Gauss Method Research Articles

УСТРОЙСТВА ДЛЯ ЭКСПЕРИМЕНТАЛЬНОГО ОПРЕДЕЛЕНИЯ НАГРУЖЕННОСТИ РАБОЧИХ ОРГАНОВ МТА МЕТОДОМ ТЕНЗОМЕТРИРОВАНИЯ

The efficiency of a field fleet unit is largely determined by the characteristics of the traction resistance of an agricultural machine, which are of a probabilistic nature and require experimental determination. The disadvantage of the known devices that provide measurements during the volumetric dynamometry of the working bodies of tillage machines and implements is the complexity of the design and a significant measurement error. To increase the accuracy of measurements of the dynamic loading of the working bodies of tillage machines, it is necessary to improve the measuring instruments that ensure the reduction of the error. The experimental determination of the loading of the working bodies requires the use of force sensors acting on the working bodies in different planes. Force transducers are mainly manufactured using strain gages, which are glued directly to parts with various cross-sections. To determine the efforts in the rods of the hydraulic linkage system of agricultural tractors use strain gauge fingers, which are cylinders with strain gages glued to them. It is shown that at equal distances l1 and l2 of the location of the sensors, the resulting signal does not depend on the l0 distance of force application on the axis of the finger. Typically, the calibration of force sensors is carried out on additional devices one by one under conditions as close as possible to real ones. The best calibration results will be when all sensors (strain gauge fingers) are installed in their original places and all sensors are calibrated at once. To do this, the intermediate plate is fixed with the help of tension fingers on the hydraulic tractor levers. Eyebolts are installed on the plate, to which a known force Р is applied three times and readings are taken from each of the strain gauge fingers. By solving the resulting system of equations, for example, using the Gauss method in the MathCad environment, all the calibration coefficients are simultaneously found. Additionally, a device is proposed, the kinematic diagram of which allows the use of a single measuring device, which simplifies the technology of measuring the force on the working body and increases its accuracy.

Computing Methods in the Analysis of Road Accident Reconstruction Uncertainty

The study is dedicated to the problem of uncertainty in the analysis of accident situations in road traffic. The term “uncertainty” is generally known when used with reference to measurement techniques, but its application to the analyses of accident situations in road traffic, including accident reconstruction, is a relatively new field of knowledge. The objectives of this work include the presentation and examination of selected aspects related to the taking of uncertainty into account when analysing the course of an accident and making the necessary calculations. Apart from the scientific objectives, an important utilitarian goal may also be pointed out. The data and methods presented may be used by automotive technology experts in their accident reconstruction work. The paper shows seven methods that enable the taking into account of the uncertainty of the data used for calculations, i.e. extreme values method, total differential method, higher-order total differential method, finite-difference method, Gauss method, method based on the description of stochastic processes, and Monte-Carlo method. Apart from formal (mathematical) descriptions of the methods, an example of their use for the estimation of uncertainty of selected quantities that describe an accident situation has been demonstrated. The bad and good points of individual methods have been shown in the context of the application considered.

On the Comparison of Gauss and Hybrid Methods and their Application to Calculation of Definite Integrals

As is known there is the wide class of methods for calculation of the definite integrals constructed by the well-known scientists as Newton, Gauss, Chebyshev, Cotes, Simpson, Krylov and etc. It seems that to receive a new result in this area is impossible. The aim of this work is the applied some general form of hybrid methods to computation of definite integral and compares that with the Gauss method. The generalization of the Gauss quadrature formula have been fulfilled in two directions. One of these directions is the using of the implicit methods and the other is the using of the advanced (forward-jumping) methods. Here have compared these methods by shown its advantages and disadvantages in the results of which have recommended to use the implicit method with the special structure. And also are constructed methods, which have applied to calculation of the definite integral with the symmetric bounders. As is known, one of the popular methods for calculation of the definite integrals with the symmetric bounders is the Chebyshev method. Therefore, here have defined some relations between of the above mentioned methods. For the application constructed, here methods are defined the necessary conditions for its convergence. The received results have illustrated by calculation the values for some model integral using the methods with the degree p ≤ 8.

Simulations of Currents in Conditions of Electrical Damage in the Windings of a Single-Phase Furnace Transformer with a Split Secondary Winding

A mathematical model is suggested to simulate currents under conditions of electrical damage in the windings of a single-phase furnace transformer with a split secondary winding. A method to reduce the number of equations in this mathematical model with many splits has been developed, which allows one to use iterative methods, e.g., the Gauss method, to determine currents in the windings. The effectiveness of the obtained model has been experimentally verified. It has been shown that the error in modeling the currents in the windings of a single-phase furnace transformer with secondary winding splitting during a short circuit in the primary winding does not exceed 10–15%. On the basis of the method to reduce the number of equations in a mathematical model, mathematical models have been developed; they allow one to simulate the currents in the windings of a furnace transformer during the short circuit in a short network with the break in a splitting conductor and under the normal load mode.

Explicit Symmetric Exponential Integrators for Charged-Particle Dynamics in a Strong and Constant Magnetic Field

In this paper, we derive and analyze symmetric exponential integrators for charged-particle dynamics in a strong and constant magnetic field. We first present the scheme of exponential integrators and then establish the symmetry conditions for the methods. Explicit symmetric exponential integrators up to order four are constructed on the basis of the symmetry conditions. In order to show the remarkable performances of new symmetric methods in comparison with symmetric Runge–Kutta methods including the 2-stage Gauss method, two numerical experiments are carried out and the numerical results demonstrate the super numerical behavior.

On the Comparison of Gauss and Hybrid Methods and their Application to Calculation of Definite Integrals

As is known there is the wide class of methods for calculation of the definite integrals constructed by the well-known scientists as Newton, Gauss, Chebyshev, Cotes, Simpson, Krylov and etc. It seems that to receive a new result in this area is impossible. The aim of this work is the applied some general form of hybrid methods to computation of definite integral and compares that with the Gauss method. The generalization of the Gauss quadrature formula have been fulfilled in two directions. One of these directions is the using of the implicit methods and the other is the using of the advanced (forward-jumping) methods. Here have compared these methods by shown its advantages and disadvantages in the results of which have recommended to use the implicit method with the special structure. And also are constructed methods, which have applied to calculation of the definite integral with the symmetric bounders. As is known, one of the popular methods for calculation of the definite integrals with the symmetric bounders is the Chebyshev method. Therefore, here have defined some relations between of the above mentioned methods. For the application constructed, here methods are defined the necessary conditions for its convergence. The receive results have illustrated by calculation the values for some model integral using the methods with the degree p  8.

Calculation of pressure losses during pneumatic transport

A method for calculating pressure losses during pneumatic transport is considered with the aim of choosing the most favorable operating mode, which allows to reduce the consumption of compressed air. It has been established that the generally accepted formulas for calculating pressure losses along the path, calculated as the sum of the pressure losses for pure gas and material, do not explain the fact of the initial decrease in pressure losses with increasing air flow rate, and then their growth. The importance of accounting for changes in air density and air flow velocity along the length of the pipe, as well as the need to take into account the overlap of part of the pipeline section with material in the case of high concentrations, is shown. It is proposed to determine the pressure loss using the Bernoulli equation in integral form, for example, the Gauss method using eight nodes. The applicability of the method for calculating both straight and inclined sections of pipelines is established. The experimental results of studying and analyzing the proposed isothermal model in a vacuum pneumatic transport laboratory installation are presented. An example of calculating the pressure loss for an industrial pneumatic conveying system with a pipe length of 450 m and an inner diameter of 147 mm at various flow rates of the material is shown. Ill. 4. Ref. 21. Tab. 1.

Approaches to automated land subdivision using binary search algorithm in zoning applications

The planned development of urban areas depends on zoning applications. Although zoning practices are performed using different techniques, the parcelling operations that shape the future view of the city are the same. Preparing the parcelling plans is an important step that has a direct impact on ownership structure and reallocation. Parcelling operations are traditionally handled manually by a technician. This is a serious problem in terms of time and cost. In this study, by taking the zoning legislation, the production of a pre-land subdivision plan has been automatically performed for a region of Konya, which is one of the major cities in Turkey. The parcelling processes have been performed in three different ways: the first parcelling technique is parcelling with edge values, the second is parcelling with area values and the third is parcelling using both edge and area values together. For the entire parcelling process, the area of the parcel has been calculated using the Gauss method. Moreover, to effectively determine the boundaries and to calculate the parcel area in the parcelling process, the binary search technique has been used in all the methods. The experimental results show that the parcelling operations were carried out very quickly and successfully.

The Analysis and Shooting Reconstruction of The Ricocheted Shotgun Pellet Wounds

In this study, a 35 years old man was killed with the ricocheted shotgun pellets wounds from the asphalt road surface. The Public Prosecutor to define the ricochet point requested the ricocheted pellet pattern examination in two different ricochet distances. The ten ricochet tests were performed for at 2 meters (point A) and 1 meter (point B) from the target, and the pellet distribution and pattern area were calculated using the gauss method. Then, the test and autopsy results were compared in the pellet number, calculated pellet pattern area and ricocheted angle. Furthermore, the similarity of the pellet number and the pellet pattern areas were examined using the two-tailed Mann-Whitney U test. In this study, 81 pellets recovered from the victim's body and the distribution pattern of pellets area was 2134 cm2. At the ricochet point A, the average number of pellets on the target was 82.1 and the distribution pattern of pellets area was 2700 cm2. At the ricochet point B, the average number of pellets on the target was 132.6 and the distribution pattern of pellets area was 4928 cm2. According to the two-tailed Mann-Whitney U test, there was low-level similarity (p<0.05 level Sig. 0.023; z=-2.424) on the pellet pattern area between autopsy and the ricochet point A. However, as regards the pellet number on the target, there was a similarity (p<0.05 level Sig. 0.481; z=-0.808) between autopsy and the ricochet point A. Test results showed that the pellet ricochet occurred two meters from the victim.

Modeling of a salt gradient solar pond under Moroccan climate taking into account double-diffusive convection

This study concerns a numerical investigation of the influence of Soret and Dufour effects on heat transfer and the amount of heat stored inside a salt gradient solar pond (SGSP) under meteorological conditions of Tangier (Morocco). This SGSP consists of a square cavity open and filled with saline water. The bottom and two lateral walls of this cavity are insulated and impermeable. The Navier-Stokes, thermal energy and diffusion equations are solved using a finite volume scheme and Gauss method. The link between pressure and velocity fields is assumed by the SIMPLE algorithm. The numerical code developed in FORTRAN 95 language was validated with numerical and experimental published works. Results show that Soret effects have not an effect on thermal performance of the SGSP compared to Dufour effect. This last one influences the temperature distribution inside the SGSP and causes a great reduction of the amount of heat stored.

Features of formation of the temperature field in a well with a multilayer system during oil degassing

As is known, well thermometry is one of the main methods for diagnosing the condition of a well and formation while monitoring the development of oil and gas fields. Despite the widespread use of thermometry in the oil and gas industry, today the question of the formation of the temperature field in a well with a multilayer system under conditions of oil degassing remains poorly understood. Therefore, it is relevant to study thermohydrodynamic processes that form the temperature field in the well-reservoir system based on a mathematical model of the movement of a two-phase flow of oil and gas, taking into account oil degassing. Based on a numerical model describing the non-isothermal movement of a two-phase fluid in a well with a multi-layer system, the article discusses the problem of selecting soda oil from a well that operates two layers. The formation of the temperature field is due to the influence of the adiabatic and Joule-Thomson effects, as well as the heat of gas degassing. The system of differential equations is solved numerically, and discretization by the control volume method. The resulting system of nonlinear equations is linearized by the Newton-Raphson method, the roots of which are found by the Gauss method from the LAPACK library. The paper considers the features of temperature effects in a well with a two-layer system for various durations of pressure reduction at the wellhead, oil saturation pressure with gas, as well as the functional dependences of the relative phase permeabilities of phases on saturation. It is shown that a sharp decrease in pressure at the wellhead leads to significant cooling in the first hour of well operation. The results can be used in the planning of field geophysical studies and interpretation of the results of temperature studies of wells in the production of gas-cut oil.

Forming a methodology of basic matrices in the study of poorly conditioned linear systems

Algorithm of the basic matrix method for analysis of properties of the system of linear arithmetic equation (SLAE) in various changes introduced in the model, in particular, when including-excluding a group of rows and columns (based on framing) without re-solving the problem from beginning has been improved. Conditions of compatibility (incompatibility) of restrictions were established and vectors of the fundamental solution system in a case of compatibility were established. Influence of accuracy of representing the model elements (mantis length, order value, thresholds of machine zero and overflow) and variants of computation organization on solution properties was studied. Specifically, effect of magnitude and completeness of rank was studied on an example of a SLAE with a poorly conditioned constraint matrix. A program was developed for implementation of conducting calculations using the basic matrix methods (BMM) and Gauss method, that is, long arithmetic was used for models with rational elements. Algorithms and computer-aided implementation of Gaussian methods and artificial basic matrices (as a variant of the basic matrix method) in MATLAB and Visual C++ environments with the use of the technology of exact calculation of the method elements, first of all, for poorly conditioned systems with different dimensions were proposed.Using as an example Hilbert matrices, which are characterized as inconvenient matrices, an experiment was conducted to analyze properties of a linear system at different dimensions, accuracy of the input data and computation scenarios. Formats (exact and inexact) of representation of model elements (mantis length, order value, thresholds of machine zero and overflow) as well as variants of organization of basic computation operations during calculation and their influence on solution properties have been developed. In particular, influence of rank magnitude and completeness was traced on an example of an SLAE with a poorly conditioned constraint matrix

A Technique of Calculation of the Dynamic Mode in Computer Simulations of Electric Power Systems with an Arbitrary Configuration

The algorithmic implementation of calculations of the dynamic mode during computer simulation of electric power systems with arbitrary topology has been considered. This is more easily done when the electric circuit graph is used in full form (when just one circuit element can be placed at one branch) despite its redundancy. In this case, the graph branches can be connected with the elements of an electric circuit. Mathematical models of elements, which are based on the differential equations in the Park–Gorev axes rotating along with the rotor of a synchronous generator, allow one to eliminate variable coefficients in most differential equations of elements. In the computer simulations, the method of calculations based on the equations for nodal voltages is used. An algorithm to create conductivity and current matrices according to the technique of element-to-element contribution is given. The algorithm to determine an inverse matrix in the equation for nodal voltages is based on the Jordan–Gauss method. To calculate the dynamic mode, it is suggested to use the fourth-order Runge–Kutta method, which allows one to obtain a relatively accurate solution using information at just one previous point. In the view of the development of information technologies, the considered approach allows one both to automated separate stages of design, calculations, and analysis of operation of the electric power system and to take into account the correlation between all the stages of life cycle of the electrotechnical equipment starting with the design stage and ending with equipment decommissioning.

Обучение студентов использованию метода Гаусса для целочисленных матриц при реализации на компьютере

The paper is written on the basis of a part of “Analysis of algorithms” course for students of the Computer science department of the Division of mathematics and mechanics of Saint Petersburg State University. The example of the computer implementation of the Gauss method illustrates the difference between the algebraic complexity (the number of arithmetic operations) of processing integers and the computational complexity which depends on the length of the input data. A formula which specifies the increase in the length of matrix coefficients, along with the implementation the Gauss method, is proved. The problems arising in the processing of large integers associated with “chopping” numbers are shown. To overcome the indicated problems, the possibility of using multi-valued integers is proposed. The upper bounds of the number of steps for processing the multivalued integers is shown to coincide with such bounds for a multi-tape Turing machine.

Full-vectorial multimode nonlinear simulations on a real-space Fourier–Gauss grid

A numerical approach for nonlinear propagation in multimode waveguides based on real-space integration of the nonlinear polarization is developed for full-vectorial simulations. The real-space integrations are performed on a Fourier–Gauss quadrature grid in accordance with an earlier scalar formulation. The method is benchmarked against an alternative approach based on summation of mode overlap integrals, which is efficient for few-mode fibers but scales unfavorably with the number of modes. For silica nanowires and birefringent step-index fibers with weak index contrast, the Fourier–Gauss method is found to be comparable or superior to the mode overlap method when 12 modes (three mode groups) are included in the simulations without accounting for mode profile dispersion.

A CLASS OF S-STEP NON-LINEAR ITERATION SCHEME BASED ON PROJECTION METHOD FOR GAUSS METHOD

Various iteration schemes are proposed by various authors to solve nonlinear equations arising in the implementation of implicit Runge-Kutta methods. In this paper, a class of s-step non-linear scheme based on projection method is proposed to accelerate the convergence rate of those linear iteration schemes. In this scheme, sequence of numerical solutions is updated after each sub-step is completed. For 2-stage Gauss method, upper bound for the spectral radius of its iteration matrix was obtained in the left half complex plane. This result is extended to 3-stage and 4-stage Gauss methods by transforming the coefficient matrix and the iteration matrix to a block diagonal form. Finally, some numerical experiments are carried out to confirm the obtained theoretical results.

METHOD OF SOLVING SYSTEM OF LINEAR ALGEBRAIC EQUATIONS WITH USE OF HIGH PERFORMANCE COMPUTING ON GPU

The article discusses the use of graphics processing units for solving large system of linear algebraic equations (SLAE). A heterogeneous multiprocessor computing platform produced by the NIIVK, whose architecture allows the integration of general‑ purpose microprocessor modules with graphics processor modules was used as an equipment for solving SLAEs. The description of the SLAE solution program, developed on the basis of the CUBLAS CUDA software interface library, is given. A method is proposed for increasing the accuracy of calculations of linear systems based on the use of a modified Gauss method. It has been established that the use of the modified Gauss method practically does not increase the program operation time with a significant increase in the accuracy of calculations. It is concluded that the use of graphics processors for solving SLAEs allows processing matrices of a larger size compared to the use of general‑purpose microprocessors.

Разработка математических моделей для анализа чувствительности колебаний деталей машин

The purpose of the paper is to develop mathematical models and algorithms based on the finite element method to analyze the sensitivity of vibrations of machine parts and structural elements to changes in design parameters. The paper presents an algorithm for sensitivity analysis based on the finite element method, as well as the calculation results of sensitivity of compressor blade free oscillations to the changes in its profile thickness. The finite element method in displacements is used. The classical Gauss method is used to solve the systems of linear algebraic equations. As a result of conducted researches the developed method of frequency sensitivity coefficients is offered for the analysis of oscillations. Based on the developed mathematical models, algorithms and methods the analysis of the rotor blade eigen oscillation sensitivity to the changes in profile thickness is analyzed. The calculation results have shown that the error of the frequency prediction by the sensitivity coefficients is small as compared to the calculation of the modified models. Consequently, the sensitivity coefficients can be used to assess possible designs of the blade. The presented calculation results demonstrate a rather small error, which allows to apply the developed analysis method of eigen frequency oscillation using sensitivity coefficients for the optimal design of the gas turbine rotor blades. The use of sensitivity coefficients provides the possibility to estimate the effect of the change in the local profile thickness on the blade oscillation spectrum. This significantly increases the efficiency of work on structure optimization when designing it.

NUMERICAL ANALYSIS OF SLOW STEADY AND UNSTEADY VISCOUS FLOW BY MEANS OF R-FUNCTIONS METHOD

Context. This article is devoted to the linear problem of the steady and unsteady flows of a viscous incompressible fluid. Objective. The purpose of this paper is to compare the previously developed methods of numerical analysis of the steady and unsteady flows of a viscous incompressible fluid. Method. The flow of a viscous incompressible fluid can be described by the system of nonlinear Navier-Stokes equations. The variables of this system are velocity, pressure, density, volume forces, and fluid viscosity. Using the stream function, the Navier-Stokes equations can be transformed to the initial-boundary problem with the differential equation of the fourth order. To solve the problem the structural variational method of R-functions and Ritz method (steady problem) or Galerkin method (unsteady problem) are used. The R-functions method allows satisfying the boundary conditions accurately and transforming them to the homogeneous, which are the prerequisite for application of Ritz or Galerkin method. The problem transforms to the solving the system of linear algebraic equations or the system of ordinary differential equations for steady and unsteady flows respectively. The matrices elements are the scalar products in the norms of the corresponding differential operators. Numerical integration was made by means of Gaussian quadratures with 16 points. Solutions of the system of linear algebraic equations and the system of ordinary differential equations were found with the help of the Gauss method and the Runge-Kutta method with an automatic step-size control respectively. The existence of a unique solution of the problems is proved. Results. The computational experiments for the problem of flows of a viscous incompressible fluid for the different rectangular domains carried out. Conclusions. The conducted experiments have confirmed that the stream function, the flow velocity, and other flow characteristics are converging to the steady state when the time is increasing. This allows us to say that the obtained methods work as expected. The further research may be devoted to the comparison of the solution methods for the non-linear problems.

Parallel Algorithm for Transient Stability Simulation Based on Sherman-Morrison Formula

Parallel computing is an effective approach to transient stability real-time simulation of large-scale power system. In this paper, a new parallel algorithm for power system transient stability simulation is proposed by combining Gauss method and the Sherman-Morrison formula. The algorithm adopts the Gauss method to convert the differential-algebraic equations into a set of nonlinear algebraic equations by multi-stages discretization, while the algebraic system is solved using rigorous Newton method. On this basis, the whole Jacobian matrix involved in Newton method is split to a block diagonal matrix and a block constant coefficients matrix according to s time points, and then based on the block diagonal matrix, the computing tasks at s time points are fully decoupled through the extended Sherman-Morrison matrix inverse formula. The proposed algorithm preserves the good convergence of rigorous Newton method and meanwhile has a high degree of parallelism both in time and in space.