A boundary element method for two dimensional linear elastic fracture analysis

A boundary element method is developed for the analysis of fractures in two-dimensional solids. The solids are assumed to be linearly elastic and isotropic, and both bounded and unbounded domains are treated. The development of the boundary integral equations exploits (as usual) Somigliana's identity, but a special manipulation is carried out to regularize certain integrals associated with the crack line. The resulting integral equations consist of the conventional ordinary boundary terms and two additional terms which can be identified as a distribution of concentrated forces and a distribution of dislocations along each crack line. The strategy for establishing the integral equations is first outlined in terms of real variables, after which complex variable techniques are adopted for the detailed development. In the numerical implementation of the formulation, the ordinary boundary integrals are treated with standard boundary element techniques, while a novel numerical procedure is developed to treat the crack line integrals. The resulting numerical procedure is used to solve several sample problems for both embedded and surface-breaking cracks, and it is shown that the technique is both accurate and efficient. The utility of the method for simulating curvilinear crack propagation is also demonstrated.     

Automated simulation of fatigue crack propagation for two-dimensional linear elastic fracture mechanics problems by boundary element method

In this paper, automated simulation of multiple crack fatigue propagation for two-dimensional (2D) linear elastic fracture mechanics (LEFM) problems is developed by using boundary element method (BEM). The boundary element method is the displacement discontinuity method with crack-tip elements proposed by the author. Because of an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single-region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Local discretization on the incremental crack extension is performed easily. Further the new adding elements and the existing elements on the existing boundaries are employed to construct easily the total structural mesh representation. Here, the mixed-mode stress intensity factors are calculated by using the formulas based on the displacement fields around crack tip. The maximum circumferential stress theory is used to predict crack stability and direction of propagation at each step. The well-known Paris’ equation is extended to multiple crack case under mixed-mode loadings. Also, the user does not need to provide a desired crack length increment at the beginning of each simulation. The numerical examples are included to illustrate the validation of the numerical approach for fatigue growth simulation of multiple cracks for 2D LEFM problems.     

The dual boundary element method: Effective implementation for crack problems

The present paper is concerned with the effective numerical implementation of the two-dimensional dual boundary element method, for linear elastic crack problems. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed-mode crack problems can be solved with a single-region formulation. Both crack surfaces are discretized with discontinuous quadratic boundary elements; this strategy not only automatically satisfies the necessary conditions for the existence of the finite-part integrals, which occur naturally, but also circumvents the problem of collocation at crack tips, crack kinks and crack-edge corners. Examples of geometries with edge, and embedded crack are analysed with the present method. Highly accurate results are obtained, when the stress intensity factor is evaluated with the J-integral technique. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed-mode conditions.     

An element implementation of the boundary face method for 3D potential problems

This work presents a new implementation of the boundary face method (BFM) with shape functions from surface elements on the geometry directly like the boundary element method (BEM). The conventional BEM uses the standard elements for boundary integration and approximation of the geometry, and thus introduces errors in geometry. In this paper, the BFM is implemented directly based on the boundary representation data structure (B-rep) that is used in most CAD packages for geometry modeling. Each bounding surface of geometry model is represented as parametric form by the geometric map between the parametric space and the physical space. Both boundary integration and variable approximation are performed in the parametric space. The integrand quantities are calculated directly from the faces rather than from elements, and thus no geometric error will be introduced. The approximation scheme in the parametric space based on the surface element is discussed. In order to deal with thin and slender structures, an adaptive integration scheme has been developed. An adaptive method for generating surface elements has also been developed. We have developed an interface between BFM and UG-NX(R). Numerical examples involving complicated geometries have demonstrated that the integration of BFM and UG-NX(R) is successful. Some examples have also revealed that the BFM possesses higher accuracy and is less sensitive to the coarseness of the mesh than the BEM.     

A new boundary element method formulation for three dimensional problems in linear elasticity

Summary A new boundary element method (BEM) formulation for planar problems of linear elasticity has been proposed recently [6]. This formulation uses a kernel which has a weaker singularity relative to the corresponding kernel in the standard formulation. The most important advantage of the new formulation, relative to the standard one, is that it delivers stresses accurately at internal points that are extremely close to the boundary of a body. A corresponding BEM formulation for three dimensional problems of linear elasticity is presented in this paper. This formulation is derived through the use of Stokes' theorem and has kernels which are only 1/r singular (wherer is the distance between a source and a field point) for the displacement equation. The standard BEM formulation for three-dimensional elasticity problems has a kernel which is 1/r ² singular.With 2 Figures     

Adaptive hp-versions of boundary element methods for elastic contact problems

The variational formulation of elastic contact problems leads to variational inequalities on convex subsets. These variational inequalities are solved with the boundary element method (BEM) by making use of the Poincaré–Steklov operator. This operator can be represented in its discretized form by the Schur-complement of the dense Galerkin-matrices for the single layer potential operator, the double layer potential operator and the hypersingular integral operator. Due to the difficulties in discretizing the convex subsets involved, traditionally only the h-version is used for discretization. Recently, p- and hp-versions have been introduced for Signorini contact problems in Maischak and Stephan (Appl Numer Math, 2005) . In this paper we show convergence for the quasi-uniform hp-version of BEM for elastic contact problems, and derive a-posteriori error estimates together with error indicators for adaptive hp-algorithms. We present corresponding numerical experiments.     

A parallel implementation of the ghost-cell immersed boundary method with application to stationary and moving boundary problems

A modified version of the previously reported ghost-cell immersed boundary method is implemented in parallel environment based on distributed memory allocation. Reconstruction of the flow variables is carried out by the inverse distance weighting technique. Implementation of the normal pressure gradient on the immersed surface is demonstrated. Finite volume method with non-staggered arrangement of variables on a non-uniform cartesian grid is employed to solve the fluid flow equations. The proposed method shows reasonable agreement with the reported results for flow past a stationary sphere, rotating and transversely oscillating circular cylinder.     

The boundary element method for nonlinear problems

In this paper a boundary-only boundary element method (BEM) is developed for solving nonlinear problems. The presented method is based on the analog equation method (AEM). According to this method the nonlinear governing equation is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. The solution of the substitute equation is obtained as a sum of the homogeneous solution and a particular one of the nonhomogeneous. The nonhomogeneous term, which is an unknown fictitious domain source distribution, is approximated by a truncated series of radial base functions. Then, using BEM the field function and its derivatives involved in the governing equation are expressed in terms of the unknown series coefficients, which are established by collocating the equation at discrete points in the interior of the domain. Thus, the presented method becomes a boundary-only method in the sense that only boundary discretization is required. The additional collocation points inside the domain do not spoil the pure BEM character of the method. Numerical results for certain classical nonlinear problems are presented, which validate the effectiveness and the accuracy of the proposed method.     

An anisotropic linear elastic boundary element formulation for masonry wall analysis

This work presents an application of a Boundary Element Method (BEM) formulation for anisotropic body analysis using isotropic fundamental solution. The anisotropy is considered by expressing a residual elastic tensor as the difference of the anisotropic and isotropic elastic tensors. Internal variables and cell discretization of the domain are considered. Masonry is a composite material consisting of bricks (masonry units), mortar and the bond between them and it is necessary to take account of anisotropy in this type of structure. The paper presents the formulation, the elastic tensor of the anisotropic medium properties and the algebraic procedure. Two examples are shown to validate the formulation and good agreement was obtained when comparing analytical and numerical results. Two further examples in which masonry walls were simulated, are used to demonstrate that the presented formulation shows close agreement between BE numerical results and different Finite Element (FE) models.     

A parallel domain decomposition boundary element method approach for the solution of large-scale transient heat conduction problems

This paper develops a parallel domain decomposition Laplace transform BEM algorithm for the solution of large-scale transient heat conduction problems. In order to tackle large problems the original domain is decomposed into a number of sub-domains, and a Laplace transform method is utilized to avoid time marching. A procedure is described which provides a good initial guess for the domain interface temperatures, and an iteration procedure is carried out to satisfy continuity of temperature and heat flux at the domain interfaces. The decomposition procedure significantly reduces the size of any single problem for the BEM, which significantly reduces the overall storage and computational burden and renders the application of the BEM to large transient conduction problems on modest computational platforms. The procedure is readily implemented in parallel and applicable to 3D problems. Moreover, as the approach described herein readily allows adaptation and integration of traditional BEM codes, it is expected that the domain decomposition approach coupled to parallel implementation should prove very competitive to alternatives proposed in the literature such as fast multipole acceleration methods that require a complete re-write of traditional BEM codes.     

A boundary element method for solving 3D static gradient elastic problems with surface energy

 A boundary element methodology is developed for the static analysis of three-dimensional bodies exhibiting a linear elastic material behavior coupled with microstructural effects. These microstructural effects are taken into account with the aid of a simple strain gradient elastic theory with surface energy. A variational statement is established to determine all possible classical and non-classical (due to gradient with surface energy terms) boundary conditions of the general boundary value problem. The gradient elastic fundamental solution with surface energy is explicitly derived and used to construct the boundary integral equations of the problem with the aid of the reciprocal theorem valid for the case of gradient elasticity with surface energy. It turns out that for a well posed boundary value problem, in addition to a boundary integral representation for the displacement, a second boundary integral representation for its normal derivative is also necessary. All the kernels in the integral equations are explicitly provided. The numerical implementation and solution procedure are provided. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. Advanced algorithms are presented for the accurate and efficient numerical computation of the singular integrals involved. Two numerical examples are presented to illustrate the method and demonstrate its merits. Received: 9 November 2001 / Accepted: 20 June 2002 The first and third authors gratefully acknowledge the support of the Karatheodory program for basic research offered by the University of Patras.     

Simple boundary element method for three-dimensional magnetostatic problems

For numerical solution of three-dimensional magnetostatic problems the applications of finite-difference (FDM) and finite-element (FEM) methods require long calculating time and large storage capacity. In some cases these requirements can be reduced by the use of boundary element methods (BEM). A simple boundary element solution, and its application to the calculation of magnetization and stray fields of magnetic bodies, is described. The results are compared with experimental stray-field measurements performed using a vibrating pick-up loop magnetometer with high geometrical resolution [14].     

Least squares element method for boundary eigenvalue problems

Linear and non-linear boundary eigenvalue problems are discretized by a new finite element like method. The reason for the new construction principle is the non-linear dependence of the dynamic stiffness element matrix on an eigenparameter. The dynamic stiffness element matrix is evaluated for a fixed number of parameters and is then elementwise replaced by a polynomial in the eigenparameter by solving least squares problems. A fast solver is introduced for the resulting non-linear matrix eigenvalue problem. It consists of a combination of bisection method and inverse iteration. The superiority of the newconstructionprinciple in comparison with the finite or dynamic element method is demonstrated finally for some numerical examples.     

A semi-analytic boundary element method for parabolic problems

A new semi-analytic solution method is proposed for solving linear parabolic problems using the boundary element method. This method constructs a solution as an eigenfunction expansion using separation of variables. The eigenfunctions are determined using the dual reciprocity boundary element method. This separation of variables-dual reciprocity method (SOV-DRM) allows a solution to be determined without requiring either time-stepping or domain discretisation. The accuracy and computational efficiency of the SOV-DRM is found to improve as time increases. These properties make the SOV-DRM an attractive technique for solving parabolic problems.     

Probabilistic crack growth analyses using a boundary element model: Applications in linear elastic fracture and fatigue problems

This paper addresses the numerical solution of random crack propagation problems using the coupling boundary element method (BEM) and reliability algorithms. Crack propagation phenomenon is efficiently modelled using BEM, due to its mesh reduction features. The BEM model is based on the dual BEM formulation, in which singular and hyper-singular integral equations are adopted to construct the system of algebraic equations. Two reliability algorithms are coupled with BEM model. The first is the well known response surface method, in which local, adaptive polynomial approximations of the mechanical response are constructed in search of the design point. Different experiment designs and adaptive schemes are considered. The alternative approach direct coupling, in which the limit state function remains implicit and its gradients are calculated directly from the numerical mechanical response, is also considered. The performance of both coupling methods is compared in application to some crack propagation problems. The investigation shows that direct coupling scheme converged for all problems studied, irrespective of the problem nonlinearity. The computational cost of direct coupling has shown to be a fraction of the cost of response surface solutions, regardless of experiment design or adaptive scheme considered.     

A regularized collocation boundary element method for linear poroelasticity

This article presents a collocation boundary element method for linear poroelasticity, based on the first boundary integral equation with only weakly singular kernels. This is possible due to a regularization of the strongly singular double layer operator, based on integration by parts, which has been applied to poroelastodynamics for the first time. For the time discretization the convolution quadrature method (CQM) is used, which only requires the Laplace transform of the fundamental solution. Furthermore, since linear poroelasticity couples a linear elastic with an acoustic material, the spatial regularization procedure applied here is adopted from linear elasticity and is performed in Laplace domain due to the before mentioned CQM. Finally, the spatial discretization is done via a collocation scheme. At the end, some numerical results are shown to validate the presented method with respect to different temporal and spatial discretizations.     

A study of the factorization fill-in for a parallel implementation of the finite element method

In this paper we investigate the additional storage overhead needed for a parallel implementation of finite element applications. In particular, we compare the storage requirements for the factorization of the sparse matrices that would occur on a parallel processor vs. a uniprocessor. This variation in storage results from the factorization fill-in. We address the question of whether the storage overhead is so large for parallel implementations that it imposes severe limitations on the problem size in contrast to the problems executed sequentially on a uniprocessor. The storage requirements for the parallel implementation are based upon a new ordering scheme, the combination mesh-based scheme. This scheme uses a domain decomposition method which attempts to balance the processors' loads and decreases the interprocessor communication. The storage requirements for the sequential implementation is based upon the minimum degree algorithm. The difference between the two storage requirements corresponds to the storage overhead attributed to the parallel scheme. Experiments were conducted on regular and irregular, 2-D and 3-D problems. The meshes were decomposed into 2–256 subdomains which can be executed on 2–256 processors, respectively. The total storage requirements or fill-in for most of the 2-D problems were less than a factor of two increase over the sequential execution. In contrast, large 3-D problems had zero increase in storage or fill-in over the sequential execution; the fill-in was less for the parallel execution than the sequential execution. Thus, we conclude that the storage overhead attributed to the use of parallel processors will not impose severe constraints on the problem size. Further, for large 3-D applications, the combination mesh-based algorithm does better than minimum degree for reducing the fill-in.     

The least-squares meshfree method for solving linear elastic problems

 A meshfree method based on the first-order least-squares formulation for linear elasticity is presented. In the authors' previous work, the least-squares meshfree method has been shown to be highly robust to integration errors with the numerical examples of Poisson equation. In the present work, conventional formulation and compatibility-imposed formulation for linear elastic problems are studied on the convergence behavior of the solution and the robustness to the inaccurate integration using simply constructed background cells. In the least-squares formulation, both primal and dual variables can be approximated by the same function space. This can lead to higher rate of convergence for dual variables than Galerkin formulation. In general, the incompressible locking can be alleviated using mixed formulations. However, in meshfree framework these approaches involve an additional use of background grids to implement lower approximation space for dual variables. This difficulty is avoided in the present method and numerical examples show the uniform convergence performance in the incompressible limit. Therefore the present method has little burden of the requirement of background cells for the purposes of integration and alleviating the incompressible locking. Received: 16 December 2001 / Accepted: 4 November 2002     

On the optimal implementation of the boundary element dual reciprocity method—multi-domain approach for 3D problems

In this paper, aspects regarding implementation of the boundary element dual reciprocity method—multi-domain approach (DRM-MD), in respect to 3D problems are reviewed. Results of numerical tests on a 3D advection–diffusion problem with non-uniform velocity field are presented. The sensitivity of the accuracy and stability of the codes to the continuity of the elements, scaling, internal DRM nodes and mesh refinement have been tested. The results show that scaling is essential and that mesh refinement and/or internal DRM nodes improve the accuracy when the non-homogeneous term of the governing equation becomes dominant. The computer code implemented with discontinuous elements offers higher accuracy, especially for advection dominant transport, but is much slower than the computer code with continuous elements. At the present stage, the discontinuous element code has the advantage of flexibility since it can solve non-homogeneous domains.