External Boundary Value Problems Research Articles

Mathematical Problems in the Theory of Bone Poroelasticity

This paper concerns with the quasi-static theory of bone poroelasticity for materials with double porosity. The system of equations of this theory based on the equilibrium equations, conservation of fluid mass, the effective stress concept and Darcy’s law for material with double porosity. The internal and external basic boundary value problems (BVPs) are formulated and uniqueness of regular (classical) solutions are proved. The single-layer and double-layer potentials are constructed and their basic properties are established. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral method) and the theory of singular integral equations.

Plane Waves and Boundary Value Problems in the Theory of Elasticity for Solids with Double Porosity

This paper concerns with the dynamical theory of elasticity for solids with double porosity. This theory unifies the earlier proposed quasi-static model of Aifantis of consolidation with double porosity. The basic properties of plane waves are established. The radiation conditions of regular vectors are given. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness theorems are proved. The basic properties of elastopotentials are given. The existence of regular (classical) solution of the external BVP by means of the potential method (boundary integral method) and the theory of singular integral equations are proved.

Boundary value problems of steady vibrations in the theory of thermoelasticity with microtemperatures

AbstractIn this paper the linear theory of steady vibrations of thermoelasticity with microtemperatures for isotropic solids with microstructure is considered. The uniqueness and existence theorems of solutions of the internal and external second boundary value problems (BVPs) by means of the boundary integral method (potential method) and the theory of singular integral equations are proved. The existence of eigenfrequencies of the internal homogeneous BVP of steady vibrations is studied. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamical problems of the theory of elasticity for solids with double porosity

AbstractIn this paper the dynamical theory of elasticity for solids with double porosity is presented. The single‐layer and double‐layer potentials are constructed and basic properties are established. The uniqueness theorems of the internal and external boundary value problems (BVPs) of steady vibrations are proved. The existence theorems of classical solution of the external BVPs by means of the boundary integral method and the theory of multidimensional singular integral equations are proved. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Boundary value problems of the theory of viscoelasticity for Kelvin‐Voight materials

AbstractIn this paper, the classical Kelvin‐Voight model of the linear theory of viscoelasticity is considered and the following results are obtained: the fundamental solution of the equation of steady vibrations is constructed, the basic properties of plane waves and elastopotentials are established, the uniqueness theorem of the internal and external boundary value problems (BVPs) are proved, the existence theorems for classical solutions of the BVPs by means of the potential method and the theory of two‐dimensional singular integral equations are proved. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Basic Theorems in the Equilibrium Theory of Thermoelasticity with Microtemperatures

In this paper we consider the linear equilibrium theory of thermoelasticity with microtemperatures and some basic results of the classical theories of elasticity and thermoelasticity are generalized. The Green's formulae in the considered theory are obtained. The uniqueness theorems of the internal and external basic boundary value problems (BVPs) are proved. The representation of Galerkin type solution is obtained and the completeness of this solution is established. The formulae of integral representations of regular vector and regular (classical) solutions are obtained. The basic properties of thermoelastopotentials and singular integral operators are presented. Finally, the existence theorems for the internal and external basic BVPs are proved by means of the potential method and the theory of singular integral equations.

Potential Method in the Linear Theory of Thermoelasticity with Microtemperatures

In the present paper the linear theory of thermoelasticity with microtemperatures is considered. The basic boundary value problems of steady vibrations are investigated using the potential method. Sommerfeld–Kupradze type radiation conditions and the basic properties of thermoelastopotentials are established. The uniqueness and existence theorems of classical solutions of the external (with respect to an unbounded domain) boundary value problems are proved.

General inverse of Stokes, Vening-Meinesz and Molodensky formulae

The operator operations between the disturbing potential and the geoidal undulation, the gravity anomaly, the deflection of the vertical are defined based on the relations among the gravity potential, the normal gravity potential and the disturbing potential. With the sphere as the boundary surface, based on the solution of the external boundary value problem for the disturbing potential by the spherical harmonics in the physical geodesy, the general inverse Stokes’ formula, the general inverse Vening-Meinesz formula and the general Molodensky’s formula are derived from the operator operations defined. The general formulae can get rid of the restriction of the classical formulae only used on the geoid. If the boundary surface is defined as the geoid, the general formulas are degen-erated into the classic ones.

A method for constructing high-order differential boundary conditions for solving external boundary value problems in geoelectromagnetism

A general approach to the construction of differential boundary conditions for vector fields satisfying the Helmholtz equation is proposed on the basis of the field expansion in multipole series and the application of annihilating operators to them. The resulting differential constraints can be used as boundary conditions in solving external boundary value problems. Examples of their application to the solution of forward geoelectric problems in three-dimensionally inhomogeneous media are examined. Their use at a finite distance from the source of an anomaly is shown to yield more accurate results than those obtained under the assumption that the anomalous field at this distance vanishes. Another effect of their application is a substantial decrease in the dimensions of the modeling domain and therefore in the time required to solve the forward problem. The “safe” distance for using the Dirichlet-type boundary conditions is estimated.

Complex Variable Method for Thermal Stress Problem

ABSTRACT The two-dimensional thermal stress problem is one of the important problems in the field of solid mechanics. In this paper, the complex variable method for solving this kind of problem is stated. The basic equations of complex variable functions in conjunction with a rational mapping function are considered. The formulation of the rational mapping function and its accuracy are indicated. The general solutions for the external force, displacement, and mixed boundary value problems under uniform heat flux and under a point heat source are stated. The Green's functions for a point heat source and sink are suggested for the three kinds of boundary value problems. The equation to obtain the stress intensity factor for the crack problems and the stress intensity of debonding for a partially debonded rigid inclusion are shown. The interaction problem between a hole and a line crack under uniform heat flux is analyzed.

Potential method in the linear theory of swelling porous elastic soils

This paper deals with the isothermal linear theory of swelling porous elastic soils in the case of fluid saturation. Internal and external boundary value problems of steady vibrations are investigated using the potential method. The uniqueness and existence theorems of classical solutions of the aforementioned problems are proved.

Construction of basis functions and their application to boundary-value problems of mechanics of continuous media

A unified method for constructing basis (eigen) functions is proposed to solve problems of mechanics of continuous media, problems of cubature and quadrature, and problems of approximation of hypersurfaces. Numerical-analytical methods are described, which allow obtaining approximate solutions of internal and external boundary-value problems of mechanics of continuous media of a certain class (both linear and nonlinear). The method is based on decomposition of the sought solutions of the considered partial differential equations into series in basis functions. An algorithm is presented for linearization of partial differential equations and reduction of nonlinear boundary-value problems, which are reduced to systems of linear algebraic equations with respect to unknown coefficients without using traditional methods of linearization.

GREEN'S FUNCTION FOR A THERMOMECHANICAL MIXED BOUNDARY VALUE PROBLEM OF AN INFINITE PLANE WITH AN ELLIPTIC HOLE

This article derives the Green's function for a thermomechanical mixed boundary value problem of an infinite plane with an elliptic hole under a pair of heat source and sink. To derive the Green's function in closed form, the Cauchy integral method and a basic Green's function for an external force boundary value problem with a pair of heat source and sink are employed. Illustrative numerical results for temperature, heat flux, and stress along the hole edge and stress intensity factors when the hole collapses into a crack are presented graphically.

POTENTIAL METHOD IN THE LINEAR THEORY OF BINARY MIXTURES OF THERMOELASTIC SOLIDS

This article deals with the diffusion model of the linear theory of binary mixtures of the thermoelastic solids. A wide class of external (with respect to an unbounded domain) boundary value problems of steady oscillations is investigated using the potential method. Sommerfeld-Kupradze-type radiation conditions are established. The uniqueness and existence theorems of classical solutions of the aforementioned problems are proved.

Combination of the finite and the boundary element methods for an external boundary value problem of the Poisson equation

A numerical algorithm for an external Dirichlet problem of the Poisson equation is considered. The domain O extending to infinity is divided into a bounded subdomain O0 and the unbounded subdomain O1. The finite and the boundary element methods are applied to the boundary value problems in the bounded and the unbounded subdomains, respectively. An iterative scheme using the Dirichlet‐Neumann map on the interface ?O1 is presented. The convergence of the scheme is mathematically guaranteed. A simple numerical example shows the effectiveness of our scheme.

Green’s Function for a Heat Source in an Infinite Region With an Arbitrary Shaped Hole

In two-dimensional thermoelasticity, Green’s functions of the external force boundary value problem are derived for an infinite plane with an arbitrary shaped hole under adiabatic and isothermal boundary conditions subjected to heat sources in two cases as follows. One is the case of a heat source and a heat sink arbitrarily located within the plane, the other is the case of a heat source arbitrarily located within the plane and a heat sink at infinity. Complex stress functions, temperature function, a rational mapping function, and the thermal dislocation method are used for the analysis. In analytical examples, distributions of temperature, heat flux, and stresses are shown for an infinite plane with a rectangular hole under adiabatic and isothermal boundary conditions.

Complex velocity potential for an ellipse and a circle in translation and rotation

This paper investigates the analytical solutions of two-dimensional Neumann-type external boundary-value problem for an ellipse and a circle in complex analysis treatment. The transformation of harmonics between two planes is expanded in Laurent series. The circle theorem is extended so as to suit to the system of the ellipse and the circle. By means of Basset's process, the harmonic expressions of the system are found in the form of recurrence formulae which are suitable for numerical computations. The complex velocity potential of the system in translation and rotation is represented by the summation of the harmonic expressions.

Applied aspects of the theory of continuation of solutions of wave-scattering boundary-value problems

The article compares two approaches to continuation of solutions of external boundary-value problems for the Helmholtz equation outside the original region in the solution space — analytical continuation and the generalized image method. The comparison focuses on naturalness and ease of use of the mathematical model for the construction of numerical algorithms.

Finite abelian group methods in iterative algorithms for solving boundary-value problems with a boundary surface of complex structure

We consider external boundary-value problems for the Laplace equation on surfaces of complex structure. Various iterative computational schemes are constructed for numerical solution of the corresponding integral equations using set-theoretical group methods. Simulation results of electron-optical images are reported.

On the Uniqueness Theorems for the External Problems of the Couple-Stress Theory of Elasticity

Abstract A formula is obtained for the asymptotic representation of solutions of the basic equations of the couple-stress theory of elasticity. The formula is used in proving the uniqueness theorems of the external boundary value problems.