate solution is getting more accurate when the sought probabilities depend on time weaker. We propose a method of the exact solution of the above mentioned system of differential equations based on the use of operational calculus. In this case, the system of linear differential equations is transformed into a system of linear algebraic equations for the Laplace images of unknown probabilities. The use of matrix calculus made it possible to write down the obtained results in a compact form and to use effective numerical algorithms of linear algebra for further calculations. The model is illustrated by the example of solving the problem of technical support for the march of a battalion tactical group column, including wheeled and tracked vehicles. The boundaries of the validity of the results of a simpler approximate solution are established.


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