АПОСТЕРІОРНИЙ МЕТОД ОЦІНЮВАННЯ ПОХИБКИ І РОЗПАРАЛЕЛЕННЯ ОБЧИСЛЕНЬ ДЛЯ ОДНОГО КЛАСУ ЗАДАЧ ЕЛЕКТРОННОЇ ОПТИКИ

The electron-optical systems are basic components of the modern research complexes by the help of which complicated physical processes related to motion of the charged particles in the corresponding potential fieldsare are studied. The real electron-optical systems have a great number of the charged electrodes of the complicated configuration. Therefore, the use of economic collocation method under the conditions of cobbed-permanent approximation of the sought density requires the numeral solving the systems of linear algebraic equations of large dimensions with the densely filled matrices. In the one model example, the а posteriori method of error evaluation and parallelization of the procedures for a class of problems of electronic optics is considered. It is also considered that the connected open surface where boundary conditions are set obtains the Abelian group of symmetry of the sixteenth order. Such problems arise in the mathematical modeling of electronic optics systems. The general method based on integral equation method was improved. Specificity of the problem was taken into account by use of group theory apparatus. This article shows how using the apparatus of the group theory it is possible to solve an initial problem by the help of the sequence of the sixteen independent integral equations, where the integration is realized only on one of the congruent constituents of the surface. It creates the conditions for parallel processes of problem solution in general. The procedure of parallelization was realized with the help of the most popular means of OpenMP. The collocation method for obtaining approximate values of needed "density of charge distribution" in the particular two-dimensional integral equations is used. To take into account the singular way of solving the problem in the circuit of the open surface the a posteriori method of error evaluation is created. The account of specificity of the open-circuit surfaces allows to decrease the amount of the controlled special points considerably and in the best case to deal only with one. It also substantially simplifies the algorithm of calculations. To prove the reliability and estimation of the technique efficiency the number of numerical experiments is carried out. For the representation of the electrostatic field equipotential lines are used, thus, solutions are analysed by the help of distribution of lines of even potential.На модельном примере рассмотрены апостериорный метод оценки погрешности и процедура распараллеливания для численного решения одного класса задач электронной оптики. Учитывается тот факт, что поверхность, на которой определены краевые условия, обладает абелевой группой симметрии шестнадцатого порядка. Усовершенствована общая методика, которая основывается на методе интегральных уравнений. Специфику смоделированной проблемы учтено на основании аппарата теории групп. Используя аппарат теории групп, удалось свести исходную задачу к решению последовательности шестнадцати независимых интегральных уравнений, где интегрирование ведется только по одной из конгруэнтных составляющих поверхности. Это создало все предпосылки для распараллеливания процедуры решения задачи в целом. Процедуру распараллеливания проведено с использованием программного средства OpenMP. Для получения приближенных значений искомой "плотности распределения зарядов" в соответствующих двумерных интегральных уравнениях использован метод коллокации. С целью учета сингулярного поведения решения в окрестности контура разомкнутой поверхности построен апостериорный метод оценки погрешности. Для подтверждения уместности и оценки эффективности методики проведен ряд численных экспериментов.На модельному прикладі розглянуто апостеріорний метод оцінювання похибки і процедуру розпаралелення для чисельного розв'язування одного класу задач електронної оптики. Враховано той факт, що поверхня, на якій визначено крайові умови, володіє абелевою групою симетрії шістнадцятого порядку. Вдосконалено загальну методику, яка ґрунтується на методі інтегральних рівнянь. Специфіку змодельованої проблеми враховано на підставі апарату теорії груп. Використовуючи апарат теорії груп, вдалось звести вихідну задачу до розв'язування послідовності шістнадцяти незалежних інтегральних рівнянь, де інтегрування ведеться лише по одній з конгруентних складових поверхні. Це створило всі передумови до розпаралелення процедури розв'язування задачі загалом. Процедуру розпаралелення проведено з використанням програмного засобу OpenMP. Для отримання наближених значень шуканої "густини розподілу зарядів" у відповідних двовимірних інтегральних рівняннях використано метод колокації. З метою врахування сингулярної поведінки розв'язку в околі контуру розімкненої поверхні побудовано апостеріорний метод оцінювання похибки. Для підтвердження доречності та оцінки ефективності методики проведено ряд чисельних експериментів

On the symplectic structure deformations related to the Monge–Ampère equation on the Kähler manifold P2(C)

We analyze the cohomology structure of the fundamental two-form deformation related to a modified Monge–Ampère type on the complex Kähler manifold P2(C). Based on the Levi-Civita connection and the related vector-field deformation of the fundamental two-form, we construct a hierarchy of bilinear symmetric forms on the tangent bundle of the K\"{a}hler manifold P2(C), that generate Hermitian metrics on it and corresponding solutions to the Monge–Ampère-type equation. The classical fundamental two-form construction on the complex Kähler manifold P2(C) is generalized and the related metric deformations are discussed

Electrode potential in the weld element along the creative cylinder filled with a hydrogen-containing environment

The work examines the electrode potential in the tube element of the cylinder welded along the generator. The pipe element is filled with a hydrogen-containing medium and is under the influence of internal and external pressures. Analytical-mathematical relations are proposed. Appropriate calculations have been mad

Investigation of mathematical models for vibrations of one dimensional environments with considering nonlinear resistance forces

In this paper we consider important classes of one dimensional environments, bending stiffness of which can be neglected. It is impossible to apply approximate analytical method of solution of mathematical models of dynamic processes. So justification of existence and uniqueness of solutions, carried out a qualitative their evaluation, based on numerical analysis are considering in this paper. Also the features of dynamic processes of some of examined class of systems are analyzed. Methods of qualitative study of oscillations for restricted and unrestricted bodies under the influence of the resistance forces, described in this paper are based on the general principles of the theory of nonlinear boundary value problems – Galerkin method and the method of monotonicity. Scientific novelty consists in generalization these methods of studying for nonlinear problems at new classes of oscillating systems, justification of solution correctness for specified mathematical models that have practical application in real engineering vibration systems

Qualitative methods for research of transversal vibrations of semi-infinite cable under the action of nonlinear resistance forces

The aim of paper is to study the solution of the problem of nonlinear transverse vibrations of elastic elongated body under the force of resistance in unbounded domain. Such problems have applications in various technical systems - vibration of pipelines, railways, long bridges, electric lines, optical fibers. Unboundedness of the area creates more fundamental difficulties in the study of the problem. For the considered models of nonlinear oscillations have no general analytical techniques for determining the dynamic characteristics of the oscillatory process. Therefore it is sugges ted to use qualitative methods of the theory of nonlinear boundary value problems to obtain correct problem solution conditions (existence and uniqueness of the solution). In the paper conditions of the correctness of the solution of mathematical model for these nonlinear systems (sufficient conditions of the existence and uniqueness in the class of locally integrable functions) are obtained. Methods of qualitative study of semi-infinite cable vibrations under the forces of resistance based on general principles of the theory of nonlinear boundary value problems - method of monotony and Galerkin method. Scientific novelty of the work lies in particular in the generalization of methods of studying nonlinear problems on a new class of oscillatory systems In unbounded domains, justifying the correctness of the solution with specified mathematical model, which has practical applications in real engineering oscillatory systems. The technique allows not only for proving the correctness of the model solution, but also has an opportunity in its study to apply various approximate methods

Influence of some speed parameters on the dynamics of nonlinear flexural vibrations of a drill column

We investigate the influence of the motion of fluid flushing the cutter of a well drilling column, and the angular rotational velocity upon dynamic characteristics of its flexural vibrations. We take into account the nonlinear elastic features of column material. As a base of the research we took the Galerkin method and the Van der Pol method. Combining those two methods made possible to obtain the relations describing the main parameters of the dynamical process In both nonresonance and resonance case

Metoda i sposób pomiaru małych wartości rezystancji elektrycznej

The paper presents the results of developing simulation of a system for measuring small quantities of electrical resistance using technology of microelectromechanical systems (MEMS). It also offers the analysis and evaluation of the results. The authors established the dependence of residual voltage (as the output parameter of measuring system) on controlled resistance (as input parameter of measuring system core) during the process of balancing the electric bridge circuit.Artykuł przedstawia wyniki symulacji opracowanego systemu do pomiaru małych wartości rezystancji elektrycznej wykorzystującego układy MEMS. Przedstawiono takie analizy oraz ocenę wyników. Opisano zależność napięcia szczątkowego (jako parametr wyjściowy systemu pomiarowego) od rezystancji mierzonej (jako parametr wejściowy systemu pomiarowego) podczas procesu równoważenia mostka