A Green's function time‐domain boundary element method for the elastodynamic half‐plane

The transient Green's function of the 2‐D Lamb's problem for the general case where point source and receiver are situated beneath the traction‐free surface is derived. The derivations are based on Laplace‐transform methods, utilizing the Cagniard–de Hoop inversion. The Green's function is purely algebraic without any integrals and is presented in a numerically applicable form for the first time. It is used to develop a Green's function BEM in which surface discretizations on the traction‐free boundary can be saved. The time convolution is performed numerically in an abstract complex plane. Hence, the respective integrals are regularized and only a few evaluations of the Green's function are required. This fast procedure has been applied for the first time. The Green's function BEM developed proved to be very accurate and efficient in comparison with analogue BEMs that employ the fundamental solution. Copyright © 1999 John Wiley & Sons, Ltd.     

Multiple pole residue approach for 3D BEM analysis of mathematical degenerate and non‐degenerate materials

In this paper we develop an alternative boundary element method (BEM) formulation for the analysis of anisotropic three‐dimensional (3D) elastic solids. Our implementation is based on the derivation of explicit expressions for the fundamental solution displacements and tractions, of general validity for any class of anisotropic materials, by means of Stroh formalism and Cauchy's residue theory. The resulting fundamental solution remains valid for mathematical degenerate cases when Stroh's eigenvalues are coincident, meanwhile it does not exhibit numerical instabilities for quasi‐degenerate cases when Stroh's eigenvalues are nearly equal. A multiple pole residue approach is followed, leading to general explicit expressions to evaluate the traction fundamental solution for poles of m‐multiplicity. Despite the existence of general displacement solutions in the literature, and for the sake of completeness, the same approach as for the traction solution is considered to derive the displacement fundamental solution as well. Based on these solutions, an explicit BEM approach for the numerical solution of 3D linear elastic problems for solids with general anisotropic behavior is presented. The analysis of cracked anisotropic solids is also considered. Details on the numerical implementation and its validation for degenerate cases are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Wave propagation in a simplified modelled poroelastic continuum: fundamental solutions and a time domain boundary element formulation

In finite element formulations for poroelastic continua a representation of Biot's theory using the unknowns solid displacement and pore pressure is preferred. Such a formulation is possible either for quasi‐static problems or for dynamic problems if the inertia effects of the fluid are neglected. Contrary to these formulations a boundary element method (BEM) for the general case of Biot's theory in time domain has been published (Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach. Lecture Notes in Applied Mechanics. Springer: Berlin, Heidelberg, New York, 2001.). If the advantages of both methods are required it is common practice to couple both methods. However, for such a coupled FE/BE procedure a BEM for the simplified dynamic Biot theory as used in FEM must be developed. Therefore, here, the fundamental solutions as well as a BE time stepping procedure is presented for the simplified dynamic theory where the inertia effects of the fluid are neglected. Further, a semi‐analytical one‐dimensional solution is presented to check the proposed BE formulation. Finally, wave propagation problems are studied using either the complete Biot theory as well as the simplified theory. These examples show that no significant differences occur for the selected material. Copyright © 2005 John Wiley & Sons, Ltd.     

A semi‐implicit time‐stepping model for frictional compliant contact problems

In this paper, we formulate a semi‐implicit time‐stepping model for multibody mechanical systems with frictional, distributed compliant contacts. Employing a polyhedral pyramid model for the friction law and a distributed, linear, viscoelastic model for the contact, we obtain mixed linear complementarity formulations for the discrete‐time, compliant contact problem. We establish the existence and finite multiplicity of solutions, demonstrating that such solutions can be computed by Lemke's algorithm. In addition, we obtain limiting results of the model as the contact stiffness tends to infinity. The limit analysis elucidates the convergence of the dynamic models with compliance to the corresponding dynamic models with rigid contacts within the computational time‐stepping framework. Finally, we report numerical simulation results with an example of a planar mechanical system with a frictional contact that is modelled using a distributed, linear viscoelastic model and Coulomb's frictional law, verifying empirically that the solution trajectories converge to those obtained by the more traditional rigid‐body dynamic model. Copyright © 2004 John Wiley Sons, Ltd.     

Compression of time‐generated matrices in two‐dimensional time‐domain elastodynamic BEM analysis

This paper describes a new scheme to improve the efficiency of time‐domain BEM algorithms. The discussion is focused on the two‐dimensional elastodynamic formulation, however, the ideas presented apply equally to any step‐by‐step convolution based algorithm whose kernels decay with time increase. The algorithm presented interpolates the time‐domain matrices generated along the time‐stepping process, for time‐steps sufficiently far from the current time. Two interpolation procedures are considered here (a large number of alternative approaches is possible): Chebyshev–Lagrange polynomials and linear. A criterion to indicate the discrete time at which interpolation should start is proposed. Two numerical examples and conclusions are presented at the end of the paper. Copyright © 2004 John Wiley & Sons, Ltd.     

The 3‐D BEM implementation of a numerical Green's function for fracture mechanics applications

The use of Green's functions has been considered a powerful technique in the solution of fracture mechanics problems by the boundary element method (BEM). Closed‐form expressions for Green's function components, however, have only been available for few simple 2‐D crack geometry applications and require complex variable theory. The present authors have recently introduced an alternative numerical procedure to compute the Green's function components that produced BEM results for 2‐D general geometry multiple crack problems, including static and dynamic applications. This technique is not restricted to 2‐D problems and the computational aspects of the 3‐D implementation of the numerical Green's function approach are now discussed, including examples. Copyright © 2000 John Wiley & Sons, Ltd.     

The boundary contour method for piezoelectric media with linear boundary elements

This paper presents a development of the boundary contour method (BCM) for piezoelectric media. First, the divergence‐free property of the integrand of the piezoelectric boundary element is proved. Secondly, the boundary contour method formulation is derived and potential functions are obtained by introducing linear shape functions and Green's functions (Computer Methods in Applied Mechanics and Engineering 1998; 158 : 65) for piezoelectric media. The BCM is applied to the problem of piezoelectric media. Finally, numerical solutions for illustrative examples are compared with exact ones and those of the conventional boundary element method (BEM). The numerical results of the BCM coincide very well with the exact solution, and the feasibility and efficiency of the method are verified. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite element stress formulation for wave propagation

Based on a variational principle due to Gurtin, for linear elastodynamics, a finite element method in terms of stresses is developed for wave propagation problems. The finite element equations are simultaneous integral equations in time, with the peculiarity that they are equivalent to simultaneous linear differential equations with zero initial conditions. Written as differential equations, the finite element equations are of the form where [H] is a symmetric positive-semidefinite matrix, [Q] is a symmetric positive-definite matrix and the stresses are represented by {s?}. This is, of course, the same form as the equations for the displacement formulation. As a demonstration of the validity of the formulation numerical results are compared with a solution for a triangular-shaped strip load applied to an elastic half-space as a ramp function of time. This solution is obtained by numerical integration of the exact solution for Lamb's problem of a line load suddenly applied to a half-space. The agreement is found to be generally very good. As a further example, the case of a plate subject to uniform tension on two ends and containing a hole in the centre is presented. The results are found to be reasonable in that the characteristic stress concentration occurs near the hole, and away from the hole the results are similar to the solution for an infinitely wide plate.     

Boundary element methods for boundary condition inverse problems in elasticity using PCGM and CGM regularization

For an isotropic linear elastic body, only displacement or traction boundary conditions are given on a part of its boundary, whilst all of displacement and traction vectors are unknown on the rest of the boundary. The inverse problem is different from the Cauchy problems. All the unknown boundary conditions on the whole boundary must be determined with some interior points' information. The preconditioned conjugate gradient method (PCGM) in combination with the boundary element method (BEM) is developed for reconstructing the boundary conditions, and the PCGM is compared with the conjugate gradient method (CGM). Morozov's discrepancy principle is employed to select the iteration step. The analytical integral algorithm is proposed to treat the nearly singular integrals when the interior points are very close to the boundary. The numerical solutions of the boundary conditions are not sensitive to the locations of the interior points if these points are distributed along the entire boundary of the considered domain. The numerical results confirm that the PCGM and CGM produce convergent and stable numerical solutions with respect to increasing the number of interior points and decreasing the amount of noise added into the input data.     

Solving the advection–diffusion equation on unstructured meshes with discontinuous/mixed finite elements and a local time stepping procedure

Explicit schemes are known to provide less numerical diffusion in solving the advection–diffusion equation, especially for advection‐dominated problems. Traditional explicit schemes use fixed time steps restricted by the global CFL condition in order to guarantee stability. This is known to slow down the computation especially for heterogeneous domains and/or unstructured meshes. To avoid this problem, local time stepping procedures where the time step is allowed to vary spatially in order to satisfy a local CFL condition have been developed. In this paper, a local time stepping approach is used with a numerical model based on discontinuous Galerkin/mixed finite element methods to solve the advection–diffusion equation. The developments are detailed for general unstructured triangular meshes. Numerical experiments are performed to show the efficiency of the numerical model for the simulation of (i) the transport of a solute on highly unstructured meshes and (ii) density‐driven flow, where the velocity field changes at each time step. The model gives stable results with significant reduction of the computational cost especially for the non‐linear problem. Moreover, numerical diffusion is also reduced for highly advective problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Eigenvalue analysis for acoustic problem in 3D by boundary element method with the block Sakurai–Sugiura method

This paper presents accurate numerical solutions for nonlinear eigenvalue analysis of three-dimensional acoustic cavities by boundary element method (BEM). To solve the nonlinear eigenvalue problem (NEP) formulated by BEM, we employ a contour integral method, called block Sakurai–Sugiura (SS) method, by which the NEP is converted to a standard linear eigenvalue problem and the dimension of eigenspace is reduced. The block version adopted in present work can also extract eigenvalues whose multiplicity is larger than one, but for the complex connected region which includes a internal closed boundary, the methodology yields fictitious eigenvalues. The application of the technique is demonstrated through the eigenvalue calculation of sphere with unique homogenous boundary conditions, cube with mixed boundary conditions and a complex connected region formed by cubic boundary and spherical boundary, however, the fictitious eigenvalues can be identified by Burton–Miller's method. These numerical results are supported by appropriate convergence study and comparisons with close form.     

GPU‐accelerated boundary element method for Helmholtz' equation in three dimensions

Recently, the application of graphics processing units (GPUs) to scientific computations is attracting a great deal of attention, because GPUs are getting faster and more programmable. In particular, NVIDIA's GPUs called compute unified device architecture enable highly mutlithreaded parallel computing for non‐graphic applications. This paper proposes a novel way to accelerate the boundary element method (BEM) for three‐dimensional Helmholtz' equation using CUDA. Adopting the techniques for the data caching and the double–single precision floating‐point arithmetic, we implemented a GPU‐accelerated BEM program for GeForce 8‐series GPUs. The program performed 6–23 times faster than a normal BEM program, which was optimized for an Intel's quad‐core CPU, for a series of boundary value problems with 8000–128000 unknowns, and it sustained a performance of 167 Gflop/s for the largest problem (1 058 000 unknowns). The accuracy of our BEM program was almost the same as that of the regular BEM program using the double precision floating‐point arithmetic. In addition, our BEM was applicable to solve realistic problems. In conclusion, the present GPU‐accelerated BEM works rapidly and precisely for solving large‐scale boundary value problems for Helmholtz' equation. Copyright © 2009 John Wiley & Sons, Ltd.     

Initial conditions contribution in a BEM formulation based on the convolution quadrature method

This work presents a two‐dimensional boundary element method (BEM) formulation for the analysis of scalar wave propagation problems. The formulation is based on the so‐called convolution quadrature method (CQM) by means of which the convolution integral, presented in time‐domain BEM formulations, is numerically substituted by a quadrature formula, whose weights are computed using the Laplace transform of the fundamental solution and a linear multistep method. This BEM formulation was initially developed for scalar wave propagation problems with null initial conditions. In order to overcome this limitation, this work presents a general procedure that enables one to take into account non‐homogeneous initial conditions, after replacing the initial conditions by equivalent pseudo‐forces. The numerical results included in this work show the accuracy of the proposed BEM formulation and its applicability to such kind of analysis. Copyright © 2006 John Wiley & Sons, Ltd.     

Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two‐dimensional elasticity

For a plane elasticity problem, the boundary integral equation approach has been shown to yield a non‐unique solution when geometry size is equal to a degenerate scale. In this paper, the degenerate scale problem in the boundary element method (BEM) is analytically studied using the method of stress function. For the elliptic domain problem, the numerical difficulty of the degenerate scale can be solved by using the hypersingular formulation instead of using the singular formulation in the dual BEM. A simple example is shown to demonstrate the failure using the singular integral equations of dual BEM. It is found that the degenerate scale also depends on the Poisson's ratio. By employing the hypersingular formulation in the dual BEM, no degenerate scale occurs since a zero eigenvalue is not embedded in the influence matrix for any case. Copyright © 2002 John Wiley & Sons, Ltd.     

BEM for crack‐inclusion problems of plane thermopiezoelectric solids

The problem of interactions between an inclusion and multiple cracks in a thermopiezoelectric solid is considered by boundary element method (BEM) in this paper. First of all, a BEM for the crack–inclusion problem is developed by way of potential variational principle, the concept of dislocation, and Green's function. In the BE model, the continuity condition of the interface between inclusion and matrix is satisfied, a priori, by the Green's function, and not involved in the boundary element equations. This is then followed by expressing the stress and electric displacement (SED) and elastic displacements and electric potential (EDEP) in terms of polynomials of complex variables ξ_t and ξ_k in the transformed ξ‐plane in order to simulate SED intensity factors by the BEM. The least‐squares method incorporating the BE formulation can, then, be used to calculate SED intensity factors directly. Numerical results for a piezoelectric plate with one inclusion and a crack are presented to illustrate the application of the proposed formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

On a regularized method of fundamental solutions coupled with the numerical Green's function procedure to solve embedded crack problems

The method of fundamental solutions (MFS) is applied to solve linear elastic fracture mechanics (LEFM) problems. The approximate solution is obtained by means of a linear combination of fundamental solutions containing the same crack geometry as the actual problem. In this way, the fundamental solution is the very same one applied in the numerical Green's function (NGF) BEM approach, in which the singular behavior of embedded crack problems is incorporated. Due to severe ill-conditioning present in the MFS matrices generated with the numerical Green's function, a regularization procedure (Tikhonov's) was needed to improve accuracy, stabilization of the solution and to reduce sensibility with respect to source point locations. As a result, accurate stress intensity factors can be obtained by a superposition of the generalized fundamental crack openings. This mesh-free technique presents good results when compared with the boundary element method and estimated solutions for the stress intensity factor calculations.     

Stress analyses around holes in composite laminates using boundary element method

A three-dimensional (3D) boundary element method (BEM) is developed for the analysis of composite laminates with holes. Instead of using Kelvin-type Green's functions of anisotropic infinite space, 3D layered Green's functions with the materials of each layer being generally anisotropic, derived recently in the Fourier transform domain, are implemented into a 3D BEM formulation. A novel numerical algorithm is designed to calculate layered Green's functions efficiently. It should be noted that since layered Green's functions satisfy exactly the continuity conditions along the interfaces and top and bottom free surfaces a priori, the model becomes truly 2D and discretization is only needed along the hole surface and prescribed traction and/or displacement boundaries. To test the validity and accuracy of the proposed method, the present layered BEM formulation is applied to the problem of an infinite anisotropic plate with a circular hole where the analytical solution is available. It is found that even with a very coarse mesh, the present BEM can predict the hoop stress very accurately along the hole surface. The BEM formulation is then applied to analyze two composite laminates (90/0)_s and (−45/45)_s, under a remote in-plane strain, that have been studied previously with different approaches. For the (90/0)_s case, the hoop stresses along the hole surface predicted by the present layered BEM formulation are in very close agreement with the previous results. For the (−45/45)_s case, however, it is found that a nearly converged solution (less than 5% convergence by doubling the mesh) by the present method is at significant variance with the previous ones that are lack-of-convergence checks. It can be expected that for designing the bolted joints of composites with many layers, a computational tool developed based on the present techniques would be robust and offer a much better solution with regard to accuracy, versatility and design cycle time.     

Large deflection analysis of elastic space membranes

In this paper a solution to the problem of elastic space (initially non‐flat) membranes is presented. A new formulation of the governing differential equations is presented in terms of the displacements in the Cartesian co‐ordinates. The reference surface of the membrane is the minimal surface. The problem is solved by direct integration of the differential equations using the analogue equation method (AEM). According to this method the three coupled non‐linear partial differential equations with variable coefficients are replaced with three uncoupled equivalent linear flat membrane equations (Poisson's equations) subjected to unknown sources under the same boundary conditions. Subsequently, the unknown sources are established using a procedure based on the BEM. The displacements as well as the stress resultants are evaluated at any point of the membrane from their integral representations of the solution of the substitute problems, which are used as mathematical formulae. Several membranes are analysed which illustrate the method and demonstrate its efficiency and accuracy as compared with analytical and existing numerical methods. Copyright © 2005 John Wiley & Sons, Ltd.     

The first-kind and the second-kind boundary integral equation systems for solution of frictionless contact problems

An original approach to the numerical solution of displacement boundary integral equation (BIE) and traction hypersingular boundary integral equation (HBIE) by the boundary element method (BEM) for contact problems is given. The main point is to show, how the contact conditions are used to formulate the first-kind and the second-kind BIE systems in the case of frictionless two-body elastic contact. The solution of the first-kind BIE is performed by symmetric Galerkin BEM; the second-kind BIE is solved by an appropriate collocation BEM. The contact problem in itself is solved by the method of subsequent approximations of contact region. Both forms of BIE system are compared in several numerical examples. This comparison is made for different kinds of contact problem. The major emphasis is put on the evaluation of contact pressure. The obtained results are compared with referenced numerical and with the analytical ones.