Line-spring boundary element method for a surface cracked plate

In the published line-spring boundary element method, the effect of bending is not considered. Therefore it cannot be used to deal with the problem of a surface cracked plate. Taking the advantage of the line-spring model and the boundary element method the authors present a new line-spring boundary element method in which the effect of membrane force and bending moment in a Reissner plate is taken into account. The implementation of the method is discussed in detail. The stress intensity factors for several example problems concerning surface cracked plates are calculated. Comparisons are made with the Newman-Raju solutions. The results show that the proposed method is efficient.     

Iterative techniques for 3-D boundary element method systems of equations

A key issue in the boundary element method (BEM) is the solution of the associated system of algebraic equations whose matrices are dense, nonsymmetric and sometimes ill conditioned. For large scale tridimensional problems, direct methods like Gauss elimination become too expensive and iterative methods may be preferable. This paper presents a comparison of the performances of some iterative techniques based on conjugate gradient solvers as conjugate gradient squared (CGS) and bi-conjugate gradient (Bi-CG) that seem to have the potential to be efficient and competitive for BEM algebraic systems of equations, specially when used with an appropriate preconditioner. A comparison with the direct application of the conjugate gradient method to the normalized systems of equations (CGNE and CGNR) is also presented.     

Computation of 3-D stress singularities for multiple cracks and crack intersections by the scaled boundary finite element method

Among the various possible ways of dealing with notch and crack situations, the scaled boundary finite element method [SBFEM, (Wolf and Song in Finite element modelling of unbounded structures. Wiley, Chichester, 1996; Wolf in The scaled boundary finite element method. Wiley, Chichester, 2003)] has been adopted in this work. This method has been proved to be versatile, much less time consuming than the finite element method and generates highly accurate numerical predictions in cases of structures with notches and cracks. The SBFEM gives the advantage of boundary element method by reducing one dimension in modelling the structures but the mathematical formulations are more related to conventional displacement based finite element method. This method requires a certain scalability of the given structure with respect to a point called similarity center. Like in the case of the boundary element method, the structure needs to be discretized only at the surface where standard displacement based isoparametric finite element formulations are adequate. Unlike in the boundary element method, however, no fundamental solution is required by the scaled boundary finite element method. The similarity or scalability of the method requires separation of coordinates such that in the radial direction (i.e. scaling direction) it yields simple differential equations that can be solved analytically. So this approach can be considered as a semi-analytical method. Several two-dimensional examples have been analysed for crack and notch situations that are well known cases in fracture mechanics. A number of three-dimensional cases have been considered for different crack configurations that yield high order of singularity. The results, according to the authors’ knowledge are up to now unpublished in the open literature. Parametric studies are conducted for structures with bi-material interfaces.     

Dynamic analysis of 3-D structures by a transformed boundary element method

In this work, an advanced implementation of the direct boundary element method applicable to periodic (steadystate) and transient dynamic problems involving three-dimensional structures of arbitrary shape and connectivity is presented. Interior, exterior and halfspace type of problems can all be solved by the present method. The discussion first focuses on the formulation of the method, followed by material pertaining to the fundamental singular solutions and to the isoparametric boundary elements used for discretizing the surface of the problem. Subsequently, numerical integration techniques and the solution algorithm are introduced. This methodology has been incorporated in a versatile, general purpose computer program. Finally, the stability and high accuracy of this dynamic analysis technique are established through comparisons with available analytical and numerical results.     

Dynamic analysis of cracks using boundary element method

This paper presents a procedure for dynamic stress intensity factor computations using traction singular quarter-point boundary elements. Cracks in a complete space, a half-space and a finite body loaded by steady state waves are studied. Curves for elastodynamic stress intensity factors vs frequency are presented. Transient stress intensity factors are computed by means of Fourier transform. The results are compared with other authors and shown to be accurate in all cases. The dynamic stress intensity factors are computed in a very direct and easy way to implement. This versatile procedure allows for the study of problems with complex geometry that include one or several cracks.     

A 3-D indirect boundary element method for bounded creeping flow of drops

The simulation of the flow of emulsions in porous media presents formidable challenges, due to the extremely complex evolving geometry. Methods based on boundary integral equations, suitable for creeping flows, reduce the effort dedicated to geometry representation, but can become computationally expensive. An efficient indirect boundary integral formulation representing deformable drops in a bounded Stokes flow, resulting in a set of Fredholm integral equations of the second kind, is presented. The boundary element method (BEM) based on the formulation employs an accurate numerical integration scheme for the singular kernels involved, an effective and accurate curvature and normal calculation method, and an adaptive remeshing method to simulate interfacial deformation of drops. Two benchmark problems are used to assess the accuracy of the method, and to investigate its behavior for large problems. The method is found to provide accurate results combined with well-posedness, making it suitable for use in accelerated fast multipole method algorithms.     

The direct boundary element method in plate bending

The direct boundary element method based on the Rayleigh-Green identity is employed for the static analysis of Kirchhoff plates. The starting point is a slightly modified version of Stern's equations. The focus is on the implementation of the method for linear elements and a Hermitian interpolation for the deflection w. The concept of element matrices is developed and the Cauchy principal values of the singular integrals are given in detail. The treatment of domain integrals, the handling of internal supports, the properties of the solution and the effect of singularities are discused. Numerical examples illustrate the various techniques. In the appendix the influence functions for the second and third derivatives of the deflection w are given.     

The 3-D thermoelastic boundary element method: semi-analytical integration for subparametric triangular elements

In this paper a semi-analytical integration scheme is described which is designed to reduce the errors that can result with the numerical evaluation of integrals with singular integrands. The semi-analytical scheme can be applied to quadratic subparametric triangular elements for use in thermoelastic problems. Established analytical solutions for linear isoparametric triangular elements are combined with standard quadrature techniques to provide an accurate integration scheme for quadratic subparametric triangular elements. The use of subparametric elements provides an efficient means for coupling thermal and elastostatic analyses. It is possible for the same mesh to be employed, with linear isoparametric elements used for thermal analysis and quadratic subparametric elements used for deformation analysis. Numerical tests are performed on simple test problems to demonstrate the advantages of the semi-analytical approach which is shown to be orders of magnitude more accurate than standard quadrature techniques. Moreover, the expected increased accuracy with subparametric elements is also demonstrated. © 1998 John Wiley & Sons, Ltd.     

Application of boundary element method to 3-D problems of coupled thermoelasticity

This paper deals with a new boundary element method for analysis of the quasistatic problems in coupled thermoelasticity. Through some mathematical manipulation of the Navier equation in elasticity, the heat conduction equation is transformed into a simpler form, similar to the uncoupled-type equation with the modified thermal conductivity which shows the coupling effects. This procedure enables us to treat the coupled thermoelastic problems as an uncoupled one, A few examples are computed by the proposed BEM, and the results obtained are compared with the analytical ones available in the literature, whereby the accuracy and versatility of the proposed method are demonstrated.     

A boundary element method for calculating the K field for cracks along a bimaterial interface

A multi-region boundary element method (BEM) based on the modified crack closure method (CCM) is developed to obtain the energy release rate G for cracks in homogeneous materials and along a bimaterial interface. The energy release rate obtained using the CCM are compared with that obtained using the crack opening displacement (COD) method. A combination of these methods allows us to determine the phase angle and therefore the complex stress intensity factor K for crack problems. We access the accuracy of our BEM by comparing its results with known analytic solutions and previous FEM results in the literature. Computations are also carried out for the asymmetric double cantilever beam (ADCB) specimen, which has been used to determine fracture toughness of polymer/polymer and polymer/nonpolymer interfaces. An auxiliary K _A field method to evaluate K is also discussed.This work is supported by the Cornell Materials Science Center (MSC) which is funded by the National Science Foundation (DMR-MRL program).     

Computation of weight functions for three-dimensional cracks by the boundary element method

A boundary element procedure is presented for calculating weight functions for three-dimensional cracks with a smooth front. The weight function for the particular point at the crack front is represented as a sum of regular and singular parts. The known weight function for a circular crack in an infinite body is used as the singular part. The boundary integral equation is formulated for the regular part of the weight function in the vicinity of considered crack front point and for the whole weight function for the rest of the body. A discretized form of the boundary integral equation is given. Some examples are provided to test the accuracy of the proposed procedure.     

Stiffened plate bending analysis by the boundary element method

 In this work, the plate bending formulation of the boundary element method (BEM) based on the Kirchhoff's hypothesis, is extended to the analysis of stiffened elements usually present in building floor structures. Particular integral representations are derived to take directly into account the interactions between the beams forming grid and surface elements. Equilibrium and compatibility conditions are automatically imposed by the integral equations, which treat this composite structure as a single body. Two possible procedures are shown for dealing with plate domain stiffened by beams. In the first, the beam element is considered as a stiffer region requiring therefore the discretization of two internal lines with two unknowns per node. In the second scheme, the number of degrees of freedom along the interface is reduced by two by assuming that the cross-section motion is defined by three independent components only. Received 6 November 2000     

A new boundary element method for mixed boundary value problems involving cracks and holes: Interactions between rigid inclusions and cracks

This paper derives a new boundary integral equation (BIE) formulation for plane elastic bodies containing cracks and holes and subjected to mixed displacement/ traction boundary conditions, and proposes a new boundary element method (BEM) based upon this formulation. The basic unknown in the formulation is a complex boundary function H(t), which is a linear combination of the boundary traction and boundary displacement density. The present BIE formulation can be related directly to Muskhelishvili's formalism. Singular interpolation functions of order r ^–1/2 (where r is the distance measured from the crack tip) are introduced such that singular integrand involved at the element level can be integrated analytically. By applying the BEM, the interaction between a rigid circular inclusion and a crack is investigated in details. Our results for the stress intensity factor are comparable with those given by Erdogan and Gupta (1975) and Gharpuray et al. (1990) for a crack emanating from a stiff inclusion, and with those by Erdogan et al. (1974) for a crack in the neighborhood of a stiff inclusion.     

A boundary element method for elastic buckling analysis of assembled plate structures

In this paper, we first discuss the integral equation formulation for the buckling problem of a single plate, using the biharmonic fundamental solution for the plate bending problems. The so called boundary-element method previously proposed by the senior author is applied to the numerical solution of the resulting set of integral equations. The total set of simultaneous equations are derived for nodal unknowns taken out of the whole domain, and reduced to an algebraic set of eigenvalue equations. The proposed method is method to the solution of elastic buckling of assembled plate structures. A few examples are computed and results obtained are compared with other solutions to demonstrate the potential usefulness of the proposed method.     

Evaluations of the T-stress for interface cracks by the boundary element method

The path independent M-integral is applied to computation of the T-stress for interface cracks between dissimilar materials. The unique relation between the M-integral and T-stress is found for a properly selected auxiliary solution. The problem of a semi-infinite interface crack between dissimilar materials loaded by a point force applied to the crack tip in a direction parallel to the interface suits such an auxiliary solution. A new subregion boundary element method is applied to solve the given bimaterial interface crack problem. Numerical results for centre-cracked plate, single-edged notch and double-edged notch specimens are included.     

Stabilized finite element approximations for heat conduction in a 3-D plate with dominant thermal source

The present work studies the finite element approximation for the heat transfer process in an opaque three-dimensional plate with a temperature-dependent source dominating the conductive operator. The adopted mechanical model assumes the existence of a heat transfer from/to the plate following Newton's law of cooling. The numerical simulations performed have attested the instability of the classical Galerkin method when subjected to very high source-dominated regimen. Usual strategies in the Engineering practice of dealing with this shortcoming proved to be inefficient. A Gradient-Galerkin/Least-Squares formulation was adopted in the numerical simulations as a remedy for the Galerkin's instability when subjected to those regimen. Received 25 May 1998     

3-D open boundary magnetic field analysis using infinite elementbased on hybrid finite element method

A method for analyzing 3-D open-boundary magnetic field problems using infinite elements has been developed. The infinite problem has the advantage that the bandwidth of the coefficient matrix and the number of unknown variables are reduced. Moreover, no experience is necessary in determining decay parameters. The effectiveness of the infinite-element method is illustrated by the accuracy and the CPU time obtained when various boundary conditions are applied     

Dual boundary element analysis of closed cracks

A two-dimensional boundary element method for analysis of closed or partially closed cracks under normal and frictional forces is developed. The single domain dual formulation is used. As a contact problem is non-linear due to the friction phenomena at the crack interface and also because of the boundary conditions which may change during the loading, it is formulated in an incremental and iterative fashion. The stress intensity factors are calculated with the J-integral method. Also crack growth is considered. Several benchmark cases have been analysed to verify the results given by the method. The stress intensity factors and crack paths calculated are similar to those given in the literature. © 1997 John Wiley & Sons, Ltd.     

A fast multipole boundary element method for solving the thin plate bending problem

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.