Exact boundary integral transformation of the thermoelastic domain integral in BEM for general 2D anisotropic elasticity

In the direct formulation of the boundary element method, body-force and thermal loads manifest themselves as additional volume integral terms in the boundary integral equation. The exact transformation of the volume integral associated with body-force loading into surface ones for two-dimensional elastostatics in general anisotropy, has only very recently been achieved. This paper extends the work to treat two-dimensional thermoelastic problems which, unlike in isotropic elasticity, pose additional complications in the formulation. The success of the exact volume-to-surface integral transformation and its implementation is illustrated with three examples. The present study restores the application of BEM to two-dimensional anisotropic elastostatics as a truly boundary solution technique even when thermal effects are involved.     

Analytical integrations in 2D BEM elasticity

In the context of two‐dimensional linear elasticity, this paper presents the closed form of the integrals that arise from both the standard (collocation) boundary element method and the symmetric Galerkin boundary element method. Adopting polynomial shape functions of arbitrary degree on straight elements, finite part of Hadamard, Cauchy principal values and Lebesgue integrals are computed analytically, working in a local coordinate system. For the symmetric Galerkin boundary element method, a study on the singularity of the external integral is conducted and the outer weakly singular integral is analytically performed. Numerical tests are presented as a validation of the obtained results. Copyright © 2001 John Wiley & Sons, Ltd.     

A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids

This paper presents a single-domain boundary element method (BEM) analysis of fracture mechanics in 2D anisotropic piezoelectric solids. In this analysis, the extended displacement (elastic displacement and electrical potential) and extended traction (elastic traction and electrical displacement) integral equations are collocated on the outside boundary (no-crack boundary) of the problem and on one side of the crack surface, respectively. The Green's functions for the anisotropic piezoelectric solids in an infinite plane, a half plane, and two joined dissimilar half-planes are also derived using the complex variable function method. The extrapolation of the extended relative crack displacement is employed to calculate the extended `stress intensity factors' (SIFs), i.e., K_I, K_II, K_III and K_IV. For a finite crack in an infinite anisotropic piezoelectric solid, the extended SIFs obtained with the current numerical formulation were found to be very close to the exact solutions. For a central and inclined crack in a finite and anisotropic piezoelectric solid, we found that both the coupled and uncoupled (i.e., the piezoelectric coefficient e_ijk=0) cases predict very similar stress intensity factors K_I and K_II when a uniform tension σ_yy is applied, and very similar electric displacement intensity factor K_IV when a uniform electrical displacement D_y is applied. However, the relative crack displacement and electrical potential along the crack surface are quite different for the coupled and uncoupled cases. Furthermore, for a inclined crack within a finite domain, we found that while a uniform σ_yy (=1 N m⁻²) induces only a very small electrical displacement intensity factor (in the unit of Cm^−3/2), a uniform D_y (=1 C m⁻²) can produce very large stress intensity factors (in the unit of Nm^−3/2).     

Efficient calculation of internal results in 2D elasticity BEM

The solution of boundary integral equations in their discretized form requires an accurate treatment of regular as well as singular integrals. The regular integrals are usually solved numerically using Gauss quadrature. Since these integrations make up the major part of the numerical work the choice of the appropriate Gauss order is essential to an accurate and efficient boundary element analysis. Thus, a considerable number of publications is dealing with the subject of choosing a Gauss order suitable to gain efficiency without loosing accuracy. The guidelines determining the choice of the appropriate Gauss order is usually called an integration criterion. This paper presents a study on this topic with emphasis on the accuracy of internal results in 2D elasticity. First the necessity for a new integration criterion is shown. Then a new criterion is derived. This new criterion and various existing criteria from the literature are applied to a standard benchmark problem. The superior performance of the novel criterion is demonstrated.     

Revisit of degenerate scales in the BIEM/BEM for 2D elasticity problems

The boundary integral equation method in conjunction with the degenerate kernel, the direct searching technique (singular value decomposition), and the only two-trials technique (2 × 2 matrix eigenvalue problem) are analytically and numerically used to find the degenerate scales, respectively. In the continuous system of boundary integral equation, the degenerate kernel for the 2D Kelvin solution in the polar coordinates is reviewed and the degenerate kernel in the elliptical coordinates is derived. Using the degenerate kernel, an analytical solution of the degenerate scales for the elasticity problem of circular and elliptical cases is obtained and compared with the numerical result. Further, the triangular case and square case were also numerically demonstrated.     

A 2D time-domain BEM for transient wave scattering analysis by a crack in anisotropic solids

A two-dimensional (2D) time-domain boundary element method (BEM) is presented in this paper for transient analysis of elastic wave scattering by a crack in homogeneous, anisotropic and linearly elastic solids. A traction boundary integral equation formulation is applied to solve the arising initial-boundary value problem. A numerical solution procedure is developed to solve the time-domain boundary integral equations. A collocation method is used for the temporal discretization, while a Galerkin-method is adopted for the spatial discretization of the boundary integral equations. Since the hypersingular boundary integral equations are first regularized to weakly singular ones, no special integration technique is needed in the present method. Special attention of the analysis is devoted to the computation of the scattered wave fields. Numerical examples are given to show the accuracy and the reliability of the present time-domain BEM. The effects of the material anisotropy on the transient wave scattering characteristics are investigated.     

A hypersingular time‐domain BEM for 2D dynamic crack analysis in anisotropic solids

A hypersingular time‐domain boundary element method (BEM) for transient elastodynamic crack analysis in two‐dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack‐faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time‐stepping scheme is obtained to compute the unknown boundary data including the crack‐opening‐displacements (CODs). Special crack‐tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time‐domain BEM. Copyright © 2008 John Wiley & Sons, Ltd.     

Efficient computation of the Green's function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis

An alternative scheme to compute the Green's function and its derivatives for three dimensional generally anisotropic elastic solids is presented in this paper. These items are essential in the formulation of the boundary element method (BEM); their evaluation has remained a subject of interest because of the mathematical complexity. The Green's function considered here is the one introduced by Ting and Lee [Q. J. Mech. Appl. Math. 1997; 50: 407–26] which is of real-variable, explicit form expressed in terms of Stroh's eigenvalues. It has received attention in BEM only quite recently. By taking advantage of the periodic nature of the spherical angles when it is expressed in the spherical coordinate system, it is proposed that this Green's function be represented by a double Fourier series. The Fourier coefficients are determined numerically only once for a given anisotropic material; this is independent of the number of field points in the BEM analysis. Derivatives of the Green's function can be performed by direct spatial differentiation of the Fourier series. The resulting formulations are more concise and simpler than those derived analytically in closed form in previous studies. Numerical examples are presented to demonstrate the veracity and superior efficiency of the scheme, particularly when the number of field points is very large, as is typically the case when analyzing practical three dimensional engineering problems.     

Three‐dimensional BEM for scattering of elastic waves in general anisotropic media

Based on the full‐space Green's functions, a three‐dimensional time‐harmonic boundary element method is presented for the scattering of elastic waves in a triclinic full space. The boundary integral equations for incident, scattered and total wave fields are given. An efficient numerical method is proposed to calculate the free terms for any geometry. The discretization of the boundary integral equation is achieved by using a linear triangular element. Applications are discussed for scattering of elastic waves by a spherical cavity in a 3D triclinic medium. The method has been tested by comparing the numerical results with the existing analytical solutions for an isotropic problem. The results show that, in addition to the frequency of the incident waves, the scattered waves strongly depend on the anisotropy of the media. Copyright © 2003 John Wiley & Sons, Ltd.     

A simple a-posteriori error estimation for adaptive BEM in elasticity

In this paper, the properties of various boundary integral operators are investigated for error estimation in adaptive BEM. It is found that the residual of the hyper-singular boundary integral equation (BIE) can be used for a-posteriori error estimation for different kinds of problems. Based on this result, a new a-posteriori error indicator is proposed which is a measure of the difference of two solutions for boundary stresses in elastic BEM. The first solution is obtained by the conventional boundary stress calculation method, and the second one by use of the regularized hyper-singular BIE for displacement derivative. The latter solution has recently been found to be of high accuracy and can be easily obtained under the most commonly used C ⁰ continuous elements. This new error indicator is defined by a L ₁ norm of the difference between the two solutions under Mises stress sense. Two typical numerical examples have been performed for two-dimensional (2D) elasticity problems and the results show that the proposed error indicator successfully tracks the real numerical errors and effectively leads a h-type mesh refinement procedure.     

A non-linear transformation applied to boundary layer effect and thin-body effect in BEM for 2D potential problems

The accurate numerical evaluation of nearly singular integrals plays an important role in many engineering applications. In general, these include evaluating the solution near the boundary or treating problems with thin domains, which are respectively named the boundary layer effect and the thin-body effect in the boundary element method. Although many methods of evaluating nearly singular integrals have been developed in recent years with varying degrees of success, questions still remain. In this article, a general non-linear transformation for evaluating nearly singular integrals over curved two-dimensional (2D) boundary elements is employed and applied to treat boundary layer effect and thin-body effect occurring in 2D potential problems. The introduced transformation can remove or damp out the rapid variations of nearly singular kernels and extremely high accuracy of numerical results can be achieved without increasing other computational efforts. Extensive numerical experiments indicate that the proposed transformation will be more efficient, in terms of the necessary integration points and central processing unit-time, compared to previous transformation methods, especially for dealing with thin-body problems.     

Computation of 3D MEMS electrostatics using a nearly exact BEM solver

The electrostatic properties of thin plate shaped structures relevant to the micro-electro-mechanical systems (MEMS) have been computed using a nearly exact boundary element method (BEM) solver. The solver uses closed form expressions for three-dimensional potential and force fields due to uniform sources/sinks distributed on finite flat surfaces. The expressions have been validated and, being analytical, have been found to be applicable throughout the physical domain. The solver has been applied to compute accurately and efficiently the charge densities on thin plate shaped conductors as used in MEMS components. We have presented results for the model problem of parallel plate capacitors and compared them with results obtained from several other BEM based solvers.     

A time-domain collocation-Galerkin BEM for transient dynamic crack analysis in anisotropic solids

This paper presents a time-domain boundary element method (BEM) for transient elastodynamic crack analysis in homogeneous and linear elastic solids of general anisotropy. A finite crack subjected to a transient loading is investigated. Two-dimensional (2D) generalized plane-strain or plane-stress condition is considered. The initial-boundary value problem is described by a set of hypersingular time-dependent traction boundary integral equations (BIEs), in which the crack-opening displacements (CODs) are unknown quantities. The hypersingular time-domain BIEs are first regularized to weakly singular ones by using spatial Galerkin method, which transfers the derivatives of the fundamental solutions to the unknown CODs and the weight functions. To solve the time-domain BIEs numerically, a time-stepping scheme is developed. The scheme applies the collocation method for temporal discretization of the time-domain BIEs. As spatial shape-functions, two different functions are implemented. For elements away from crack-tips, linear spatial shape-function is used, while for elements near the crack-tips a special ‘crack-tip shape-function’ is applied to describe the local ‘square-root’ behavior of the CODs at the crack-tips properly. Special attention of the analysis is devoted to the numerical computation of the transient elastodynamic stress intensity factors for cracks in general anisotropic and linear elastic solids. Numerical examples are presented to verify the accuracy of the present time-domain BEM.     

A new boundary integral equation method for cracked 2-D anisotropic bodies

By using integration by parts to the traditional boundary integral formulation, a traction boundary integral equation for cracked 2-D anisotropic bodies is derived. The new traction integral equation involves only singularity of order 1/r and no hypersingular term appears. The dislocation densities on the crack surface are introduced and the relations between stress intensity factors and dislocation densities near the crack tip are induced to calculate the stress intensity factors. The boundary element method based on the new equation is established and the singular interpolation functions are introduced to model the singularity of the dislocation density (in the order of ) for crack tip elements. The proposed method can be directly used for the 2-D anisotropic body containing cracks of arbitrary geometric shapes. Several numerical examples demonstrate the validity and accuracy of BEM based on the new boundary integral equation.     

A comparison between isoparametric Lagrangian elements in 2D BEM

A comparison between a family of isoparametric C°-continucus elements in the context of two-dimensional elastostatics using the direct boundary element method is presented. The elements studied use Lagrangian shape functions of orders quadratic, cubic and quartic. The relative efficiencies of these elements from the viewpoint of accuracy of solution as well as cost effectiveness are examined through certain selected problems. It emerges from the present study that the quartic element is more reliable in accuracy than the lower members of the family without being necessarily more expensive.     

A boundary element method solution for anisotropic nonhomogeneous elasticity

A new BEM approach is presented for the plane elastostatic problem for nonhomogeneous anisotropic bodies. In this case the response of the body is described by two coupled linear second order partial differential equations in terms of displacement with variable coefficient. The incapability of establishing the fundamental solution of the governing equations is overcome by uncoupling them using the concept of analog equation, which converts them to two Poisson’s equations, whose fundamental solution is known and the necessary boundary integral equations are readily obtained. This formulation introduces two additional unknown field functions, which physically represent the two components of a fictitious source. Subsequently, they are determined by approximating them globally with radial basis functions series. The displacements and the stresses are evaluated from the integral representation of the solution of the substitutes equations. The presented method maintains the pure boundary character of the BEM. The obtained numerical results demonstrate the effectiveness and accuracy of the method.     

A comparative study of three BEM for transient dynamic crack analysis of 2-D anisotropic solids

Three different boundary element methods (BEM) for transient dynamic crack analysis in two-dimensional (2-D), homogeneous, anisotropic and linear elastic solids are presented. Hypersingular traction boundary integral equations (BIEs) in frequency- domain, Laplace-domain and time-domain with the corresponding elastodynamic fundamental solutions are applied for this purpose. In the frequency-domain and the Laplace-domain BEM, numerical solutions are first obtained in the transformed domain for discrete frequency or Laplace-transform parameters. Time-dependent results are subsequently obtained by means of the inverse Fourier-transform and the inverse Laplace-transform algorithm of Stehfest. In the time-domain BEM, the quadrature formula of Lubich is adopted to approximate the arising convolution integrals in the time-domain BIEs. Hypersingular integrals involved in the traction BIEs are computed through a regularization process that converts the hypersingular integrals to regular integrals, which can be computed numerically, and singular integrals which can be integrated analytically. Numerical results for the dynamic stress intensity factors are presented and discussed for a finite crack in an infinite domain subjected to an impact crack-face loading.     

Boundary integral equations and the boundary element method for fracture mechanics analysis in 2D anisotropic thermoelasticity

This paper develops the Somigliana type boundary integral equations for fracture of anisotropic thermoelastic solids using the Stroh formalism and the theory of analytic functions. In the absence of body forces and internal heat sources, obtained integral equations contain only curvilinear integrals over the solid’s boundary and crack faces. Thus, the volume integration is eliminated and also there is no need to evaluate integrals over the contours in the mapped temperature domain as it was done before. In addition to finite solids, the case of an infinite anisotropic medium with a remote thermal load is also studied. The dual boundary element method for fracture of anisotropic thermoelastic solids is developed based on the obtained boundary integral equations. Presented numerical examples show the validity and efficiency of the obtained equations in the analysis of both finite and infinite solids with cracks.