Interior Boundary Value Problems Research Articles

Time-Fractional Cattaneo-Type Thermoelastic Interior-Boundary Value Problem Within A Rigid Ball

The paper discusses the solution of an interior-boundary value problem of one-dimensional time-fractional Cattaneo-type heat conduction and its stress fields for a rigid ball. The interior value problem describes the dependence of the boundary conditions within the ball's inner plane at any instant with a prescribed temperature state, in contrast to the exterior value problem, which relates the known surface temperature to boundary conditions. A single-phase-lag equation with Caputo fractional derivatives is proposed to model the heat equation in a medium subjected to time-dependent physical boundary conditions. The application of the finite spherical Hankel and Laplace transform technique to heat conduction is discussed. The influence of the fractional-order parameter and the relaxation time is examined on the temperature fields and their related stresses. The findings show that the slower the thermal wave, the bigger the fractional-order setting, and the higher the period of relaxation, the slower the heat flux propagates.

Direct boundary method toolbox for some elliptic problems in FreeHyTE framework

FreeHyTE Direct Boundary Method Toolbox is a new computational framework for the solution of interior and exterior boundary value problems in two dimensions using three classes of direct methods: the Boundary Element Method, the Method of Fundamental Solutions and the Trefftz-Herrera Method. The toolbox, currently including solvers for Laplace and Helmholtz boundary value problems, is straightforward to use, featuring a simple graphical user interface and automatic mesh generators, and amenable to extension, as it provides modular computational procedures, directly applicable to other types of boundary elements and differential equations. The toolbox supports the definition of simply or multiply-connected domains, boundary elements of any order, complex wavenumbers, and Dirichlet, Neumann and Robin boundary conditions. FreeHyTE Direct Boundary Method Toolbox is freely distributed under the GNU General Public License and supported by manuals to quickly get new users started.

Green's function of an interior boundary-value problem for a fractional ordinary differential equation with constant coefficients

Green's function of an interior boundary-value problem for a fractional ordinary differential equation with constant coefficients

The Neumann problem in plane deformations of a micropolar elastic solid with micropolar surface effects

We consider the Neumann problem in a theory of plane micropolar elasticity incorporating micropolar surface effects. The incorporation of surface elasticity utilizes the Eremeyev–Lebedev–Altenbach shell model, leading to a set of second-order boundary conditions describing the separate micropolar elasticity of the surface. The Neumann problem is of particular interest, since the question of solvability is complicated by the fact that the corresponding systems of homogeneous singular integral equations admit nontrivial solutions that affect the solvability of both the interior and exterior Neumann boundary value problems. We overcome this difficulty by constructing integral representations of the solutions based on specifically constructed auxiliary matrix functions leading to uniqueness and existence theorems in appropriate classes of smooth matrix functions.

On Solving Second-Order Linear Elliptic Equations

A method is presented for solving interior boundary-value problems for second-order elliptic equations by transition to ray variables. The domain is divided into cells within which the coefficients and sources have the smoothness and continuity properties necessary for the existence of a regular classical solution in the cell. The finite discontinuities of the coefficients (if any) are located on the cell boundaries. The regular solution in the cell is sought in the form of a superposition of the contributions made by volume and boundary sources placed on the rays arriving at the given point from the cell boundaries. Next, a finite analytic scheme for the numerical solution of the boundary value problem in a domain with discontinuous coefficients and sources is constructed by matching the regular solutions emerging from cells at the cell boundaries. The scheme exhibits no hard dependence of the accuracy of approximation on the sizes and shape of the cells, which is inherent in finite-difference schemes.

Existence and integral representation of solutions for plane deformations of a micropolar elastic solid with surface elasticity

AbstractWe consider a linear theory of elastic boundary reinforcement of a micropolar elastic solid subjected to plane‐strain deformations. The reinforcement consists of a thin micropolar elastic coating bonded to part of the boundary of the solid. The elastic properties of the coating incorporate both classical and micropolar bending, extension and twisting effects. Interior and exterior mixed boundary problems are formulated and analyzed using the boundary integral equation method. The boundary value problems are reduced to systems of singular integro‐differential equations to which Noether‐type theorems are shown to apply. We consider also the corresponding boundary value problems based on an alternative lower‐order shell model of the reinforcement. Finally, existence and uniqueness results are presented for the corresponding interior and exterior boundary value problems in the appropriate classical function spaces.

Twofold Domain Decomposition Method for the Analysis of Multiscale Composite Structures

A nonconformal twofold domain decomposition method (TDDM) based on the hybrid finite element method-boundary element method (FEM-BEM) is proposed for analyzing 3-D multiscale composite structures. The proposed TDDM starts by partitioning the composite object into a closed exterior boundary domain and an interior volume domain. The interior and exterior boundary value problems are coupled to each other through the Robin transmission conditions (RTCs). Both domains are then independently decomposed into subregions to facilitate computation. Specifically, FEM-DDM with the second order transmission conditions (SOTCs) is employed for the interior domain, and BEM-discontinuous Galerkin (BEM-DG) based on the combined field integral equation (CFIE) is applied for the exterior boundary domain. The proposed TDDM allows for nonconformal discretization between any touching subdomains. Without the introduction of a stabilization term that relies on a line integral over intersection of nonmatching meshes and relevant terms involving surface-line integrals, the proposed TDDM provides an effective domain decomposition (DD) preconditioner for the global system. Numerical examples are presented, and the comparisons of the simulation results with FEM-BEM confirm the validity and accuracy of TDDM. Moreover, its ability to model practical large-scale and multiscale targets is also demonstrated.

Numerical test for the Neumann problem of interior BVP of plane elasticity

Following the concept of the patch test in FEM (finite element method), several numerical tests are suggested for the Neumann problem of interior BVP (boundary value problem) of plane elasticity by using different techniques. We choose the elliptic plate as an object in the test. Along the boundary of the plate, two groups of boundary traction are assumed. The first group is derived from the constant stress solution (or constant strain solution), and the second group is derived from the linear stress distribution. From displacement solution, we can evaluate the normal peripheral stress along the boundary, which is compared with the result from an exact solution. In addition, several error estimation factors are defined to evaluate the achieved accuracy. Extensive numerical examples and computed results are provided.

Uniqueness of solution for plane deformations of a micropolar elastic solid with surface effects

We present a linear theory describing plane-strain deformations of a micropolar elastic solid which incorporates the additional contribution of surface micropolar elasticity. This theory provides a more comprehensive account of size effects in solids allowing for a more accurate representation of, for example, the mechanical properties of micro-/nanostructures. The surface model is represented by a thin micropolar reinforcing elastic shell of separate micropolar elasticity, perfectly bonded to the micropolar bulk and capable of both extension and bending. We present two different versions of the surface shell model: one which gives rise to a fourth-order surface theory and another which, via a particular form of the Kirchhoff–Love kinematic assumption, results in a second-order surface model. In each case, we formulate interior and exterior mixed boundary value problems and show that they can have at most one smooth solution despite the presence of boundary conditions of order equal to or higher than that of the governing equations. This result is essential in establishing that the corresponding mixed boundary value problems are well-posed in certain classes of smooth matrix functions.

A New Non-Overlapping Domain Decomposition Method for a 3D Laplace Exterior Problem

We propose a method for solving three-dimensional boundary value problems for Laplace’s equation in an unbounded domain. It is based on non-overlapping decomposition of the exterior domain into two subdomains so that the initial problem is reduced to two subproblems, namely, exterior and interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To match the solutions on the interface between the subdomains (the sphere), we introduce a special operator equation approximated by a system of linear algebraic equations. This system is solved by iterative methods in Krylov subspaces. The performance of the method is illustrated by solving model problems.

A method of boundary equations for unsteady hyperbolic problems in 3D

We consider interior and exterior initial boundary value problems for the three-dimensional wave (d'Alembert) equation. First, we reduce a given problem to an equivalent operator equation with respect to unknown sources defined only at the boundary of the original domain. In doing so, the Huygens' principle enables us to obtain the operator equation in a form that involves only finite and non-increasing pre-history of the solution in time. Next, we discretize the resulting boundary equation and solve it efficiently by the method of difference potentials (MDP). The overall numerical algorithm handles boundaries of general shape using regular structured grids with no deterioration of accuracy. For long simulation times it offers sub-linear complexity with respect to the grid dimension, i.e., is asymptotically cheaper than the cost of a typical explicit scheme. In addition, our algorithm allows one to share the computational cost between multiple similar problems. On multi-processor (multi-core) platforms, it benefits from what can be considered an effective parallelization in time.

Scattering problems of the SH Wave by Using the Null-Field Boundary Integral Equation Method

The scattering problem of seismic waves is an important issue for studying earthquake engineering. In this paper, the null-field boundary integral equation approach was used in conjunction with degenerate kernels and eigenfunction expansion to solve the SH-wave scattering problem of a circular or an elliptical-arc hill. The original problem is divided into subdomains by taking a free-body diagram. One region is an interior boundary value problem. The other is a canyon scattering problem. For the boundary value problem, not only a simply connected domain (elliptical-arc hill problem) but also a doubly connected domain (a circular-arc hill problem containing a circular tunnel or a circular inclusion) is considered. The canyon scattering problem may be addressed in an infinite domain with an artificial boundary of a full plane such that the degenerate kernel can be fully utilized. The null-field integral equation method is used to match boundary conditions. Numerical results are compared favorably with the available data.

Amended influence matrix method for removal of rigid motion in the interior BVP for plane elasticity

A conventional complex variable boundary integral equation (CVBIE) in plane elasticity is provided. After using the Somigliana identity between a particular fundamental stress field and a physical stress field, an additional integral equality is obtained. By adding both sides of this integral equality to both sides of the conventional CVBIE, the amended boundary integral equation (BIE) is obtained. The method based on the discretization of the amended BIE is called the amended influence matrix method. With this method, for the Neumann boundary value problem (BVP) of an interior region, a unique solution for the displacement can be obtained. Several numerical examples are provided to prove the efficiency of the suggested method.

On the removal of rigid body motions in the interior BVP of plane elasticity by using node support in conjunction with least square technique

This paper provides an effective technique for removing the rigid body motions in the interior traction BVP (boundary value problem) of plane elasticity. In the technique, we assume three additional node support conditions. After substituting those conditions in the algebraic equations derived from discretization of BIE (boundary integral equation), we will obtain the algebraic equations where the number of the equations is larger than the number of unknowns. The mentioned algebraic equations can be solved by using the least square technique. Finally, the displacement vector can be evaluated. A stress component derived from the suggested technique is compared to the exact one. Excellent accuracy has been found in the suggested technique.

An inverse scattering problem with generalized oblique derivative boundary condition

Consider the scattering of long ocean tidal waves by an island taking into account the influence of daily rotation of the Earth, which is modeled by an exterior boundary value problem for the two-dimensional Helmholtz equation with generalized oblique derivative boundary condition. In this paper, we are concerned with a corresponding inverse scattering problem which is to reconstruct the unknown obstacle (island) from the far-field data. After proving the unique solvability of the direct scattering problem in a suitable function space required for our inverse scattering problem, we establish the linear sampling method (LSM) for reconstructing the boundary of the obstacle from the far-field data. To clarify the validity of such a sampling-type method which essentially depends on the solvability of an interior boundary value problem, we show that, except a discrete set of wave numbers, such an interior problem has a unique solution. Finally, some numerical examples are presented to demonstrate the efficiency of the reconstruction scheme.

Numerical solution of Eshelby's elastic inclusion problem in plane elasticity by using boundary integral equation

This paper provides a numerical solution for Eshelby's elastic inclusions in an plane elasticity based on the complex variable boundary integral equation (CVBIE) method. An inclusion with arbitrary shape is embedded in the infinite matrix. In the inclusion, the constant eigenstains are assumed. No remote loading is applied to the matrix. The displacements from the assumed eigenstrains are evaluated exactly, which in turn are the generalized loading in the problem. The proposed problem is reduced to solve interior and exterior boundary value problems simultaneously. For the elliptical inclusion case, the computed stresses along the interface of the inclusion side are nearly uniform. One more numerical example is devoted to a square-type inclusion with round corner.

A method for solving an exterior three-dimensional boundary value problem for the Laplace equation

We develop and experimentally study the algorithms for solving three-dimensionalmixed boundary value problems for the Laplace equation in unbounded domains. These algorithms are based on the combined use of the finite elementmethod and an integral representation of the solution in a homogeneous space. The proposed approach consists in the use of the Schwarz alternating method with consecutive solution of the interior and exterior boundary value problems in the intersecting subdomains on whose adjoining boundaries the iterated interface conditions are imposed. The convergence of the iterative method is proved. The convergence rate of the iterative process is studied analytically in the case when the subdomains are spherical layers with the known exact representations of all consecutive approximations. In this model case, the influence of the algorithm parameters on the method efficiency is analyzed. The approach under study is implemented for solving a problem with a sophisticated configuration of boundaries while using a high precision finite elementmethod to solve the interior boundary value problems. The convergence rate of the iterations and the achieved accuracy of the computations are illustrated with some numerical experiments.

A new approach to investigation of Maxwell equations in spherical coordinates

In this article general solution to the Maxwell equations in spherical coordinates is constructed. The method of expansion of unknown functions into series of spherical harmonics is used. It is shown that the exterior boundary value problems for the Maxwell equations have the unique oriented solutions, and the interior boundary value problems have non-trivial solutions in the case of resonance. Necessary and sufficient solvability conditions of over-determined boundary value problems for the Maxwell equations are found.

Inverse scattering approach with measurement of only total electric field

ABSTRACTAn inverse scattering approach using only total electric field data is developed for reconstructing electrical parameters of unknown scatterers. Two interior boundary value problems are set up whose solutions coincide with each other if and only if a medium inside the observation surface is the same as that of the original scattering problem. A cost functional of misfit between the two solutions is introduced to solve the inverse problem and its gradient is derived explicitly. Numerical examples are provided to assess the proposed inversion approach. © 2015 Wiley Periodicals, Inc. Microwave Opt Technol Lett 57:137–141, 2015

Properties of integral operators in complex variable boundary integral equation in plane elasticity

This paper investigates properties of integral operators in complex variable boundary integral equation in plane elasticity, which is derived from the Somigliana identity in the complex variable form. The generalized Sokhotski-Plemelj`s formulae are used to obtain the BIE in complex variable. The properties of some integral operators in the interior problem are studied in detail. The Neumann and Dirichlet problems are analyzed. The prior condition for solution is studied. The solvability of the formulated problems is addressed. Similar analysis is carried out for the exterior problem. It is found that the properties of some integral operators in the exterior boundary value problem (BVP) are quite different from their counterparts in the interior BVP.