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The paper deals with the numerical implementations of local integral equation formulation for the solution of two-dimensional (2-d) problems in linear elastic media with continuously variable Young’s modulus. Two kinds of the finite element based interpolation are developed for approximation of field variables on local sub-domains around nodal points. One of these approximations can be classified as meshless, since the elements are generated automatically from the predefined nodal points on the analyzed domain. The other one utilizes the predefined mesh of elements. Besides the element based interpolations we present also the meshless Point Interpolation Method. The accuracy, convergence, numerical stability and efficiency of the proposed techniques are tested by numerical examples with using the exact benchmark solutions.

In this paper we present the advanced form of the boundary integral equations for solution of transient heat conduction problems. The BIE are free of Cauchy principal value integrals in the boundary element formulation with time dependent fundamental solutions. This is made independently of the order of the approximation within boundary elements.

We have implemented numerically and tested two schemes for computation of internal stresses in 2-d stationary thermoelasticity. As expected, both the nonregularized and regularized schemes fail as the internal point approaches the boundary

A meshless boundary element method (BEM) for stress analysis in two-dimensional (2-D), isotropic, continuously non-homogeneous, and linear elastic functionally graded materials (FGMs) is presented in this paper. It is assumed that Young’s modulus has an exponential variation, while Poisson’s ratio is taken to be constant. Since no fundamental solutions are yet available for general FGMs, fundamental solutions for isotropic, homogeneous, and linear elastic solids are applied, which results in a boundary-domain integral formulation. Normalized displacements are introduced in the formulation, which avoids displacement gradients in the domain-integrals. The radial integration method (RIM) is used to transform the domain-integrals into boundary integrals along the global boundary. The normalized displacements appearing in the domain-integrals are approximated by a series of prescribed basis functions, which are taken as a combination of radial basis functions and polynomials in terms of global coordinates. Numerical examples are presented to verify the accuracy and the efficiency of the present meshless BEM.

This paper is a comparative study for various numerical implementations of local integral equations developed for stress analysis in plane elasticity of solids with functionally graded material coefficients. Besides two meshless implementations by the point interpolation method and the moving least squares approximation, the element based approximation is also utilized. The numerical stability, accuracy, convergence of accuracy and cost efficiency (assessed by CPU-times) are investigated in numerous test examples with exact benchmark solutions.

The aim of this paper is to present the construction and closed form expression of the fundamental solutions of partial differential operators given by a product of metaharmonic by polyharmonic operators. The inverse operator of such a product of operators is decomposed into a power series of inverse metaharmonic and polyharmonic operators for which the fundamental solutions are available in the literature.

Publisher Summary This chapter introduces an advanced computational method for transient heat conduction analysis in a homogeneous 3D axisymmetric body. The method is based on the local boundary integral equations with moving least square (MLS) approximation of the temperature and heat flux. The integral formulations are given in the Laplace transform domain. The Stehfest algorithm is applied for the numerical Laplace inversion to obtain the time-dependent solution. The boundary-domain formulation can be easily implemented for a simple circular subdomain to which the local boundary integral equation is related. Contrary to the conventional boundary integral equation methods, all integrands in the present formulation are regular. Thus, no special integration techniques are required to evaluate the integrals. An important advantage of the present method is that it can be equally well applied to transient heat conduction analysis in inhomogeneous solids.

Piezoelectric materials offer many possibilities in advanced engineering structures due to their inherent coupling effects between mechanical and electrical fields and are widely applied in smart devices and structures like transducers, actuators and sensors [2]. An important application of piezoelectric materials is related to layered or laminated composites because they can be optimized to satisfy the high-performance requirements according to different in-service conditions. Beside cracks inside homogeneous domains, one of the most dominant failure mechanisms in layered or laminated composites is the interface failure. Interface cracks and interface debonding may be induced by the mismatch of the mechanical, electrical and thermal properties of the material constituents during the manufacturing process and the in-service loading conditions. This paper presents a hypersingular symmetric Galerkin boundary element method (SGBEM) for crack analysis in two-dimensional (2D), layered and linear piezoelectric solids. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

CCSD(T) ground state potential curves of Pb···RG systems (RG = He, Ne and Ar) are presented and the importance of the inclusion of spin-orbit effects is discussed. The closed-shell character of the Pb atom at the two-component relativistic level of relativistic theory leads to shallower potential energy curves compared to scalar relativistic open-shell calculations. The pressure-independent cross-diffusion coefficients pD12 have been simulated using the extrapolated two-component CCSD(T) ground state potential curves. The diffusion coefficients from scattering theory are compared with simulations based on molecular dynamics (MD) using the velocity autocorrelation function (VACF) and the Einstein equation. A correction for the proper assessment of the uncertainty in the VACF is proposed. The acceleration of the MD simulation of Pb in RG diffusion is proposed utilizing the RG in Pb diffusion. The dU[TQ]Z/CCSD(T) potential curve of Pb···He (De = 8.667 cm(-1), re = 4.683 Å) supports only one vibrational level. The anharmonicity of this potential is compared to the potential of He···He which also supports only one vibrational level. The comparison is based on the mean square separations of the vibrational wave function.

This paper discusses various electromagnetic boundary conditions on the crack-faces in two-dimensional magnetoelectroelastic materials. For this purpose, a meshless method based on the local Petrov?Galerkin approach is developed to solve the initial-boundary value problems of two-dimensional cracked magnetoelectroelastic solids with nonlinear electrical and magnetic boundary conditions on the crack-faces. A Heaviside step function as the test function is applied in the weak form to derive local integral equations. Nodal points are spread on the analyzed domain and each node is surrounded by a small circle for simplicity. The spatial variations of the displacements, electric and magnetic potentials are approximated by the moving least-squares (MLS) scheme. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time-stepping method. An iterative solution algorithm is developed to consider nonlinear electromagnetic crack-face boundary conditions.

The boundary integral equations, which give the relation between the crack opening displacement and traction on the surface of a crack embedded in an infinite isotropic elastic body are formulated. The integral equations are transformed into spherical and cylindrical coordinates in the cases of cracks curved in the shape of spherical and cylindrical surfaces respectively, so that these boundary integral equations may be converted into a system of algebraic equations by the boundary element method. The dependence of stress-intensity factors on the curvature of crack has been numerically calculated for the spherical crack with circular contour under a constant load.

This paper presents a new regularized integral representation of the stress tensor at internal points, and contains derived formulae expressing stresses at internal or boundary points in terms of nodal values of boundary displacements and forces. Comparison is made for accuracy of stresses computed by regularized and non-regularized integral representations.

The present paper is concerned with the influence of couple stresses on the stress concentration around a penny-shaped crack in an infinite body within the framework of both the micropolar and couple stress (indeterminate) theory. Taking into account the symmetry of the boundary value problem considered this one is converted into solving dual integral equations, and finally into a Fredholm integral equation of the second kind. Approximating the kernel the latter is solved analytically.

In the paper the boundary integral equation method for solving the problem of a plate resting on a two-parameter foundation is presented. Three boundary value problems are considered, and three cases are to be distinguished as regards a characteristic parameter of the foundation. The boundary integral equation solution for a clamped circular plate under a concentrated load is compared to the analytical one. Die Randintegralmethode wird zur Losung eines Plattenproblems angewandt, wobei die Platte auf einem Untergrund ruht, der durch zwei Parameter beschrieben wird. Es werden drei Randwertprobleme betrachtet und drei Falle behandelt, die sich hinsichtlich eines charakteristischen Parameters des Untergrundes unterscheiden. Die mit der Randintegralmethode erhaltene Losung fur eine eingespannte Kreisplatte mit Einzellast wird mit der analytischen Losung verglichen.

In this paper the non-perturbative theory of crack dislocations is developed on the basis of an advanced boundary integral formulation of crack problems. The distribution of crack dislocations is determined by the boundary integro-differential equations derived from the regularized integral representation of the stress field. The method is applicable to the cracked body with any crack profile and the topics discussed include the formulation of cracks problems in: (i) elastostatics; (ii) elastodynamics; and (iii) thermoelasticity. In der vorliegenden Arbeit wird die Theorie der Risversetzungen auf der Basis einer qualifizierten Randintegralformulierung von Risproblemen entwickelt. Die Verteilung der Risversetzungen wird von der Randintegrodifferentialgleichung, die aus der regularisierten Integraldarstellung des Spannungsfeldes abgeleitet wird, bestimmt. Dir Methode ist auf jede Risform anwendbar, und es werden Risprobleme in der Elastostatik, Elastodynamik und Thermoelastizitat diskutiert.

This paper presents conversion of the volume integral of temperature gradients to a surface integral in the integral formulation of boundary value problems of uncoupled thermoelasticity. Particular classes of problems such as thermal stresses, quasi-static problems of uncoupled thermoelasticity, and stationary problems of thermoelasticity are considered. The improved formulation makes numerical computation more accurate and less formidable. The integral formulations in uncoupled thermoelasticity are given in the Laplace transform domain as well as in the time domain.

The general formulation of the transient elastodynamic second boundary value problem in an isotropic linear elastic body with a crack of arbitrary shape by combining the boundary integral equation method and the Laplace transform with respect to time is presented in this paper. Both finite and infinite elastic bodies are considered. A numerical solution of the transformed boundary integral equations is proposed.

A vector boundary integral equation formulation is presented for two-dimensional problems of elastodynamics. The BIE on which numerical work is based is written in a form entirely free of Cauchy principal value integrals. This method is used for computation of dynamic stress intensity factors for cracks in plane strain problems, when a finite body is subjected to dynamic loading with Heaviside-function time dependence.

An integral equation formulation for three dimensional anisotropic elastostatic boundary value problems is presented. Both the finite and infinite anisotropic body with a crack are considered. The boundary integral equation can be solved numerically for the unknown surface tractions and displacements in a well-posed boundary value problem. Once all boundary quantities are known, the field solution is given by a Somigliana type integral formula. Isotropic elastostatic boundary value problems are included as a special case of this general formulation.

The boundary integral equation formulation of thermoelasticity problems from part I is applied to crack problems in both finite and infinite thermoelastic bodies. For a flat crack in an infinite body the normal and tangential crack opening displacement are decoupled. Transient and steady state problems of thermoelasticity, as well as stationary problems, are considered.

The meshless Petrov–Galerkin method (MLPG) is applied to plate bending analysis in 1D orthorhombic quasicrystals (QCs) under static and transient dynamic loads. The Bak and elasto-hydrodynamic models are applied for phason governing equation in the elastodynamic case. The phason displacement for the orthorhombic QC in the first-order shear deformation plate theory depends only on the in-plane coordinates on the mean plate surface. Nodal points are randomly distributed over the mean surface of the considered plate. Each node is the center of a circle surrounding this node. The coupled governing partial differential equations are satisfied in a weak-form on small fictitious subdomains. The spatial variations of the phonon and phason displacements are approximated by the moving least-squares (MLS) scheme. After performing the spatial MLS approximation, a system of ordinary differential equations (ODEs) for nodal unknowns is obtained. The system of the ODEs of the second order is solved by the Houbolt finite-difference scheme. Our numerical examples demonstrate clearly the effect of the coupling parameter on both static and dynamic phonon/phason deflections.

The meshless local Petrov-Galerkin (MLPG) method is used to analyze transient dynamic problems in 3D axisymmetric piezoelectric solids with continuously inhomogeneous material properties. Both mechanical and thermal loads are considered here. A 3D axisymmetric body is created by rotation of a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduces the original 3D boundary value problem into a 2D problem. The cross section is covered by small circular sub-domains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function, in order to derive local integral equations on the boundaries of the chosen sub-domains, called local boundary integral equations (LBIE). These integral formulations are either based on the Laplace transform technique or the time-difference approach. The local integral equations are non-singular and take a very simple form, despite of inhomogeneous and anisotropic material behaviour across the analyzed structure. Spatial variation of all physical fields (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the sub-domain by means of the moving least-squares (MLS) method. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.

This paper presents the integral representations of the displacement and rotation fields of the micropolar continuum, and the regularized integral representations of the traction and couple vector. Thus the boundary integral equations and the boundary integro-differential equations are derived. In addition, the formulation for crack problems is given by both the boundary integral equations and the boundary integro-differential equations.

A meshless method based on the local Petrov–Galerkin approach is proposed, to solve initial-boundary value problems of magneto-electro-elastic solids with continuously varying material properties. Stationary and transient thermal problems are considered in this paper. The mechanical 2-D fields are described by the equations of motion with an inertial term. Nodal points are spread on the analyzed domain, and each node is surrounded by a small circle for simplicity. The spatial variation of displacements, electric and magnetic potentials is approximated by the moving least-squares (MLS) scheme. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time stepping method.

This work investigates the dynamic stress intensity factor for a crack in an infinite elastic medium. Two classes of special problems of thermoelasticity are considered. These are the problems of classical elastodynamics and quasi-static problems of uncoupled thermoelasticity. The boundary value problems in these fields are recasted in an integral form. The boundary integro-differential equations are solved by the boundary element method in the Laplace transformed domain. The method is illustrated by two numerical examples for harmonic and impact load in elastodynamics and harmonically varying temperature load in thermoelasticity.

The paper addresses the three-dimensional problem on steady-state vibrations of an elastic body consisting of two perfectly joined dissimilar half-spaces with an elliptic mode I crack located in one of the half-spaces normally to the interface. The problem is reduced to a boundary integral equation for the crack opening function. The integration domain of the equation is bounded by the crack domain, and the interaction between the crack and the interface is described by a regular kernel. The equation is solved using the mapping method. Numerical results are obtained for the case where the surfaces of the elliptic crack are subjected to harmonic loading with constant amplitude. The dependences of the stress intensity factors on the wave number are presented for various relationships among the mechanical constants that ensure the absence of near-surface waves

The conventional boundary element method (BEM) requires a domain integral in unsteady thermal stress analysis with heat generation and/or an initial temperature distribution. In this paper, it is shown that the three-dimensional unsteady thermal stress problem can be solved effectively using the triple-reciprocity boundary element method without internal cells. In this method, the distributions of heat generation and initial temperature are interpolated using integral equations and higher order time-dependent fundamental solutions. A new computer program was developed and applied for solving several test problems.