Method Of Discrete Singularities Research Articles

A review of the problem of modeling the aerodynamics of small-sized ekranoplanes

A characteristic trend in the development of modern transport systems is the widespread introduction of unmanned technologies. One of the new vehicles that are promising for use as high-speed unmanned aerial vehicles are ekranoplanes (WIG crafts). Such devices use the principle of dynamic support above the surface. Therefore, the processes of aerodynamics are decisive for their reliable and effective operation. The presented work considers the problem of modeling aerodynamic processes during the movement of the vehicle near the surface. In contrast to studies in an unrestricted flow, the presence of a surface leads to a number of features that determine the need for special approaches to both physical and mathematical experiments. It is established that each of the methods can ensure the similarity of the modeled processes to the real ones within the defining limits. A brief review of methods of numerical modeling of ekranoplan aerodynamics is presented. The approaches, which based on the Navier-Stokes equations and method of discrete singularities, are considered. There are examples of calculation of the processes of wing aerodynamics herein. The advantages and disadvantages of viscous and inviscid models for research are shown.

NUMERICAL MODELLING OF ELECTROMAGNETIC WAVE SCATTERING ON GRATINGS WITH ONE-DIMENSIONAL QUASI-FRACTAL PERIOD STRUCTURE

Numerical modelling of the properties of E-polarised and H-polarised waves scattered on periodic screened quasi-fractal gratings is carried out. The location of the band system at each period is determined by the principle of constructing a generalized symmetric Cantor set at a certain step of the algorithm. A mathematical model of the problems based on systems of boundary singular integral equations of the first kind was used in the study. These systems of equations were obtained using the method of parametric representations of singular and hypersingular integral operators. The systems of singular integral equations were solved numerically using the computational schemes of the method of discrete singularities. The solutions of these equations are used to obtain the main characteristics of the electric and magnetic fields. This experiment proved the possibility of using the MDS computational scheme to analyse systems containing 8 – 16 bands at different distances from each other. Graphs of the dependence of harmonic amplitudes on the wavenumber, point plots of absolute values of all non-zero harmonics at resonant wavenumber values, and maps of electric and magnetic field components in the region above the grating were obtained. It is confirmed that the overall field structure in the case of normal incidence is significantly influenced by all harmonics with absolute numbers from 0 to 50. The harmonics had a large number of resonances that were observed at different values of the wavenumber. This led to a complex structure of the isolines of absolute values of the scattered electric and magnetic field amplitudes in the region above the structure, and a significant difference in amplitude values with small changes in coordinates. In the future, it is planned to carry out computer simulations for imperfectly conducting structures and compare the results with the numerical results for the ideal case considered in this paper. The proposed structure may be of interest for the design of multimode broadband antennas.

КОМП’ЮТЕРНЕ МОДЕЛЮВАННЯ РОЗСІЮВАННЯ Е-ПОЛЯРИЗОВАНИХ ХВИЛЬ НА ЕКРАНОВАНІЙ ПРЕДКАНТОРОВІЙ СИСТЕМІ РЕФЛЕКТОРІВ

Computer simulation of electromagnetic wave scattering on a shielded periodic reflector system, with a special period structure is considered. The electrodynamic reflective structure consists of a screen and a periodic system of strips. The cross-section of the strips is a system of segments formed by the principle of constructing a generalized Cantor set at each of the periods. Іt was used a mathematical model based on the reduction of the initial boundary value problems for the Helmholtz equation to systems of singular integral equations. The choice of this approach did not require analytical calculations when changing the number of lattice elements. Using this approach, it was not necessary to make changes to the computer implementation of the algorithm. The computer experiment has proved the possibility of applying the discrete singularities method computational scheme for the considered structure. It is proved that the structure constructed according to the proposed principle has the properties of broadband and multimode for the studied values of the structure parameters. In particular, for the studied structures, the resonance occurred when the wavenumber was changed from 0 to 50, with a significant increase in amplitude occurring at harmonics with numbers up to 40. This phenomenon led to a complex structure of the lines of the absolute values of the amplitude of the scattered electric field above the grating. The presence of small areas with an increase in the amplitude of the scattered field by a factor of 5-6 compared to the amplitude the incident electromagnetic wave was observed. These properties show practical interest in the use of this type of antennas to improve electronic camouflage and tactical radio communications. In the future, it is planned to conduct computer simulations for imperfectly conducting structures and compare the results with the numerical results for the ideal case discussed in this paper.

Numerical modeling of the performance of a well with an arbitrary piecewise-smooth external boundary in an anisotropic heterogeneous reservoir

The extraction of oil and gas from productive reservoirs of complex geological structures is constantly increasing. Thus, it is obvious that reliable mathematical models able to manage such porous media are of particular interest. In this paper, a model of filtration flow to a well located in an anisotropic heterogeneous soil formation, with a piecewise-smooth external contour, is numerically studied. The permeability of soil, as a porous medium, is characterized by a second-rank tensor. Soil heterogeneity is modeled by a power function of one of the coordinates. We consider a particular case with a positive exponent. The solution of the stated boundary problem of well operation in a porous reservoir causes significant mathematical difficulties associated with the complicated form of the basic equation. The formulation of the problem is reduced to a canonical form, which greatly simplifies the solution. To do this, we turn on the auxiliary plane using homeomorphic (affine) transformation. The problem is reduced to a system that includes a Fredholm type integral equation and an integral relation. The results were obtained using the discrete singularity method. It was found that the algorithm convergence is slightly better under weak anisotropy. The influence of anisotropy and heterogeneity of the soil on the production in the case of the productive reservoir with a rectangular boundary was assessed. The anisotropy can greatly change the production rate compared to the rate of a well in an isotropic soil. Permeability tensor components are located on the main diagonal and have a major impact on the rate of the flow in the anisotropic and heterogeneous soil. This method can be used to solve different problems of fluid filtration in an anisotropic porous medium.

Математична модель дифракції хвиль на системі стрічок з різними значеннями поверхневого імпедансу

A mathematical model of diffraction of E-polarized and H-polarized waves on a finite system of not perfectly conducting tapes is obtained. The value of the surface impedance on the two sides of the stripes is different. The initial boundary value problem for the Helmholtz equation with boundary conditions of the third kind was reduced to a system of boundary integral equations. This system of boundary integral equations consists of singular integral equations of the first kind and integral equations of the second kind with a logarithmic singularity. The method of parametric representation of integral operator was used to perform transformations. The values of the physical characteristics of the process are expressed through the solutions of the obtained systems of integral equations. Numerical solution of these equations is performed using a computational scheme based on the discrete singularities method.

Calculation of the Aeroelastic Characteristics of a Wing during Vibrations in a Subsonic Flow by the Discrete Singularity Method

The problem of determining the total and generalized aerodynamic forces for an airplane wing during vibrations in a compressible flow with allowance for the real frequencies of its natural vibrations is considered. The need to develop operational numerical methods for solving the problem, which is reduced to a Neumann-type boundary problem for a scalar Helmholtz equation, is substantiated. The results of numerical calculations of the wing of the Yak-130 aircraft are presented.

NUMERICAL SOLUTION OF A SYSTEM OF SINGULAR INTEGRAL EQUATION WITH HILBERT AND CAUCHY KERNELS

In the paper a specific system of first kind singular integral equations with the Hilbert and Cauchy kernels arising when solving some problems of electrostatics and electrodynamics is studied. The method of discrete singularities is applied for constructing its discrete mathematical model, which is a system of n linear algebraic equations. Under the additional smoothness assumptions on the right-hand parts of the equations of the initial system and regularity assumptions on the kernels of the integrals in them the obtained system of linear algebraic equations is proved to admit a unique solution for n sufficiently large. The rate of convergence of the solution of the discrete problem to the exact solution of the system of singular integral equations is estimated.

Математическое моделирование работы скважины в случае двумерной фильтрации в анизотропном неоднородном пласте

Поставлена плоская (двумерная) задача о математическом моделировании работы скважины в анизотропном неоднородном пласте грунта с раздельной анизотропией и неоднородностью, когда контур питания произвольный. Рассматривается совершенная скважина, когда она полностью вскрывает пласт своей рабочей частью (фильтром). Проницаемость грунта характеризуется тензором второго ранга, компоненты которого моделируются степенной функцией координат. Гомеоморфным аффинным преобразованием координат эта задача приводится к каноническому виду, что значительно упрощает ее исследование. Получено в конечном виде аналитическое решение задачи о дебите скважины с конкретным эллиптическим контуром питания, а также в случае, когда контур питания удален в бесконечность. В случае произвольного гладкого контура питания задача о дебите редуцирована к системе сингулярного интегрального уравнения и интегрального соотношения, которая решена численно методом дискретных особенностей. Исследовано влияние на дебит анизотропии, неоднородности пласта и формы контура питания. A flat (two-dimensional) problem has been posed on the mathematical modeling of well in an anisotropic inhomogeneous reservoir of soil with separate anisotropy and heterogeneity when the power contour is arbitrary. The considered well completely opens the formation with its working part (filter). Such a well is called perfect. The permeability of the soil is characterized by a second-rank tensor whose components are modeled by a power function of the coordinates. With a homeomorphic affine transformation of coordinates, this problem is reduced to a canonical form which greatly simplifies its study. An analytical solution of the problem of well production with an elliptical power contour is obtained in the final form as well as in the case when the power contour is removed to infinity. In the general case, the problem is reduced to a system of integral equations and the integral relation. The results were obtained in the general case using the discrete singularities method. The influence on the flow rate of anisotropy, heterogeneity of the reservoir and the shape of the power contour was studied.

Liquid oscillation in a cylindrical-conical shell under the action of vertical and horizontal excitation

Vibrations of an ideal incompressible fluid in shells of revolution have been considered. The shells of revolution under consideration include cylindrical and conical parts. It is assumed that the shell is subjected to vertical and horizontal excitations. The liquid in the shells is supposed to be an ideal and incompressible one. The fluid flow is the irrotational. Therefore the velocity potential that satisfies the Laplace equation exists. The non-penetration conditions are applied to the wetted surfaces of the shell and the kinematic and dynamic conditions on the free surface have been considered. The liquid pressure as the function of the velocity potential is defined using the Bernoulli equation. The problem of determining the fluid pressure is reduced to solving a singular integral equation. The numerical solution of the equation has been obtained by the method of discrete singularities. The method of simulating the free and forced oscillations of the fluid in the shells of revolution has been developed.

Method of discrete singularities in problems of filler vibrations in fuel tanks under overloads and at low gravity

The paper is about free vibration problems in fuel tanks with different levels of fillers both under overloads and low gravity. The fuel tanks are considered as rigid shells of revolution. The liquid, contained in the tanks, is supposed to be an ideal and incompressible one, and its flow, induced by external excitations, is irrotational. The problem of free axisymmetric harmonic oscillations of the fluid-filled rigid compound shell is considered. The mixed boundary value problem is formulated, and boundary conditions are received on the boundaries of the fluid domain. The non-penetration conditions is formulated on the rigid shell boundaries, and on the free liquid surface there are kinematic and dynamic boundary conditions. The flow fluctuations are described by using the velocity potential that for ideal and incompressible liquids satisfies the Laplace equation. For its solution, the integral representation is in use. But there are two unknown functions, the velocity potential and the function describing the shape and position of the free surface during time. The transformation of the boundary conditions leads to eliminating one of these unknown functions. So, the system of boundary singular integral equations is obtained to determine the velocity potential. The discrete singularity method and the boundary element method are applied for its numerical solution. The problem of determining own modes and frequencies is solved by using the techniques, where the surface tension effects are neglected. Then these modes are considered as basic functions to determine the modes and frequencies of the liquid taking into account the surface tension. Thus, in this work the method is developed which takes into account the surface tension effect on the frequency of fluid fluctuations in the rigid tank under low gravity conditions. The surface of the interaction between liquid media and gas is considered as a thin membrane, whose thickness is neglected. The fluid pressure on this surface is determined by the Laplace-Young equation. The developed method is useful for the investigation of free and forced fluid oscillations in rigid compound shells with arbitrary meridians.

Method of singular integral equations in liquid vibration problems for coaxial shells

The paper deals with the problem of free vibrations of an ideal incompressible fluid in coaxial shells of revolution. It is assumed that the motion of the fluid is irrotational that allows us to introduce the velocity potential. In these suppositions the potential is satisfied to Laplace equation. The boundary conditions are formulated on the wetted surfaces of the shells and on the free liquid surface. The non-penetration conditions are applied to the wetted surfaces. On the free surface we consider dynamical and kinematical boundary conditions. The dynamical condition consists in equality of the liquid pressure on the free surface to the atmospheric one. The kinematic condition requires that total time derivative of the free surface elevation will be equal to zero at any instant. Regarding the potential of velocities, a boundary value problem is formulated that is further reduced to the eigenvalue problem. To solve the boundary value problem for the Laplace equation, the boundary element method is used in a direct formulation. The axial symmetric form of the shells allows us to reduce the obtained system of singular equations to one-dimensional equations. The kernels in singular operators of obtained integral equations are expressed on terms of elliptical integrals of the first and second kinds, and have the logarithmic singularities. The special numerical technique is elaborated to treat with such kind integral equations. The resulting one-dimensional singular equation is solved by the method of discrete singularities. The integration region contains the free surface of the fluid that in the case of coaxial shells is a ring. So, the possibility of using the boundary integral equation approach coupled with application of the discrete singularities method is established to solution of the singular integral equation with incoherent boundaries. A numerical study has been carried out that made it possible to determine the frequencies and modes of the liquid sloshing in the shells for different ratios of the inner and outer radii of cylindrical coaxial shells. The obtained modes of natural vibrations will be used for numerical simulation of forced liquid vibrations in the tanks and reservoirs.

Singular integral equations in diffraction by multilayer grating of graphene strips in the THz range

Diffraction of the H-polarized electromagnetic wave by multilayer graphene grating in the THz range is considered. The scattered field is represented in the spectral domain via unknown spectral functions. The mathematical model is based on the graphene surface impedance and the method of singular integral equations. The numerical solution is obtained by the Nystrom-type method of discrete singularities. The scattering and absorption characteristics as functions of frequency and distance between layers are studied.

Modeling of Graphene Planar Grating in the THz Range by the Method of Singular Integral Equations

AbstractDiffraction of the$H$- polarized electromagnetic wave by the planar graphene grating in the THz range is considered. The scattering and absorption characteristics are studied. The scattered field is represented in the spectral domain via unknown spectral function. The mathematical model is based on the graphene surface impedance and the method of singular integral equations. The numerical solution is obtained by the Nystrom-type method of discrete singularities.

Algorithms of the Discrete Singularity Method for Computing Technologies

Algorithms of the discrete singularity method for computing technologies are considered. The algorithms transform discrete integral representations with discontinuous functions and change the order of singularities in the system of discrete singularities. The results of transformation allow us to correctly calculate the values of the functions and their derivatives under parametric dependence of characteristic functions on time. The algorithms of computing technologies are applicable for both two-dimensional and three-dimensional hydrodynamic problems of non-stationary detached flow.

ДИФРАКЦИЯ Е-ПОЛЯРИЗОВАННОЙ ВОЛНЫ НА БЕСКОНЕЧНОЙ ПЕРИОДИЧЕСКОЙ ЛЕНТОЧНОЙ РЕШЕТКЕ В ОТСУТСТВИЕ ОДНОЙ ЛЕНТЫ

PACS number: 41.20.Jb Purpose: The E-polarized wave diffraction by an infinite periodic strip grating without a single strip is considered. Design/methodology/approach: The total field is found as a sum of field of infinite periodical grating and field induced by the removal of a single strip. The problem is reduced to the singular integral equations with additional conditions. Findings: The directional patterns and field distribution in the domain above the grating are represented. Conclusions: The effective algorithm for study of the field which appeared as a result of absence of a single strip is suggested. Key words: infinite periodic grating, integral equation, diffraction Manuscript submitted 30.05.2016 Radio phys. radio astron. 2016, 21(3): 189-197 REFERENCES 1. SHESTOPALOV, V. P., 1971. The method of the Riemann-Hilbert problem in the theory of electromagnetic wave diffraction and propagation . Kharkiv: Kharkiv State University Press (in Russian). 2. SHESTOPALOV, V. P., LYTVYNENKO, L. M., MASALOV, S. A. and SOLOGUB, V. G., 1973. Wave diffraction by gratings . Kharkiv: Kharkiv State University Press,(in Russian). 3. SOLOGUB, V. G., 1975. On some method for studying the problem of diffraction by a finite number of strips in the same plane. Dokl. AN USSR . Ser. A . no. 6, pp. 549–552 (in Russian). 4. LYTVYNENKO, L. M. and PROSVIRNIN, S. L., 2012. Wave diffraction by periodic multilayer structures . Cambridge: Cambridge Scientific Publishers. 5. LYTVYNENKO, L. M., KALIBERDA, M. E. and POGARSKY,S. A., 2013. Wave diffraction by semi-infinite venetian blind type grating. IEEE Trans. Antennas Propag . vol. 61, no. 12, pp. 6120–6127. DOI: https://doi.org/10.1109/TAP.2013.2281510 6. KALIBERDA, M. E., LYTVYNENKO, L. M. and POGARSKY,S. A., 2015. Diffraction of H-polarized electromagnetic waves by a multi-element planar semi-infinitegrating. Telecommunications and Radio Engineering . vol. 74, no. 9, pp. 753–767. DOI: https://doi.org/10.1615/TelecomRadEng.v74.i9.10 7. NEPA, P., MANARA, G. and ARMOGIDA, A., 2005. EM scattering from the edge of a semi-infinite planar strip grating using approximate boundary conditions. IEEE Trans.Antennas Propag . vol. 53, no. 1, pp. 82–90. DOI: https://doi.org/10.1109/TAP.2004.840523 8. GANDEL, YU. V., 1986. The method of discrete singularities in problems of electrodynamics. Voprosy Kibernetiki . no. 124, pp. 166–183 (in Russian) 9. GANDEL, YU. V., 2010. Boundary-value problems for the Helmholtz equation and their discrete mathematical models. J. Math. Sci. vol. 171, no. 1, pp. 74–88. DOI: https://doi.org/10.1007/s10958-010-0127-3 10. ZAGINAYLOV, G. I., GANDEL, Y. V., KAMYSHAN, O. P.,KAMYSHAN, V. V., HIRATA, A., THUMVONGSKUL, T. and SHIOZAWA, T., 2002. Full-wave analysis of the field distribution of natural modes in the rectangular waveguide grating based on singular integral equation method. IEEE Trans. Plasma Sci . vol. 30, no. 3. pp. 1151–1159. DOI: https://doi.org/10.1109/TPS.2002.801613 11. ZAMYATIN, YE. V. and PROSVIRNIN, S. L., 1986. Diffraction of electromagnetic waves by an array with small random fluctuations of the dimensions. Sov. J. Commun. Technol. Electron . vol. 31, no. 3, pp. 43–50. 12. ABRAMOWITZ, M. and STEGUN, I. A., eds., 1964. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables . Numder 55 in National Bureau of Standards Applied Mathematics Series. U. S. Government Printing Office, Washington, D. C. 13. FELSEN, L. B. and MARCUVITS, N., 1973. Radiation and Scattering of Waves . Englewood Cliffs, N.J.: Prentice-Hall.

Energy modeling method as applied to diffraction of stationary plane elastic waves

The energy method uses reciprocity works theorem and transformation of occurring hypersingular integrals for a system of equations consisting of singular integral equations of the first and second kind. Algorithms for the numerical solution are obtained by the method of discrete singularities. Precision of calculations is controlled due to algorithmic diversity. Application to the numerical modeling of diffraction of stationary plane waves on finite and periodic systems of different inhomogeneities is shown

Analytical and numerical modeling of the work of a perfect well in anisotropic homogeneous soil formation

The extraction of oil and gas reservoirs in complex geological structures is constantly increasing. Thus, it is obvious that the reliable mathematical models governing such porous media must be of interest. The problem is concerned with the work of a well in anisotropic soil formation in the case of a random smooth (piecewise-smooth) external boundary of reservoir. The permeability of the soil is described by an asymmetric second-rank tensor. To find a solution to the problem is difficult because of the complicated form of the basic equation. The solution can be simplified if the equation is transformed into the Laplace equation. To do this, we turn on the auxiliary plane using homeomorphic (affine) transformation. In a special case when the external boundary of reservoir of the auxiliary plane takes the form of a circle, we arrive at a closed-form (analytical) solution. These analytic solutions are useful in testing numerical codes. In the general case, the problem is reduced to a system of integral equations with Cauchy kernel and the integral relation. The system is solved by the method of discrete singularities. Convergence of the algorithm is studied. Convergence solutions are improved with an increase in the number of division points. The convergence solutions with a piecewise-smooth (square) external boundary of reservoir are slightly worse compared with the smooth (circular) ones. Soil anisotropy greatly affects the production rate and can strongly change the production rate compared to the rate of the well in an isotropic soil. Permeability tensor components located on the main diagonal have a major impact on the rate of flow in the anisotropic soil. This method can be used to solve different problems in filtering anisotropic porous medium.

504 基本的な三次元角柱の付加質量と付加減衰の高速解法と実験(GS5 流体工学(2),研究討論セッション)

In many fluid-structure interaction problems, we often needed to consider the fluid forces caused by arbitral accelerated motions of fluid and/or structures. Such forces have been understood using the concept of the "added mass" or the "virtual mass." In the present study, We propose a simple and efficient method to specify fluid-force coefficients of a three-dimensional oscillating object in viscous fluid. In this method, we consider incompressible and viscous fluid under the assumption of infinitesimal amplitude of a three-dimensional object, and properly modify the Navier-Stokes equations into linear equations, namely, the Brinkman equations. The solving method is based on a discrete singularity method (DSM). Then, we apply the DSM to many three-dimensional objects. As a result, we can specify the fluid forces of square-cross-section cylinders at aspect ratio b/a = 0.5-5.0. Moreover, we conduct the experiment, and we can confirm the amplitude effects.

Discrete Mathematical Model of the Problem of Diffraction for E -Polarized Waves on Slots in the Impedance Plane

We consider the problem of diffraction for E -polarized plane monochromatic waves on slots in the impedance plane. A discrete mathematical model of the boundary integral equations of this problem is constructed. We also perform a numerical experiment based on the use of the efficient numerical method of discrete singularities.

Mathematical Model of Wave Scattering by an Impedance Grating1

This article develops a numerical-analytical method for solving the problem of scattering acoustic and electromagnetic waves by an impedance grating. This problem leads to a third-type boundary problem for the two-dimensional Helmholtz equation with additional conditions. The boundary value problem is reduced to boundary integral equations, and the discrete singularities method is modified to numerically solve them. Dependencies of integral characteristics of solutions to the problem on frequency are obtained.