Fundamental Boundary Value Problems Research Articles

On divergence type linear and quasilinear equations in the complex plane

This is a survay of our recent results concerning divergence type linear and quasilinear equations in the complex plane. It contains a number of existence, representation and regularity theorems for solutions of fundamental boundary value problems for such equations. The degeneration case of uniform ellipticity is also covered by means of BMO, VMO, Calderon-Zigmund, Lehto and Orlicz technic.

The Effect of Hydrodynamics on the Power Efficiency of a Toroidal Oscillating Water Column Device

This study tries to identify the effect of hydrodynamics on the absorbed wave power using a toroidal Oscillating Water Column (OWC) device. To this end, the fundamental hydrodynamic boundary value problem for an arbitrary shaped toroidal OWC device of revolution with vertical axis is solved. The described method is based on the discretization of the flow field around the device by means of ring-shaped macro-elements, in each of which axisymmetric eigenfunction expansions for the velocity potential is made. The solution sought for the corresponding diffraction and radiation velocity potentials is achieved using Galerkin’s method. The applied formulation is generic and can be employed for arbitrary configurations of toroidal OWCs. Therefore, the numerical results shown and discussed in the present paper, in terms of the hydrodynamic loads and the air volume flows inside the OWC chamber, concern different types of OWCs. From the present analysis, it is demonstrated that the absorbed wave power by the examined toroidal devices is strongly affected by the geometrical parameters of the device; thus, these should be properly considered towards the maximization of the device’s wave power efficiency.

Topological sensitivity analysis for identification of voids under Navier’s boundary conditions in linear elasticity

This article is concerned with a geometric inverse problem related to the two-dimensional linear elasticity system. Thereby, voids under Navier’s boundary conditions are reconstructed from the knowledge of partially over-determined boundary data. The proposed approach is based on the so-called energy-like error functional combined with the topological sensitivity method. The topological derivative of the energy-like misfit functional is computed through the topological-shape sensitivity method. Firstly, the shape derivative of the corresponding misfit function is presented briefly from previous work Méjri (2018 J. Inverse Ill-Posed Problems). Then, an explicit solution of the fundamental boundary-value problem in the infinite plane with a circular hole is calculated by the Muskhelishvili formulae. Finally, the asymptotic expansion of the topological gradient is derived explicitly with respect to the nucleation of a void. Numerical tests are performed in order to point out the efficiency of the developed approach.

Solution of Periodic Boundary-Value Problems of the Spatial Theory of Elasticity in the Vector Form

We discuss boundary-value problems for the system of equations of the spatial theory of elasticity in the class of double-periodic functions and obtain a general solution of the system. We distinguish six types of elementary Floquet waves and examine their energy characteristics. We consider fundamental boundary-value problems in the half-space in the vector form. The diffraction problem for an elastic wave on a periodic system of defects in the vector form is reduced to the paired summator functional equation.

Water wave trapping in a long array of bottomless circular cylinders

This study tries to identify wave trapping situations by engaging and properly combining two well established phenomena: (i) the trapped modes induced by arrays of cylinders and (ii) the pumping trapped modes which are known to occur in moonpools. To this end, the fundamental hydrodynamic boundary value problem for arrays of bottomless cylinders was solved using standard domain decomposition. The method employed expansions of the solutions for the velocity potentials in polar harmonics combined with the eigenfunction expansions technique. The solution sought for the velocity potentials is achieved using the “direct” method of approach which accordingly requires the employment of a sophisticated matrix manipulation process.The elaboration of the concerned concept was motivated by three basic tasks: (i) to identify whether arrays of truncated and bottomless cylinders indeed preserve the occurrence of Neumann, Dirichlet and near trapped modes, extensively investigated for bottom-seated cylinders; (ii) to examine whether the expected pumping modes in moonpools modify the characteristics of the hydrodynamic resonance regimes (trapped modes) in the open liquid space between the cylinders and vice versa and (iii) to explore the possibility to suggest relevant configurations as parts of integrated mechanisms for practical applications, focusing a fortiori to clusters of hydrodynamically interacting Oscillating Water Columns (OWCs).The method developed is generic and can be employed for arbitrary configurations of multi-body arrays accommodating bottomless cylinders with uneven geometrical characteristics. Trapped modes are identified numerically as peaks in loading and this fact has been explicitly demonstrated in rows of cylinders. Therefore, the numerical results shown and discussed in the present are based on a specific in-line array that has been investigated in the past for bottom-seated cylinders. The investigated subject, i.e. whether the combined wave trapping induced by the examined configuration could be conceived as an efficient water wave power extraction mechanism is approached and discussed through dedicated computations of the free-surface displacements in the moonpools.

Method of deformation analysis for composite structures of soils and masonry stones

The coupled numerical manifold method (NMM) and discontinuous deformation analysis (DDA) are enhanced to simulate deformations of continuous soil and discontinuous masonry structures. An elasto-plastic NMM-DDA is formulated that incorporates elasto-plastic constitutive laws into incremental forms of the equation of motion. A node-based uniform strain element is applied to avoid volumetric locking, which often occurs in conventional NMM-DDA. The proposed method is applied to three fundamental boundary value problems: a beam bending problem, a bearing capacity problem of a footing, and a bearing capacity problem of a masonry structure. The method is verified through comparisons with conventional solutions.

A novel nonreflecting boundary condition for unsteady flow

SUMMARYA novel nonreflecting boundary condition, which converges to the specified time‐dependent boundary condition within any degree of accuracy, is introduced for the numerical simulation of hyperbolic systems and validated against the solution of two fundamental boundary value problems in fluids. First, transonic nozzle flow with backward acoustic disturbance is considered. Using high‐order aeroacoustic numerical schemes, the proposed nonreflecting boundary condition yields results that are in excellent agreement with those obtained using conventional nonreflecting boundary conditions based on the method of characteristics as well as with the results of the exact solution. The novel nonreflecting boundary condition, implemented into a semi‐analytical solution algorithm of unsteady bubbly cavitating nozzle flows, is also validated against results obtained using a Lagrangian finite volume scheme. Copyright © 2013 John Wiley & Sons, Ltd.

Modeling the effects of stress concentrations on the initiation of matrix cracking in woven CMCs

In this study, we employ detailed three-dimensional (3D) finite element models of plain and satin weave ceramic matrix composites (CMCs) as needed to establish the stress concentration around existing voids and their effect on the elastic response of these complex material systems. Fundamental 3D elasticity boundary value problems addressing the response of these materials under a combination of remote biaxial tension, in-plane shear, and thermal loading are utilized to characterize the matrix micro-stresses in the vicinity of large-scale macroscopic voids. The combined stress fields are then used to assess the conditions for the initiation of matrix cracking in such regions of high-stress concentration. Comparison of model results with available experimental data is discussed. Extensive parametric studies have yielded broad matrix cracking loci, which may be critical for the reliable use of woven CMCs in engineering applications.

Boundary algebraic equations for lattice problems

Procedures for constructing boundary integral equations equivalent to linear boundary-value problems governed by partial differential equations are well established. In this paper, it is demonstrated how these procedures can be extended to linear boundary-value problems defined on lattices and governed by algebraic (‘difference’) equations. The boundary equations that arise are then themselves algebraic equations. Such ‘boundary algebraic equations’ (BAEs) are derived for fundamental boundary-value problems defined on both perfect lattices and lattices with defects. It is demonstrated that key advantages of representing a continuum boundary-value problem as an equation on the boundary, such as favourable spectral properties and minimal problem size, are preserved in the lattice environment. Certain spectral properties of BAEs are established rigorously, whereas others are supported by numerical experiments.

Piezoelectric hollow cylinder with thermal gradient

The coupling nature of piezoelectric materials has acquired wide applications in electric-mechanical and electric devices. Recent advances in smart structures technology have led to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary value problems. In this paper, an analytic solution to the axisymmetric problem of a radially polarized, radially orthotropic piezoelectric hollow cylinder with thermal gradient is developed. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. The stress and potential field distributions in the cylinder are obtained numerically for two piezoceramics. It is shown that the hoop stresses in a cylinder composed of these materials can be decreased throughout the cross-section by applying an appropriate set of boundary conditions.

Stress in piezoelectric hollow sphere with thermal gradient

The piezoelectric phenomenon has been exploited in science and engineering for decades. Recent advances in smart structures technology have led to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary value problems. In this paper, we develop an analytic solution to the axisymmetric problem of a radially polarized, spherically isotropic piezoelectric hollow sphere. The sphere is subjected to uniform internal pressure, or uniform external pressure, or both and thermal gradient. There is a constant thermal difference between its inner and outer surfaces. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. Finally, the stress distributions in the sphere are obtained numerically for two piezoceramics.

Steady-state vibrations of an unbounded linear piezoelectric medium

We consider fundamental (Dirichlet and Neumann-type) boundary value problems in a theory of generalized plane strain for the steady-state vibrations of an infinite piezoelectric medium with transversely isotropic symmetry (6 mm). Using integral equation methods with the appropriate Sommerfeld-type radiation conditions, we prove existence and uniqueness results for the corresponding exterior boundary value problems. Exact solutions are obtained in the form of integral potentials.

Steady-state vibrations for the state of generalized plane strain in a linear piezoelectric medium

In this paper, steady-state vibrations are studied for the state of generalized plane strain in a linear piezoelectric medium. The fundamental boundary value problems are stated for vibrations in a prismatic piezoelectric body with arbitrary cross-section composed of material with the more general, tetragonal 4 ¯ , symmetry . Along with general theory, we present some other useful results concerning the behavior of the matrix of fundamental solutions. The conditions for the critical values of the field quantities on the boundary are derived.

Antiplane shear deformations in a linear theory of elasticity with microstructure

In this paper, we solve fundamental boundary value problems in a theory of antiplane elasticity which includes the effects of material microstructure. Using the real boundary integral equation method, we reduce the fundamental problems to systems of singular integral equations and construct exact solutions in the form of integral potentials.

The steady motion of a circular crack along the interface of a layer and a half-space

An exact analytical solution is given of the direct axisymmetric steady dynamic problem in the theory of elasticity of the motion, with an arbitrary constant subsonic velocity ν = c along the interface of a rigidly coupled layer 0 ≤ z ≤ H and a half-space z ≤ 0, of a circular transverse shear crack 0 ≤ r ≤ r0 + ct (r0 ≤ 0) with and without a cavity at its tip. Using Hankel transformation in terms of biwave potentials, a general solution of the fundamental equations of motion in the theory of elasticity and the basic solutions of the first fundamental boundary-value problem are separately constructed for the layer and the half-space for the case of arbitrary normal and shear stresses in the plane of separation z = 0 in a moving cylindrical system of coordinates r1 = r + ct, z1 = z. A special regularization of the main solution is carried out which ensures the convergence of the integrals for all stresses and displacements while preserving the high accuracy of the solution to whatever level may be desired [1, 2]. On the basis of the main solutions, a mathematical formulation is given of the mixed problem of the motion of a transverse shear crack with a cavity at the tip and its reduction to a system of three singular integral equations with Cauchy kernels which allows of regularization by the Carleman-Vekua method in terms of the closed solution of the corresponding characteristic system of singular integral equations. When the width of the cavity vanishes, one of the equations of the system solves the problem of a transverse shear crack without a cavity. Criteria are established for the existence of a cavity and its absence as a function of the elastic and velocity characteristics of the layer and half-space and the velocity of motion of the crack c.

The Stress Response of Radially Polarized Rotating Piezoelectric Cylinders

Recent advances in smart structures technology have lead to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary value problems. In this paper, we develop an analytic solution to the axisymmetric problem of an infinitely long, radially polarized, radially orthotropic piezoelectric hollow circular cylinder rotating about its axis at constant angular velocity. The cylinder is subjected to uniform internal pressure, or a constant potential difference between its inner and outer surfaces, or both. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. The stress distribution in the tube is obtained numerically for a specific piezoceramic of technological interest, namely PZT-4. For the special problem of a uniformly rotating solid cylinder with traction-free surface and zero applied electric charge, explicit closed-form solutions are obtained. It is shown that for certain piezoelectric solids, stress singularities at the origin can occur analogous to those occurring in the purely mechanical problem for radially orthotropic elastic materials.

Axisymmetric contact problems for a two-layer elastic half-space with an annular or circular crack at the interface of the layers

Exact analytical solutions of three related axisymmetric mixed problems of the theory of elasticity are given. These problems are concerned with the pressure of an annular or circular punch on a two-layer half-space taking into account cohesion and the existence of an annular or circular crack at the interface of the layers. The formulation and solution of these problems are based on a special regularization of the solution of the first fundamental boundary-value problem of the theory of elasticity for a single layer when there are arbitrary normal and shear loads on its boundary planes. The solution is constructed by the Hankel transformation method which ensures that the integrals converge for all stresses and displacements.

Internally Pressurized Radially Polarized Piezoelectric Cylinders

The piezoelectric phenomenon has been exploited in science and engineering for decades. Recent advances in smart structures technology have lead to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary-value problems. In this paper, we develop an analytic solution to the axisymmetric problem of an infinitely long, radially polarized, radially orthotropic piezoelectric hollow circular cylinder. The cylinder is subjected to uniform internal pressure, or a constant potential difference between its inner and outer surfaces, or both. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. The stress distributions in the cylinder are obtained numerically for two typical piezoceramics of technological interest, namely PZT-4 and BaTiO3. It is shown that the hoop stresses in a cylinder composed of these materials can be made virtually uniform throughout the cross-section by applying an appropriate set of boundary conditions.

Micro-polar theory for a periodic force on the edge of elastic honeycomb

A micro-polar theory has been used to calculate displacements and stresses generated by a fundamental boundary value problem in dynamics of elastic continua; namely, a time-harmonic line force acting at a point on the edge of a two-dimensional honeycomb half-space. An integral method is developed to solve this problem; the method combines a deformed contour of integration, similar to Cagniard–deHoop's, with the concept of Lamb's choice for cuts. The relationship between micro-inertia, excitation frequency and cell wall geometry is established; this shows that micro-inertia is more significant in honeycomb with more slender cell walls. The result also shows that the micro- and macro-rotations are almost the same at small excitation frequencies, but differ as the excitation frequency becomes larger.

Micropolar theory for two–dimensional stresses in elastic honeycomb

A micropolar theory has been used to calculate the stress field generated by a fundamental boundaryvalue problem in planar elasticity; namely, a normal line force acting on the surface of a honeyco...