Numerical Solution Of System Research Articles

Approximating piecewise nonlinearities in dynamic systems with sigmoid functions: advantages and limitations

In the industry field, the increasingly stringent requirements of lightweight structures are exposing the ultimately nonlinear nature of mechanical systems. This is extremely true for systems with moving parts and loose fixtures which show piecewise stiffness behaviours. Nevertheless, the numerical solution of systems with ideal piecewise mathematical characteristics is associated with time-consuming procedures and a high computational burden. Smoothing functions can conveniently simplify the mathematical form of such systems, but little research has been carried out to evaluate their effect on the mechanical response of multi-degree-of-freedom systems. To investigate this problem, a slightly damped mechanical two-degree-of-freedom system with soft piecewise constraints is studied via numerical continuation and numerical integration procedures. Sigmoid functions are adopted to approximate the constraints, and the effect of such approximation is explored by comparing the results of the approximate system with the ones of the ideal piecewise counter-part. The numerical results show that the sigmoid functions can correctly catch the very complex dynamics of the proposed system when both the above-mentioned techniques are adopted. Moreover, a reduction in the computational burden, as well as an increase in numerical robustness, is observed in the approximate case.

Kinetička simulacija ekspanzije vakuumske plazme izvan granica "plazma aproksimacije"

Introduction/purpose: One of the key approaches to solving an entire class of modern plasma physics problems is the so-called "plasma approximation". The most general definition of the "plasma approximation" is a theoretical approach to the electric field calculation of a system of charges under the electric quasi-neutrality condition. The purpose of this paper is to compare the results of the numerical simulation of the kinetic processes of the quasi-neutral plasma bunch expansion to the analytical solution of a similar kinetic model but in the "plasma approximation". Methods: The given results are obtained by the methods of deterministic modeling based on the numerical solution of the system of Vlasov-Poisson equations. Results: The provided comparison of the analytical expressions for the solution of kinetic equations in the "plasma approximation" and the numerical solutions of the Vlasov-Poisson equations system convincingly show the limitations of the "plasma approximation" in some important cases of the considered problem of plasma formation decay. Conclusion: The theoretical results of this work are of great importance for understanding the shortcomings of the "plasma approximation", which can manifest themselves in practical applications of computational plasma physics.

ANALYSIS OF CALCULATION METHODS, OBTAINING ESTIMATES AND ENSURING THE ACCURACY OF SIMULATION OF CONTROL OBJECTS

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.

Numerical solution for system with Atangana-Baleanu-Caputo derivative: An influenza epidemic model

In this work, we discuss the fractional three strain influenza epidemic model with vaccination for strain 1 and awareness of strain 2. We use Atangana-Baleanu in the Caputo sense. We show that the model has a unique solution, and we perform the numerical simulation with various values of derivative order. It is shown that by decreasing the value of the derivative order, the maximum value also decreases.

Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems

We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which is based on a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. The proposed method allows to use large time steps, while avoiding the solution of large linear systems, which would be required by implicit time discretization techniques. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. A novel extrapolation technique is proposed to enforce second-order accurate Dirichlet boundary conditions. We include a theoretical analysis of the scheme, along with numerical experiments which demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations.

Numerical solution of some systems of nonlinear algebraic equations

The problem of numerical solution of systems of nonlinear algebraic equations with even powers is studied in this paper. Nondifferentiable (nonsmooth) unconstrained minimization problems are associated with the system under consideration, with objective functions, based on discrete l 1 - and l ∞ - norm, respectively, by using these two norms as proximity criteria, and the considered system is solved by minimizing the sum of absolute values of residuals, in the case of l 1 - norm, or by minimizing the maximal residual, in the case of l ∞ - norm, respectively. Also, a “differentiable” unconstrained minimization problem (least squares problem) is associated with the considered system of nonlinear equations, with objective function, based on the discrete l 2 -norm, and the considered system is solved by minimizing the sum of squares of the residuals. Subgradients of the objective functions of unconstrained minimization problems are calculated, and a subgradient method for solving the associated problems is presented. The iterative gradient method for solving the corresponding “differentiable” discrete least squares problem, based on l 2 - norm, is also outlined, and gradients of the objective function of the corresponding “differentiable” unconstrained minimization problems are calculated.

Определение периода повторяемости электромагнитных процессов в аварийных режимах работы многофазных вентильных двигателей малой мощности

Increasing requirements for the reliability indicators of technical systems lead to the need to apply new approaches to the design of electric machines and electric drives as an electrical complex, one of which is a systematic approach. To provide a set of increased reliability indicators, it is proposed to use multiphase AC converter-fed motors that differ in structural and algorithmic redundancy It is shown in order to implement variants of AC converter-fed motor circuit, in addition to reliability indicators, a quantitative assessment of its energy capabilities in normal and emergency operating modes is necessary. The most important stage of the practical implementation of the numerical process in the quantitative assessment of energy capabilities is to determine the period of repeatability of electromagnetic processes. The main issues of determining the period of repeatability of quasi-steady-state electromagnetic processes of an electromechanical energy converter of low-power multiphase AC converter-fed motors in the design of systems of increased reliability are considered. This made it possible to implement the numerical solution of systems of differential equations describing the physical processes in AC converter-fed motors with a variable structure of the electromechanical converter in normal and emergency operating modes.

Computatıonal Algorıthm for the Numerıcal Solutıon of Systems of Volterra Integro-Dıfferentıal Equatıons

In this paper, we employ variational iterative method (VIM) to develop a suitable Algorithm for the numerical solution of systems of Volterra integro-differential equations. The formulated algorithm is used to solve first and second order linear and nonlinear system of Volterra integrodifferential equations which demonstrated a good numerical approach to overcome lengthen computational and integral simplification involves. Moreover, the comparison of the exact solution with the approximated solutions are made and approximate solutions p(x) q(t) proved to converge to the exact solutions p(x) q(t) respectively. The results reveal that the formulated algorithm are simple, effective and faster than analytical approach of solving Volterra integro-differential equations.

The Numerical Solutions of Systems of Nonlinear Integral Equations with the Spline Functions

The main goal of this work is to develop an effective technique for solving nonlinear systems of Volterra integral equations. The main tools are the cardinal spline functions on small compact supports. We solve a system of algebra equations to approximate the solution of the system of integral equations. Since the matrix for the algebraic system is nearly triangular, It is relatively painless to solve for the unknowns and an approximation of the original solution with high precision is accomplished. In order to enhance the accuracy, several cardinal splines are employed in the paper. Our schemes were compared with other techniques proposed in recent papers and the advantage of our method was exhibited with several numerical examples.

Численное решение систем дифференциальных уравнений комплексной модели влагообмена замкнутого аппарата и анализ результатов

Численное решение систем дифференциальных уравнений комплексной модели влагообмена замкнутого аппарата и анализ результатов

FLUCTUATIONS OF THE CARGO TRANSPORTED BY LIFTING CRANE: SIMULATION AND ANALYSIS

Introduction.Reducing fluctuations in the load transported by hoisting cranes with a flexible rope suspension of the load is an urgent task since it can significantly reduce the time taken to complete the operation of moving the load. A promising direction for reducing load fluctuations is to optimize the trajectory of movement of the load suspension upper point.Materials and methods.The paper discussed the method of mathematical simulation of plane vibrations of a load moved by a crane with a horizontally moving suspension point, using the software of the MATLAB system. For modeling, the authors used the function of the MATLAB ode45 system, intended for the numerical solution of systems of non-stationary differential equations of arbitrary order.The second-orderdifferential equation used to describe the fluctuations of the transported load and its implementation in the form of program code was presented. Moreover, the authors demonstrated the elements of program code for the analysis and visualization of simulation results.Results.The authors obtained and presented the series of graphs in the inclination angle’s changing of the cargo rope, the acceleration of the suspension point and the value of the objective function with the sinusoidal nature of the acceleration. The objective function was the sum of the absolute values of the deflection angle of the rope and the first derivative at the final moment of the suspension point’s movement with acceleration.Discussion and conclusions.As a result, the paper shows that the system with energy dissipation does not reach the zero value of the objective function even by a symmetrical nature of acceleration and deceleration of the suspension point. Therefore, it is necessary to give asymmetry to the acceleration and deceleration periods of the suspension point in order to completely absorb the residual fluctuations of the load.

METHODS MODELING SYSTEMS FOR THE IMPROVEMENT OF THEIR RELIABILITY


 
 
 
 The method of Markov’s processes for the analysis of systems with constant bounce and recovery intensities considered. The article presents calculations of the failure probability of the system for describing the various cases of redundancy of its components using Markov’s models. Expressions obtained for calculating the approximate value of the failure probability of the system and analyzed of failures to improve the reliability of the system. The Markov’s graph of transitions in the reservation of the system, which reflects its behavior, described. Analysis of the results of numerical solution of systems shows that when loaded with redundancy, the probability of failure is higher than with partially loaded, and with partially loaded - higher than with unloaded backup. A tree of errors for the system of cooling and clearing of flue gas at the reservation made by replacing "2 of 3", which has seven minimum bounce cross sections. Calculated the probability of system failure. The obtained calculations allow to analyze failures of technical systems in order to increase the reliability of their functioning.
 
 
 

On a Three-Step Efficient Fourth-Order Method for Computing the Numerical Solution of System of Nonlinear Equations and Its Applications

In this paper, a new fourth-order iterative scheme for finding the zeros of systems of nonlinear equations has been built and analyzed. Theoretical proof has been given to confirm the convergence order of the new method. The effectiveness of the proposed method is shown by the comparison of traditional as well as flops-like efficiency index with recent existing same order schemes. Numerical examples confirm that the new iterative method is efficient and gives tough competition to some existing fourth-order methods. We have also discussed the application of our proposed method for finding numerical solution of nonlinear ODE and PDE.

Numerical solution for system of nonlinear Fredholm–Hammerstein integral equations based on hybrid Bernstein Block-Pulse functions with the Gauss quadrature rule

This work provides an efficient method for solving system of nonlinear Fredholm–Hammerstein integral equations. The propose method approximates the unknown function with hybrid Bernstein Block-Pulse functions (HBPF). In order to do this, we apply these functions and then using the collocation method for the numerical solutions of this system. Furthermore, we implement this method in conjunction with the quadrature rule for converting the problem to a system of algebraic equations that can be solved easily by applying mathematical programming techniques. The merits of this method lie in the fact that, on the one hand, the problem will be reduced to a system of algebraic equations and on the other hand, the efficiency and accuracy of the hybrid Bernstein Block-Pulse functions with the Gauss quadrature rule (HBPF-GQR) for solving this system are remarkable. The existence and uniqueness of solution have been presented. Moreover, the convergence of this algorithm will be shown by preparing some theorems. Several numerical examples are presented to show the superiority and efficiency of current method in comparison with some other well-known methods.

New hybrid symmetric two step scheme with optimized characteristics for second order problems

In the present paper, for the first time in the literature, we build a new three-stages symmetric two-step finite difference pair with optimized properties. In more details the new method: (1) is of symmetric type, (2) is of two-step algorithm, (3) is of three-stages—i.e. hybrid or Runge–Kutta type, (4) it is of tenth-algebraic order, (5) it has vanished the phase-lag and its first and second derivatives, (6) it has optimized stability properties for the general problems, (7) it is a P-stable finite difference scheme since it has an interval of periodicity equal to $$\left( 0, \infty \right) $$ . The new Runge–Kutta type algorithm is builded based on the following approximations: A full theoretical analysis (local truncation error analysis, comparative error analysis and stability and interval of periodicity analysis) is given for the new builded finite difference pair. The effectiveness of the new builded hybrid scheme is evaluated on the numerical solution of systems of coupled differential equations of the Schrodinger type.

New Runge–Kutta type symmetric two-step method with optimized characteristics

In this paper and for the first time in the literature, we build a new hybrid symmetric two-step method with the following properties: (1) the new scheme is of symmetric type, (2) the new scheme is of two-step, (3) the new scheme is of five-stages, (4) the new scheme is of twelfth-algebraic order, (5) the new scheme has eliminated the phase-lag and its first, second, third, fourth and fifth derivatives, (6) the new scheme has improved stability characteristics for the general problems, (7) the new scheme is P-stable [with interval of periodicity equal to $$\left( 0, \infty \right) $$ ] and (8) the new scheme builded based on the following approximations: For the new builded scheme we give a full numerical analysis (local truncation error and stability analysis). The efficiency of the new builded scheme is examined with the numerical solution of systems of coupled differential equations of the Schrodinger type.

Numerical solution of systems of differential equations using operational matrix method with Chebyshev polynomials

ABSTRACTIn this study, we introduce an effective and successful numerical algorithm to get numerical solutions of the system of differential equations. The method includes operational matrix method and truncated Chebyshev series which represents an exact solution. The method reduces the given problem to a set of algebraic equations including Chebyshev coefficients. Some numerical examples are given to demonstrate the validity and applicability of the method. In Examples, we give some comparison between present method and other numerical methods. The obtained numerical results reveal that given method very good approximation than other methods. Moreover, the modelling of spreading of a non-fatal disease in a population is numerically solved. All examples run the mathematical programme Maple 13.

New five-stages two-step method with improved characteristics

In the present paper and for the first time in the literature, a new five-stages symmetric two-step method with improved characteristics is developed. The main properties of the new method are: For the new obtained method a full numerical analysis (error and stability analysis) is presented. The evaluation of the efficiency of the new method is based on the numerical solution of systems of coupled differential equations of the Schrodinger type.

A five–stages symmetric method with improved phase properties

In this paper, for the first time in the literature, we introduce a new five–stages symmetric two–step method with improved phase and stability properties. The characteristics of the new non linear two–step method are: (1) it is of symmetric type, (2) it is of two–step finite difference pair, (3) it is of five–stages, (4) it is of twelfth–algebraic order, (5) it has vanished the phase–lag and its first and second derivatives, (6) it has improved stability properties for the general problems, (7) it is a P-stable method since it has an interval of periodicity equal to $$\left( 0, \infty \right) $$ , (8) for the development of the new scheme, the following approximations are used: (a) an approximation developed on the first layer on the point $$x_{n-1}$$ , (b) an approximation developed on the second layer on the point $$x_{n-1}$$ , (c) an approximation developed on the third layer on the point $$x_{n-1}$$ , (d) an approximation developed on the fourth layer on the point $$x_{n}$$ and finally, (e) an approximation developed on the fifth (final) layer on the point $$x_{n+1}$$ . For the new developed method we give a full theoretical analysis (error and stability analysis). The efficiency of the new five–stages symmetric two–step method is examined by applying it on the numerical solution of systems of coupled differential equations arising from the Schrodinger equation.

Numerical Modeling of the Gas Dynamics of the Safety Valve

Рассматриваются газодинамические процессы в полости предохранительного клапана с определением силового воздействия газового потока на диск с учетом нестационарности и сжимаемости потока. Предохранительный клапан представляет собой механизм, автоматически открывающийся при повышении давления выше рабочего и закрывающийся при снижении давления до рабочего или несколько ниже него. Задачей расчета предохранительных клапанов является определение пропускной способности, подбор характеристик пружины к ним, динамических усилий, возникающих при срабатывании предохранительного клапана. Известные математические модели работы клапанов, описывающие движения диска под действием сил со стороны газа и пружины, основанные либо на применении эмпирических коэффициентов, либо на сложно реализуемом решении задачи о газодинамических процессах, осуществляемых в трехмерной постановке с использованием пакета ANSYS. В данной статье обосновывается возможность численного моделирования основных газодинамических характеристик клапана на основе решения осесимметричных уравнений газовой динамики. Для численного решения системы газодинамических уравнений с осевой симметрией применяется метод контрольного объема. Параметры газа на границах контрольных объемов определяются по методу С. К. Годунова с использованием автомодельного решения задачи о распаде произвольного разрыва. В расчете применяется структурированная ортогональная разностная сетка, построенная комплексным методом граничных элементов и отображающая криволинейную расчетную область на прямоугольник. Записана математическая модель движения диска клапана с учетом нестационарного изменения давления в емкости, в которой регулируется давление. Получены результаты численного моделирования нестационарного процесса открытия и закрытия клапана. Проведенное численное моделирование газодинамических процессов в предохранительном пружинном клапане прямого действия показало, что течение в области, расположенной выше по потоку от диска, имеет осевую симметрию. Звуковая линия, «запирающая» эту область от возмущений ниже по потоку, преобладающее время соприкасается с краем диска. Результаты эксперимента подтверждают динамику открытия клапана, рассчитанную по рассматриваемому методу.