Vector Boundary Value Problems Research Articles

On contact problems with a deformable punch and variable rheology

The paper presents for the first time one of the methods for studying and solving contact problems with a deformed stamp for those cases when there is a need to change the rheology of the stamp material. It is based on a new universal modeling method previously published by the authors, which is used in boundary value problems for systems of partial differential equations. With its help, solutions of complex vector boundary value problems for systems of differential equations can be decomposed into solutions of scalar boundary value problems for individual differential equations. Among them, the Helmholtz equations are the simplest. Solutions to scalar boundary value problems are represented as fractals, self-similar mathematical objects first introduced by the American mathematician B. Mandelbrot. The role of fractals is performed by packed block elements. The transition from systems of differential equations in partial derivatives to individual equations is carried out using the transformation of academician B. G. Galerkin or representation by potentials. It is known that the solutions of dynamic contact problems with a deformable stamp of complex rheology are cumbersome, and their study is always difficult. The problem is complicated by the presence of discrete resonant frequencies in such problems, which were once discovered by Academician I. I. Vorovich.

On the properties of topological discretization of solutions to boundary value problems

This paper seems to show for the first time that Packed block elements used in solving a boundary value problem by the block element method are elements of a discrete topological space. Since the solutions to boundary value problems belong to a discrete topological space instead of equations, it is possible to obtain a solution in a new coordinate system without having to study the boundary value problem. The block element method used in problems of continuum mechanics, which has a connection with topology, can become more effective in applications if the General properties of topological spaces studied in detail are used more deeply. One of the important properties of topology is the existence of discrete topological spaces. Their characteristic property is that an element that represents the Union of any set of elements in a topological space belongs to a discrete topological space. Belonging of discrete block elements to a topological space means that it is possible to completely cover any area with a piecewise smooth boundary and, thus, an exact solution of the boundary problem in it. In topological space, continuous geometric transformations and transitions to new coordinate systems are possible. In this paper, it is shown that Packed block elements generated by the boundary value problem for the Helmholtz equation, are elements of a discrete topological space. Given that scalar solutions of the Helmholtz equation can describe solutions to a fairly wide set of vector boundary value problems, this property also applies to solutions of more complex boundary value problems. The constructions made for the homogeneous Helmholtz equation remain valid for the inhomogeneous one.

Топологическая дискретизация решений граничных задач механики сплошных сред

This paper seems to show for the first time that Packed block elements used in solving a boundary value problem by the block element method are elements of a discrete topological space. Since the solutions to boundary value problems belong to a discrete topological space instead of equations, it is possible to obtain a solution in a new coordinate system without having to study the boundary value problem. The block element method used in problems of continuum mechanics, which has a connection with topology, can become more effective in applications if the General properties of topological spaces studied in detail are used more deeply. One of the important properties of topology is the existence of discrete topological spaces. Their characteristic property is that an element that represents the Union of any set of elements in a topological space belongs to a discrete topological space. Belonging of discrete block elements to a topological space means that it is possible to completely cover any area with a piecewise smooth boundary and, thus, an exact solution of the boundary problem in it. In topological space, continuous geometric transformations and transitions to new coordinate systems are possible. In this paper, it is shown that Packed block elements generated by the boundary value problem for the Helmholtz equation, are elements of a discrete topological space. Given that scalar solutions of the Helmholtz equation can describe solutions to a fairly wide set of vector boundary value problems, this property also applies to solutions of more complex boundary value problems.

Exact Solution of a Nonstationary Problem for the Elastic Layer with Rigid Cylindrical Inclusion

We present the exact solution of a nonstationary problem for the infinite elastic layer containing a rigid cylindrical inclusion with conditions of smooth contact imposed on the cylindrical surface. One surface of the layer is subjected to the action of an axisymmetric normal nonstationary compressive load. The other surface either is perfectly coupled with an absolutely rigid foundation or is supported by a smooth foundation without friction. To construct the fields of displacements and stresses in the layer, we successively apply the Laplace and Weber integral transformations to the axisymmetric equations of motion. This gives an inhomogeneous vector boundary-value problem for the unknown transforms of displacements. The problem is solved with the help of the matrix differential calculus. The normal stresses are analyzed on the cylindrical surface of the inclusion and on the bottom surface of the elastic layer. The obtained solution is studied in detail for the case of quasistatic vibration.

Application of the Block Element Method in a Certain Boundary Value Problem of Academician I.I. Vorovich

By the example of the boundary value problems—the vector problem (set of differential equations) and the scalar problem (single equation)—the features of their solutions by the block element method are demonstrated. In this method, the solutions can be presented in packed and unpacked forms. Here, these properties are explained in comparison by the example of two problems, the vector and scalar boundary value problems, solved for the first time exactly by the block element method. The vector boundary value problem was formulated previously by Academician I.I. Vorovich to a student, and its exact solution is given in this study.

Stability of mathematical models of the main problems of the anisotropic theory of elasticity

The boundary problems of the complex-variable function theory are effectively used while investigating equilibrium of homogeneous elastic mediums. The most complicated systems of the boundary value problems correspond to the case when an elastic body exhibits anisotropic properties. Anisotropy of the medium results in the drift of boundary conditions of the function that in general disrupts analyticity of the functions of interest. The paper studies systems of the boundary value problems with drift for analytic vectors corresponding to the primal elastic problems (first, second and mixed problems). Systems of analytic vectors with drift are reduced to equivalent systems of Hilbert boundary value problems for analytic functions with weak singularity integrators. The obtained general solution of the primal boundary value problems for the anisotropic theory of elasticity allows us to check the above problems for stability with respect to perturbations of boundary value conditions and contour shape. The research is relevant as there is necessity to apply approximate numerical methods to the boundary value problems with drift. The main research result comes to be a proof of stability of the systems of the vector boundary value problems with drift for analytic functions on the Hölder space corresponding to the primal problems of the elastic theory for anisotropic bodies in the case of change in the boundary value conditions and contour shape.

The existence, uniqueness and asymptotic estimates of solutions for third-order full nonlinear singularly perturbed vector boundary value problems

In this paper, we discuss third-order full nonlinear singularly perturbed vector boundary value problems. We first present the existence of solutions for the nonlinear vector boundary value problems without perturbation by using the upper and lower solutions method and topological degree theory. Then the existence, uniqueness and asymptotic estimates of solutions for the singularly perturbed vector boundary value problems are established by constructing appropriate a lower solution–upper solution pair, as well as analysis technique. Some known results are extended.

Investigation of the stress state of the elastic semi-strip with a transverse crack

The stress state of the semi-strip with a transverse crack is investigated. The mechanical load is applied to the semi-strip’s short edge. The initial problem is reduced to the one-dimensional vector boundary-value problem with the help of the semi-infinite Fourier transformation applied by the generalized scheme. The vector boundary-value problem is solved with the help of the matrix differential calculation apparatus and Green’s matrix-function apparatus. The solving of the problem is reduced to the solving of the system of three singular integral equations (SSIE). First singular equation contains two fixed singularities, so the special method is used for the SSIE’s solving. Stress intensity factors (SIF) are calculated for the different crack’s length.

Reduction of Vector Boundary Value Problems on Riemann Surfaces to One-Dimensional Problems

This article lays foundations for the theory of vector conjugation boundary value problems on a compact Riemann surface of arbitrary positive genus. The main constructions of the classical theory of vector boundary value problems on the plane are carried over to Riemann surfaces: reduction of the problem to a system of integral equations on a contour, the concepts of companion and adjoint problems, as well as their connection with the original problem, the construction of a meromorphic matrix solution. We show that each vector conjugation boundary value problem reduces to a problem with a triangular coefficient matrix, which in fact reduces the problem to a succession of one-dimensional problems. This reduction to the well-understood one-dimensional problems opens up a path towards a complete construction of the general solution of vector boundary value problems on Riemann surfaces.

Reduction of vector boundary value problems on Riemann surfaces to one-dimensional problems

Reduction of vector boundary value problems on Riemann surfaces to one-dimensional problems

The contact problem for an elastic semi-infinite layer

It is solved the contact problem on the action of a circular stamp with an extra-central power on the elastic semi-infinite layer. On the lateral and lower side of the layer the conditions of the smooth contact are given. The solution of the elastic problem for the semi-infinite layer when concentrated force was situated on the upper side of the layer was used. This problem was solved by the new method which is based on the reduction of Lame’s equations system to two together and one independent solved equations relatively to the new unknown functions associated with the displacements. Using of the integral transformations reduces the problem to a one-dimensional vector boundary value problem that to be solved exactly by the apparatus of the matrix differential calculus. The singular integral equation with respect to unknown contact pressure was obtained. The problem was reduced to the singular integral equation which was solved by the method of the orthogonal polynomials and was reduced to the infinite two-dimensional system of algebraic equations. These equations were solved by the reduction method. As a result the eccentricity and precipitation of the stamp that provide its forward movement were determined.

Elastic crack-tip stress field in a semi-strip

In this article the plain elasticity problem for a semi-strip with a transverse crack is investigated in the different cases of the boundary conditions at the semi-strip’s end. Unlike many works dedicated to this subject, the fixed singularities in the singular integral equation’s kernel are considered. The integral transformations’ method is applied by the generalized scheme to reduce the initial problem to a one-dimensional problem. The one-dimensional problem is formulated as the vector boundary value problem which is solved with the help of matrix differential calculations and Green’s matrix apparatus. The solution of the problem is reduced to the solving of the system of three singular integral equations. Depending on the conditions given on the short edge of the semi-strip, the constructed singular integral equation can have one, or two fixed singularities. A special method is applied to solve this equation in regard of the singularities existence. Hence the system of the singular integral equations (SSIE) is solved with the help of the generalized method. The stress intensity factors (SIF) are investigated for different lengths of crack. The novelty of this work is in the application of new approach allowing the consideration of the fixed singularities in the problem about a transverse crack in the elastic semi-strip. The comparison of the numerical results’ accuracy during the usage of the different approaches to the solving of SSIE is worked out.

Topological structure of solution sets to asymptotic $n$-th order vector boundary value problems

Topological structure of solution sets to asymptotic $n$-th order vector boundary value problems

Two-Dimensional Mixed Problem of Thermoelasticity for a Semistrip

We study the stress state of a semistrip with conditions of smooth contact imposed on one of its lateral surfaces and coupling conditions given on the other surface under the action of normal loading and temperature on the end face. The original problem is reduced to a one-dimensional vector boundary-value problem by the method of integral transformations. By using the tools of matrix differential calculus and Green matrix function, we reduce the obtained problem to a singular integral equation for the derivative of displacements on the end face of the semistrip and solve this equation by the method of orthogonal polynomials. We perform the numerical analysis of fields of displacements and stresses inside the semistrip and establish the zones of tensile stresses on the lateral face of the semistrip and the conditions of their formation.

Analysis of the properties of slow-wave structures based on coaxial resonators coupled by two arc-shaped waveguides

The dispersion and the coupling resistance of spatial harmonics of slow-wave structures formed by a chain of coaxial resonators coupled by arc-shaped waveguides are theoretically analyzed. A 3D vector boundary value problem is solved in the rigorous electromagnetic formulation with the use of projection methods. The 2D problems of excitation of a coaxial resonator are considered and amplitudes of excited waves are determined and used for solution of the 3D problem of the junction of arc-shaped waveguides and a coaxial resonator. The dispersion equation for the considered slow-wave structure is obtained. The results of calculation of the dispersion and the coupling resistance for different numbers and different configurations of coupling apertures are analyzed.

Exact analytical solution for a pie-shaped wedge thick plate under oscillating load

A finite elastic wedge-shaped thick plate is considered. One of the faces is based on a rigid base, and the other is exposed to a dynamic oscillating load through an absolutely rigid overlay. On the side faces, conditions of sliding contact are fulfilled; at its end, the stresses are given. The solution is based on a special linear transformation of Lame’s equations and application of the vector integral transformations method. The proposed approach leads to a one-dimensional vector boundary value problem for which an exact solution is constructed. The analysis of the eigenfrequency value distribution and an estimation of the edge resonance frequency are done. An analogous problem is solved for the case when at the lower wedge plate’s face the sliding conditions are given. In order to establish the possibility of separation of the lower base, a comparison of the obtained values of the stress on the bottom plate with the stresses arising in the analogous formulation of the static problem for wedge plate considering its own weight is worked out.

Axisymmetric Problem of Stressed State for a Twice Truncated Cone

We consider an axisymmetric mixed problem of stressed state for a twice-truncated cone with regard for its own weight and the conditions of smooth contact imposed on its conical surface. The application of a new integral transformation with respect to the meridian angle directly to the Lame equations reduces the problem in the space oftransforms to a one-dimensional vector boundary-value problem. The obtained problem is solved exactly with the help of the methods of matrix differential calculus. The subsequent application of the inverse integral transformations gives the final solution of the original problem. We study the solutions in special cases of the formulated problem, such as a cone with spike, a spherical dome, and a half ball. We also determine the normal stresses acting on the surface of the cone depending on its geometric parameters.

Error Estimate of Eigenvalues of Perturbed Higher-Order Discrete Vector Boundary Value Problems

This paper is concerned with the eigenvalues of perturbed higher-order discrete vector boundary value problems. A suitable admissible function space is first introduced, a new variational formula of eigenvalues is then established under certain nonsingularity conditions, and error estimates of eigenvalues of problems with small perturbation are finally given by using the variational formula. As a direct consequence, continuous dependence of eigenvalues on boundary value problems is obtained under the nonsingularity conditions. In addition, two special perturbed cases are discussed.

The axisymmetric mixed problem of elasticity theory for a cone clamped along its side surface with an attached spherical segment

The axisymmetric mixed problem of the stress state of an elastic cone, with a spherical segment attached to the base, is considered. The side surface of the cone is rigidly clamped, while the surface of the spherical segment is under a load. By using a new integral transformation over the meridial angle the problem is reduced in transformant space to a vector boundary value problem, the solution of which is constructed using the solution of a matrix boundary value problem. The unknown function (the derivative of the displacements), which occurs in the solution, is determined from the approximate solution of a singular integral equation, for which a preliminary investigation is carried out of the nature of the singularity of the function at the ends of the integration interval. Subsequent use of inverse integral transformations leads to the final solution of the initial problem. The values of the stresses obtained are compared with those that arise in the cone for a similar load, when sliding clamping conditions are specified on the side surface of the cone (for this case an exact solution of this problem is constructed, based on the known result).

Electromagnetic fields in the presence of an infinite dielectric wedge: the phased line source excitation case

Electromagnetic fields, excited by an electric phased line source in the presence of an infinite dielectric wedge, are determined by application of the Kontorovich-Lebedev transform. The Maxwell's equations together with the conditions of continuity of the tangential field components at the material interfaces are formulated as a vector boundary-value problem. By representing the field components as Kontorovich-Lebedev integrals, the problem is reduced to a system of singular integral equations for the unknown spectral functions. We construct numerical solutions to those equations that permit fields evaluation for values of the wedge refractive index, not necessarily close to unity, and for arbitrary positioned source and observer. Numerical results showing the influence of a wedge presence on the directivity of a phased line source are presented and verified through finite-difference frequency-domain simulations.