An advanced boundary element method for elasticity problems in nonhomogeneous media

Summary This paper presents a new boundary element method formulation and numerical implementation of elasticity problems in nonhomogeneous media. The fundamental solutions for elasticity in homogeneous media are employed and the nonsingular formulation is derived. A physically and mathematically meaningful elimination of internal degrees of freedom is proposed. The solution at an arbitrary point is expressed in terms of boundary displacements and tractions. The rank of the system matrix (for computation of relevant unknowns) is dependent only on the discretization of the boundary.     

T-stress solutions for two-dimensional crack problems in anisotropic elasticity using the boundary element method

The importance of a two‐parameter approach in the fracture mechanics analysis of many cracked components is increasingly being recognized in engineering industry. In addition to the stress intensity factor, the T stress is the second parameter considered in fracture assessments. In this paper, the path‐independent mutual M‐integral method to evaluate the T stress is extended to treat plane, generally anisotropic cracked bodies. It is implemented into the boundary element method for two‐dimensional elasticity. Examples are presented to demonstrate the veracity of the formulations developed and its applicability. The numerical solutions obtained show that material anisotropy can have a significant effect on the T stress for a given cracked geometry.     

A dual-reciprocity boundary element method for axisymmetric thermoelastostatic analysis of nonhomogeneous materials

The problem of determining the axisymmetric time-independent temperature and thermoelastic displacement and stress fields in a nonhomogeneous material is solved numerically by using a dual-reciprocity boundary element technique. Interpolating functions that are bounded in the solution domain but that are in relatively simple elementary forms for easy computation are constructed for treating the domain integrals in the dual-reciprocity boundary element formulation. The proposed numerical approach is successfully applied to solve several specific problems.     

Dual boundary element method for anisotropic dynamic fracture mechanics

In this work, the dual boundary element method formulation is developed for effective modelling of dynamic crack problems. The static fundamental solutions are used and the domain integral, which comes from the inertial term, is transformed into boundary integrals using the dual reciprocity technique. Dynamic stress intensity factors are computed from crack opening displacements. Comparisons are made with quasi‐isotropic as well as anisotropic results, using the sub‐region technique. Several examples are presented to assess the accuracy and efficiency of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.     

A fast dual boundary element method for 3D anisotropic crack problems

In the present paper a fast solver for dual boundary element analysis of 3D anisotropic crack problems is formulated, implemented and tested. The fast solver is based on the use of hierarchical matrices for the representation of the collocation matrix. The admissible low rank blocks are computed by adaptive cross approximation (ACA). The performance of ACA against the accuracy of the adopted computational scheme for the evaluation of the anisotropic kernels is investigated, focusing on the balance between the kernel representation accuracy and the accuracy required for ACA. The system solution is computed by a preconditioned GMRES and the preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. The effectiveness of the proposed technique for anisotropic crack problems has been numerically demonstrated, highlighting the accuracy as well as the significant reduction in memory storage and analysis time. In particular, it has been numerically shown that the computational cost grows almost linearly with the number of degrees of freedom, obtaining up to solution speedups of order 10 for systems of order 10⁴. Moreover, the sensitivity of the performance of the numerical scheme to materials with different degrees of anisotropy has been assessed. Copyright © 2009 John Wiley & Sons, Ltd.     

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     

A two-dimensional time-domain boundary element method for dynamic crack problems in anisotropic solids

A time-domain boundary element method (BEM) for transient dynamic crack analysis in two-dimensional, homogeneous, anisotropic and linear elastic solids is presented in this paper. Strongly singular displacement boundary integral equations (DBIEs) are applied on the external boundary of the cracked body while hypersingular traction boundary integral equations (TBIEs) are used on the crack-faces. The present time-domain method uses the quadrature formula of Lubich for approximating the convolution integrals and a collocation method for the spatial discretization of the time-domain boundary integral equations. Strongly singular and hypersingular integrals are dealt with by a regularization technique based on a suitable variable change. Discontinuous quadratic quarter-point elements are implemented at the crack-tips to capture the local square-root-behavior of the crack-opening-displacements properly. Numerical examples for computing the dynamic stress intensity factors are presented and discussed to demonstrate the accuracy and the efficiency of the present method.     

A dual-reciprocity boundary element method for a class of elliptic boundary value problems for non-homogeneous anisotropic media

A dual-reciprocity boundary element method is proposed for the numerical solution of a two-dimensional boundary value problem (BVP) governed by an elliptic partial differential equation with variable coefficients. The BVP under consideration has applications in a wide range of engineering problems of practical interest, such as in the calculation of antiplane stresses or temperature in non-homogeneous anisotropic media. The proposed numerical method is applied to solve specific test problems.     

Applying wavelets to improve the boundary element method for elasticity problems

This paper describes application of fast wavelet transforms in the boundary element method to solve 2D elasticity problems. Daubechies compactly supported orthogonal wavelets have been applied to compress dense and fully populated matrices arising from BEM. GMRES solver is then used to solve linear algebraic systems. A comprehensive sensitivity study is presented to answer the questions like, which order and level of D-wavelets and thresholding parameters are efficient for elasticity problems. Numerical results include a precise study on effect of applying different wavelet orders (D4, D6, D8, D10 and D12), levels and thresholds on the solution accuracy for displacements and stresses and compression ratio of sparsified matrices. The suitable order, level and thresholding parameter as well as saving in computer time and memory are presented for practical engineering problems. The results show that the proposed method is efficient for large problems.     

A complex variable boundary element method for solving plane and plate problems of elasticity

We consider the complex variable boundary element approximation of biharmonic problem on a smooth domain with various boundary conditions. Based on the Vekua's complex integral representation of the analytic function, a new boundary integral equation is formulated. The density function appearing in the integral equation is determined directly by using the boundary element method. Some plane and plate examples are presented, and the results of the numerical solutions are accurate everywhere in the solid, including the regions near the boundary.

The approach presented is only suitable for bounded simply connected regions.     

Boundary element solution for half-space elasticity or stokes problem with a no-slip boundary

We report an implementation of the Boundary Element Method (BEM) for half-space elasticity or Stokes problems with a plane interface (the boundary of the half space). With a proper choice of the singularity solution this plane interface, on which the displacement or velocity vector is zero, does not need to be discretized. For a large class of problems involving translating or rotating bodies a simplification of the boundary element formulation is possible, with a resulting improvement in the accuracy of the numerical results. The three-dimensional boundary element program was tested with the moving sphere problem and was found to be satisfactory in all cases.     

A comparison of boundary element method formulations for steady state anisotropic heat conduction problems

In this paper the boundary element method (BEM) is numerically implemented in order to solve steady state anisotropic heat conduction problems. Various types of elements, namely, constant elements, continuous and discontinuous linear elements and continuous and discontinuous quadratic elements are used. The performances of these various BEM formulations are compared for both the direct well-posed Dirichlet problem and the inverse ill-posed Cauchy problem, revealing several features of the BEM. Furthermore, previously undetermined analytical solutions for the integrals associated with linear and quadratic elements are presented.     

Complex variables boundary element method for elasticity problems with constant body force

The direct formulation of the complex variables boundary element method is generalized to allow for solving problems with constant body forces. The hypersingular integral equation for two-dimensional piecewise homogeneous medium is presented and the numerical solution is described. The technique can be used to solve a wide variety of problems in engineering. Several examples are presented to verify the approach and to demonstrate its key features. The results of calculations performed with the proposed approach are compared with available analytical and numerical benchmark solutions.     

A semi‐analytical curved element for linear elasticity based on the scaled boundary finite element method

This work introduces a semi‐analytical formulation for the simulation and modeling of curved structures based on the scaled boundary finite element method (SBFEM). This approach adapts the fundamental idea of the SBFEM concept to scale a boundary to describe a geometry. Until now, scaling in SBFEM has exclusively been performed along a straight coordinate that enlarges, shrinks, or shifts a given boundary. In this novel approach, scaling is based on a polar or cylindrical coordinate system such that a boundary is shifted along a curved scaling direction. The derived formulations are used to compute the static and dynamic stiffness matrices of homogeneous curved structures. The resulting elements can be coupled to general SBFEM or FEM domains. For elastodynamic problems, computations are performed in the frequency domain. Results of this work are validated using the global matrix method and standard finite element analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

A complex variable boundary element method for antiplane stress analysis around a crack in some nonhomogeneous bodies

Abstract

A boundary element method based on the Cauchy integral formulae, i.e. a complex variable boundary element method (CVBEM), is proposed for the numerical solution of an antiplane crack problem involving an elastic body with shear modulus that varies continuously in space. The shear modulus assumes a certain form which is quite general to allow for multiparameter fitting of its variation. The method reduces the problem to a system of linear algebraic equations and can be readily implemented on the computer. For clarity, the CVBEM formulation is first carried out for a straight crack and then its extension to include an arbitrary curved crack is indicated.     

The multiple Reciprocity boundary element method in elasticity: A new approach for transforming domain integrals to the boundary

This paper presents a new boundary element approach to transform domain integrals into equivalent boundary integrals. The technique, called the Multiple Reciprocity Method, is applied to 2-D elasticity problems and operates on domain integrals resulting from different types of body forces such as gravitational and centrifugal forces, as well as loadings due to linear and quadratic temperature distributions. Numerical examples are presented to demonstrate the accuracy and efficiency of the method.     

Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties

This paper presents the local boundary integral formulation for an elastic body with nonhomogeneous material properties. All nodal points are surrounded by a simple surface centered at the collocation point. Only one nodal point is included in each the sub-domain. On the surface of the sub-domain, both displacements and traction vectors are unknown generally. If a modified fundamental solution, for governing equation, which vanishes on the local boundary is chosen, the traction value is eliminated from the local boundary integral equations for all interior points. For every sub-domain, the material constants correspond to those at the collocation point at the center of sub-domain. Meshless and polynomial element approximations of displacements on the local boundaries are considered in the numerical analysis. Received 16 August 1999     

Modal analysis of anisotropic plates using the boundary element method

A numerical formulation for analysis of dynamic problems of thin anisotropic plates bending is presented. The bending behavior follows Kirchhoff's hypothesis. The formulation is based on the direct boundary element method. The problem is simplified by using the elastostatic fundamental solution of an infinite plate. Domain integrals arising from inertial terms are transformed into boundary integrals using the dual reciprocity technique. Boundary integrals are discretized and evaluated numerically. Natural frequencies for free vibration are obtained and the respective mode shapes are shown. The accuracy of numerical results obtained is assured by comparison with analytical or finite element results.     

A Galerkin boundary contour method for two-dimensional linear elasticity

The Boundary Contour Method (BCM) is a recent variant of the Boundary Element Method (BEM) resting on the use of boundary approximations which a-priori satisfy the field equations. For two-dimensional problems, the evaluation of all the line-integrals involved in the collocation BCM reduces to function evaluations at the end-points of each element, thus completely avoiding numerical integrations. With reference to 2-D linear elasticity, this paper develops a variational version of BCM by transferring to the BCM context the ingredients which characterize the Galerkin-Symmetric BEM (GSBEM). The method proposed herein requires no numerical integrations: all the needed double line-integrals over boundary elements pairs can be evaluated by generating appropriate “potential functions” (in closed form) and computing their values at the element end-points. This holds for straight as well as curved elements; however the coefficient matrix of the equation system in the boundary unknowns turns out to be fully symmetric only when all the elements are straight. The numerical results obtained for some benchmark problems, for which analytical solutions are available, validate the proposed formulation and the corresponding solution procedure.     

Complex fundamental solutions and complex variables boundary element method in elasticity

The CVBEM for elasticity problems in its basic formulations (direct, indirect and displacement discontinuity (DD) ones) is presented. It is based on the use of complex fundamental solutions. The complex boundary integral equations (CBIEs) arising from the basic CVBEM formulations are considered. It was shown that the developed theory includes the main CBIEs obtained by using the different approaches. Besides, new real and complex integral equations were obtained. The revealed links between real variables BEM (RVBEM) and CVBEM serve to mutual enrichment of these two approaches.