Piecewise Homogeneous Medium Research Articles

Mathematical Modeling on a Physics-Informed Radial Basis Function Network

The article is devoted to approximate methods for solving differential equations. An approach based on neural networks with radial basis functions is presented. Neural network training algorithms adapted to radial basis function networks are proposed, in particular adaptations of the Nesterov and Levenberg-Marquardt algorithms. The effectiveness of the proposed algorithms is demonstrated for solving model problems of function approximation, differential equations, direct and inverse boundary value problems, and modeling processes in piecewise homogeneous media.

Determination of the maximum mechanical stresses in the insulating material around a defect with a high dielectric permittivity in an electrostatic field

Introduction. All insulating macrohomogeneous solid materials change shape under the influence of an electric field. Problem. The presence of minor defects changes the distribution of an electric field and causes a significant concentration of mechanical stresses in a given section of the material, which, under certain circumstances, can cause partial or complete destruction of this material. Goal. The purpose of the work is to determine maximum mechanical stresses according to the von Mises criterion in insulating materials around defects with ionized air and water in an electrostatic field. Also, to analyze the influence of the following parameters on the indicated stresses: the location of the defect, the orientation angle of the semi-major axis of the defect cross-section, the ratio of semi-major and semi-minor axes, elastic and dielectric properties of the insulating material and the defect. Methodology. The study is based on the interrelated equations of electrostatics and structural mechanics for an isotropic piecewise homogeneous medium. The solution of these equations is obtained by the finite element method. Results. Graphs of dependences of maximum mechanical stresses on the ratio of semi-major and semi-minor axes of the ellipsoidal cross-section of the defect have been obtained. The minimum ratio of the greatest stresses in the insulating materials around the surface cracks and pores for ionized air has been 9.3 times for the maximum ratio of major and minor semi-axes of the cross-section of the defect considered in the work, which is 10. For a water defect, the similar ratio has been 2...5.6 times, increasing when the relative dielectric permittivity of the insulating material changes from 7 to 2. When Young’s modulus of the insulating material increases from 1 MPa to 100 GPa, the angles of the inclination of the linearized dependences of maximum mechanical stresses around bounded pores with ionized air (water) to the axis of the ratio of major and minor semi-axes of the defect cross-section have been increased by 35.9° (58.0°) and 18.6° (20.1°) at orientations of major semi-axes at angles of 0° and 45°, respectively. Originality. The numerical-field mathematical two-dimensional model has been developed for the first time, which consists of sequentially solved equations of electrostatics and structural mechanics, for the determination of the distribution of mechanical stresses in an insulating material with a liquid or gaseous defect. It has been established for the first time that the ratio of the elastic properties of the insulating material and the defect determines the angle of the inclination of the linearized dependence of the maximum mechanical stress to the axis of the ratio of major and minor semi-axes of the defect cross-section. Practical value. The types of defects that contribute to the aging of insulation materials under the combined action of an electric field and a stress field to the greatest extent have been established.

Formulation of a T-Matrix Approach for a Piecewise-Homogeneous Anisotropic Medium Excited by a Spherical Sound Wave

A piecewise-homogeneous medium, consisting of anisotropic layers, is excited by a primary spherical sound wave due to a point source lying in the exterior of the medium or in one of its layers. The direct scattering problem is formulated by means of a modified scalar Helmholtz equation incorporating the anisotropic characteristics of the problem. The T-matrix of the problem is determined analytically by means of a suitable coordinate transformation. Then, the acoustic fields in all layers are obtained. Finally, specific reductions to special cases are presented.

A Semi-Explicit Solution of Scattering From Dielectric Bodies Using the Method of Moments

This work presents a novel closed-form expression based on the Method of Moments (MoM) for solving the scattering problem in piecewise homogeneous media. Using point-matching with pulse basis functions in MoM, we derive an expression represents the scattered field as an explicit function of the permittivity value of each homogeneous part. The coefficient matrix depends on the permittivity values in the MoM solution to the scattering problem. Therefore, it requires matrix inversion whenever these values change slightly. Here, we propose a new expression where the matrix coefficients are independent of the permittivity values of each homogeneous part. Accordingly, when the geometry does not change, we propose a semi-explicit procedure for updating the solution with less computational complexity. Although the proposed method can be used for all 2D and 3D problems, the formulation details are implemented for 2D problems, TM incidence. Based on this explicit form, we derive some simple approximate expressions for three specific cases of permittivity perturbations. Our proposed solution is appropriate for microwave imaging, particularly when a priori information is utilized.

Electromagnetic wave scattering by a partially coated obstacle in a piecewise homogeneous medium

In this paper, we consider an electromagnetic obstacle scattering problem in a piecewise homogeneous medium. We assume that the incident field is the electromagnetic plane wave and the obstacle consists of metallic and nonmetallic parts. We firstly establish uniqueness and existence for the solution to the electromagnetic obstacle scattering problem. Then we prove that the electric far‐field pattern satisfies the reciprocity relation and the set of electric far‐field patterns is complete in a Hilbert space. These results have an important bearing on the solution of the inverse electromagnetic scattering problem, that is, determination of the shape of the obstacle from the knowledge of the electric far‐field patterns.

SIMULATION OF ELECTROMAGNETIC PROCESSES IN THE WORKING AREA OF SPECIAL DEVICES CONSIDERING THE INFLUENCE OF THE MAGNETIC SCREEN

The work is devoted to the creation of a modern web-oriented software platform for designing magnetic systems of special devices and electric machines of new generations that are built on the basis of permanent magnets. To solve the problem of increasing the accuracy of the performed calculations it is necessary to develop new mathematical models that take into account the shielding effect of structural elements on physical processes in electromechanical devices. To obtain mathematical expressions for the calculation of strength components of the magnetic field at given points of piecewise homogeneous space, the method of secondary sources is applied. The studied system is of a generalized design of a magnetic suspension built on the basis of the connected permanent magnets that are facing each other. The entire structure is placed in the shielded housing. Since the electromagnetic system of the device contains parts of a complex shape in its design and has non-linear characteristics of the environment, in order to obtain a mathematical model, an approach is suggested that is based on that a task of calculating the magnetic field for a piecewise homogeneous medium can be formulated in the form systems of Fredholm linear integral equations of the second kind. In accordance with the method of secondary sources, the solution to the field calculation problem is divided into two stages: first the distribution of additional sources is found, the effect of which is equivalent to the influence of a heterogeneous environment on physical processes in work zone, and, at the next stage, based on the principle of superposition of fields, the total magnetic field caused by the presence of primary and secondary sources is calculated. Based on the obtained mathematical model, a calculation algorithm was developed and a web-oriented software system for modeling magnetic field characteristics was created. Using the developed software application, numerical experiments were conducted to simulate the magnetic field in the work zone of an electrical device according to the given electrophysical and geometric design parameters. The proposed approach can be used to solve other similar problems when it is necessary to calculate the characteristics of a system containing permanent magnets in the conditions of a piecewise homogeneous environment.

Computational Modeling of Lymph Filtration and Absorption in the Lymph Node by Boundary Integral Equations

We develop a numerical method for solving three-dimensional problems of fluid filtration and absorption in a piecewise homogeneous medium by means of boundary integral equations. This method is applied to a simulation of the lymph flow in a lymph node. The lymph node is considered as a piecewise homogeneous domain containing porous media. The lymph flow is described by Darcy’s law. Taking into account the lymph absorption, we propose an integral representation for the velocity and pressure fields, where the lymph absorption imitates the lymph outflow from a lymph node through a system of capillaries. The original problem is reduced to a system of boundary integral equations, and a numerical algorithm for solving this system is provided. We simulate the lymph velocity and pressure as well as the total lymph flux. The method is verified by comparison with experimental data.

A Hybrid SIE-PDE Formulation Without Boundary Condition Requirement for Transverse Magnetic Electromagnetic Analysis

A hybrid surface integral equation partial differential equation (SIE-PDE) formulation without the boundary condition requirement is proposed to solve the transverse magnetic (TM) electromagnetic problems. In the proposed formulation, the computational domain is decomposed into two overlapping domains: the SIE and PDE domains. In the SIE domain, complex structures with piecewise homogeneous media, e.g., highly conductive media, are included. An equivalent model for those structures is constructed by replacing them with the background medium and introducing a surface equivalent electric current density on an enclosed boundary to represent their electromagnetic effects. The remaining computational domain and homogeneous background medium replaced domain consist of the PDE domain, in which inhomogeneous or non-isotropic media are included. Through combining the surface equivalent electric current density and the inhomogeneous Helmholtz equation, a hybrid SIE-PDE formulation is derived. It requires no boundary conditions, and is mathematically equivalent to the original physical model. Through careful construction of basis functions to expand electric fields and the equivalent current density, the discretized formulation is made compatible with the SIE and PDE domain interface. The accuracy and efficiency are validated through two numerical examples. Results show that the proposed SIE-PDE formulation can obtain accurate results, and significant performance improvements in terms of CPU time and memory consumption compared with the FEM are achieved.

Electromagnetic interactions of dipole distributions with a stratified medium: Power fluxes and scattering cross sections

AbstractN magnetic dipoles radiate (inside or outside) a piecewise homogeneous (layered) and isotropic medium. The power fluxes and energy processes inside and outside the medium resulting from the various interactions between participating fields are investigated by means of the introduced interaction scattering cross sections (ISCS) and interaction power fluxes (IPF). Connections between the power flux through the excitation layers and the radiation in the far field are established by specifying the law of energy conservation for the considered scattering problem. Additionally, the relation between the IPF and the ISCS is derived by applying a generalized version of the optical theorem. Physical bounds for the ISCS with respect to the minimum and maximum individual cross sections and the number of excitation dipoles and layers are derived. The large‐N behavior of the ISCS is analyzed. Variations of the ISCS and their physical bounds are numerically investigated.

About Liquid Filtration Under a Point Dam with a Vertical Weakly Permeable Film

The problem of groundwater filtration under a point dam in a piecewise homogeneous porous medium in the presence of a weakly permeable film under the dam is considered. The filtration area is considered in the form of a vertical half-plane with a horizontal line of water courses. A weakly permeable film divides the filtration area into two quadrants with different constant permeability. By the convolution method of Fourier expansions, the solution of the problem is obtained explicitly. The influence of a weakly permeable film on the filtration process is investigated. It is shown that the presence of a weakly permeable film reduces the filtration rates in the downstream.

The reciprocity gap operator for electromagnetic scattering in chiral media

We define a reciprocity gap functional for electromagnetic scattering in chiral media. We assume that a perfectly conducting scatterer is embedded in a piecewise homogeneous isotropic background chiral medium. The incident field is a spherical electromagnetic wave due to a point source located at a point contained in a chiral environment. We also define the chiral reciprocity gap operator and we prove that it is injective and it has dense range. Using these results and appropriate density properties of chiral Herglotz functions we solve an inverse scattering problem for the determination of the boundary of the scatterer from the knowledge of the tangential components of the electric and magnetic fields.

Mathematical modeling of fluid flows through the piecewise homogeneous porous medium by R-function method

The stationary fluid flow through a piecewise homogeneous porous medium is considered under the assumption that Darcy's law holds. The mathematical model of this problem is defined as an elliptic equation for the stream function, supplemented by the second-type boundary conditions at the water boundaries and the first-type boundary conditions at the impervious to liquid boundaries. The problem statement also includes the conditions of conjugation at the separation line between two soils and the unknown value of fluid discharge, which can be established from the additional integral ratio. It is proposed to use the structure-variational method of R-functions in order to numerically analyze and solve the current problem. The complete solution structure for the boundary value problem of stream function regarding the R-functions method is established, moreover, the application of the Ritz method for approximating an unspecified structural formula component is substantiated. Then, the approximate value of the fluid discharge and the approximate solution of the original problem are found from the additional integral ratio. The computational experiment was carried out with different coefficients of permeability within the area, which has the shape of the lower half ring. It is established that as the number of coordinate functions increases, the value of fluid discharge becomes constant, indicating the convergence of the proposed method.

On a Method in Dynamic Elasticity Problems for Heterogeneous Wedge-Shaped Medium

The method of analysis of steady oscillations arising in the piecewise homogeneous wedge-shaped medium composed by two homogeneous elastic wedges with different mechanical and geometric characteristics is presented. Method is based on the distributions’ integral transform technique and allows reconstructing the wave field in the whole medium by displacement oscillations given in the domain on the boundary of the medium. The problem in question is reduced to a boundary integral equation (BIA). Solvability problems of the BIA are examined and the structure of its solution is established.

Study of transmission characteristics of disk transducer in the radially propagated Rayleigh wave excitation mode

A mathematical model of a capacitive ultrasonic transducer is designed allowing to emit ultrasonic vibrations into an electrically conductive product. The influence of a polarizing electrostatic field on the Coulomb forces formation in the surface layer of a metal sample is determined. A closed solution to the electrostatics problem is obtained for a piecewise homogeneous medium in which a half-space is filled with metal having finite values of electrical conductivity and magnetic permeability. An expression is obtained for calculating the surface density of a static electric charge on the metal sample surface. As part of the mathematical model of a capacitive sensor in the mode of converting electric energy to high-frequency mechanical (ultrasonic) energy in metals, closed solutions for electrostatics and electrodynamics problems are constructed in relation to a piecewise-homogeneous medium in which a half-space is filled with metal having finite values of electrical conductivity and magnetic permeability. It is determined that a capacitive disk transducer excites forces acting normally to the surface of electrically conductive products. A quantitative assessment of the surface density of the Coulomb forces is made. The main factors determining the sensitivity of a capacitive disk transducer are specified. As an example of using the simulation results, the amplitude factor of radially propagating Rayleigh waves is calculated. The concept of the wave characteristics of the transducer in the excitation mode of ultrasonic surface waves is introduced.

Space–time discontinuous Galerkin approximation of acoustic waves with point singularities

AbstractWe develop a convergence theory of space–time discretizations for the linear, second-order wave equation in polygonal domains $\varOmega \subset{\mathbb R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space–time DG formulation developed in Moiola & Perugia (2018, A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation. Numer. Math., 138, 389–435), we (a) prove optimal convergence rates for the space–time scheme with local isotropic corner mesh refinement on the spatial domain, and (b) demonstrate numerically optimal convergence rates of a suitable sparse space–time version of the DG scheme. The latter scheme is based on the so-called combination formula, in conjunction with a family of anisotropic space–time DG discretizations. It results in optimal-order convergent schemes, also in domains with corners, with a number of degrees of freedom that scales essentially like the DG solution of one stationary elliptic problem in $\varOmega $ on the finest spatial grid. Numerical experiments for both smooth and singular solutions support convergence rate optimality on spatially refined meshes of the full and sparse space–time DG schemes.

NVESTIGATION OF TRANSMISSION CHARACTERISTICS OF DISC CAPACITY CONVERTER IN THE MODE OF EXCITATION OF ULTRASOUND RELAY WAVES

A mathematical model of a capacitive ultrasonic transducer is designed allowing to emit ultrasonic vibrations into an electrically conductive product. The influence of a polarizing electrostatic field on the Coulomb forces formation in the surface layer of a metal sample is determined. A closed solution to the electrostatics problem is obtained for a piecewise homogeneous medium in which a half-space is filled with metal having finite values of electrical conductivity and magnetic permeability. An expression is obtained for calculating the surface density of a static electric charge on the metal sample surface. The main influencing factors that determine the spatial distribution of the charge density on the product surface, which will form the radiation pattern and power of the emitted ultrasound field, are defined. As part of the mathematical model of a capacitive sensor in the mode of converting electric energy to high-frequency mechanical (ultrasonic) energy in metals, closed solutions for electrostatics and electrodynamics problems are constructed in relation to a piecewise-homogeneous medium in which a half-space is filled with metal having finite values of electrical conductivity and magnetic permeability. It is determined that a capacitive disk transducer excites forces acting normally to the surface of electrically conductive products. A quantitative assessment of the surface density of the Coulomb forces is made. The main factors determining the sensitivity of a capacitive disk transducer are specified.

Method for Calculating the Near-Surface Effect in Piecewise Homogeneous Bodies at Large Deformations Based on a Two-Level Approach

A method is proposed for calculating the near-surface effect in piecewise homogeneous bodies under large deformations based on a combined use of one-level applied and two-level carcass theories. The applied theory is used for a macromechanical continuation of the solution of the problem, but the carcass theory is used in the final part of the loading path or directly at the final loading of the body. The implementation of the problem by the carcass theory for the body as a whole ends with solution of extreme problems using a highly gradient scheme for assemblies of structural blocks near boundary surfaces of the body. The rotation-caused development of configurations of cylinders reinforced with ring fibers is studied using this method and the model of a piecewise homogeneous medium. The results obtained by the carcass theory and the model of a piecewise homogeneous medium differed only slightly, confirming the high accuracy of the analysis using the two-level approach. A higher stability of the numerical implementation of the carcass theory, in comparison with the model of a piecewise homogeneous medium, was revealed when the macromechanical continuation was carried out within the framework of this theory itself. An even higher stability of the approach was reached by implementation of the macromechanical continuation of the applied theory. Performing calculations only for nodal blocks of the medium at the micromechanical level of the carcass theory, together with application of the applied theory for the macromechanical continuation, led to an extremely high efficiency of the two-level approach. In contrast to the piecewise homogeneous medium model, this approach made it possible to study the behavior of piecewise homogeneous cylinders considering the near-surface effect up to the configurationdetermined ultimate rotation speed.

Modeling of magnetic field geometry for high precision executive devices

The paper presents the theoretical foundations of constructing a mathematical model for modeling the geometry of the magnetic field in the working area of special devices with complex geometry of the magnetic system.It is proposed to use the method of secondary sources as a method for solving the problem, in the presence of a number of different homogeneous and isotropic media in space. This allows us to divide the solution of the problem of finding the field distribution geometry into two stages: at the first stage, we find the distribution of secondary sources, the action of which is equivalent to the influence of an inhomogeneous medium, and then the total field from the primary and secondary sources is calculated.Thanks to this approach, for a piecewise-homogeneous medium, the problem of calculating the field pattern can be formulated in the form of a system of linear Fredholm integral equations of the second kind.To solve the integral equation, a transition was made to a finite system of linear algebraic equations. In this case, the geometry features of the magnetic system of the structure (boundary conditions) were taken into account. The solution of the system of algebraic equations was obtained by several methods to evaluate the simulation results.Based on the proposed method, a mathematical model was obtained and a software package was developed that implements a numerical algorithm for calculating the magnetic field geometry in the working area of a special device.

The Degenerate Scales for Plane Elasticity Problems in Piecewise Homogeneous Media Under General Boundary Conditions

The degenerate scale issue for D boundary integral equations and boundary element methods has been already investigated for Laplace equation, antiplane and plane elasticity, bending plate for Dirichlet boundary condition. Recently, the problem of Robin and mixed boundary conditions and of piecewise heterogeneous domains have been considered for the case of Laplace equation. We investigate similar questions for plane elasticity for more general boundary conditions. For interior problems, it is shown that the degenerate scales do not depend on the boundary condition. For exterior problems, the two degenerate scales (homogeneous medium) or two of them (heterogeneous medium) are tightly linked with the behavior at infinity of the solutions. The dependence of this behavior on the boundary conditions is investigated. We give sufficient conditions for the uniqueness of the solution. Numerical applications are provided and validate the set of theoretical results.

Solution of Maxwell’s Equations for Cylindrical Symmetry Waveguides

The solution of Maxwell’s equations for a piecewise homogeneous medium of cylindrical symmetry has been obtained. The parameters of the cylindrical waveguide modes have been calculated on its basis. The conclusions are confirmed by numerical calculation of the first four modes of a hollow metal waveguide operating as a mode convector.