Dislocation and point-force-based approach to the special Green's Function BEM for elliptic hole and crack problems in two dimensions

In this paper we give the theoretical foundation for a dislocation and point-force-based approach to the special Green's function boundary element method and formulate, as an example, the special Green's function boundary element method for elliptic hole and crack problems. The crack is treated as a particular case of the elliptic hole. We adopt a physical interpretation of Somigliana's identity and formulate the boundary element method in terms of distributions of point forces and dislocation dipoles in the infinite domain with an elliptic hole. There is no need to model the hole by the boundary elements since the traction free boundary condition there for the point force and the dislocation dipole is automatically satisfied. The Green's functions are derived following the Muskhelishvili complex variable formalism and the boundary element method is formulated using complex variables. All the boundary integrals, including the formula for the stress intensity factor for the crack, are evaluated analytically to give a simple yet accurate special Green's function boundary element method. The numerical results obtained for the stress concentration and intensity factors are extremely accurate. © 1997 John Wiley & Sons, Ltd.     

Green's functions for the Laplace equation in a 3-layer medium,boundary element integrals and their application to cathodic protection

We report in this paper a set of nine Green's functions for the Laplace equation for an infinite 3-layer medium in which a layer of finite width is sandwiched between two semi-infinite domains. Typical 3D plots of these Green's functions are computed and presented. Taking an offshore platform as a prime example of a structure in a 3-layer medium (atmosphere, ocean and soil), we work out the boundary element integrals using macro elements such as the tubulars. Constant elements reduce several of these boundary integrals to analytical forms. As an application, we discuss the cathodic protection modelling of offshore structures using the ‘boundary element method’.     

Boundary element modeling of cracks in piezoelectric solids

This study focuses on the application of boundary element methods for linear fracture mechanics of two-dimensional piezoelectric solids. A complete set of piezoelectric Green's functions, based on the extended Lekhnitskii's formalism and distributed dislocation modeling, are presented. Special Green's functions are obtained for an infinite medium containing a conducting crack or an impermeable crack. The numerical solution of the boundary integral equation and the computation of fracture parameters are discussed. The concept of crack closure integral is utilized to calculate energy release rates. Accuracy of the boundary element solutions is confirmed by comparing with analytical solutions reported in the literature. The present scheme can be applied to study complex cracks such as branched cracks, forked cracks and microcrack clusters.     

Finite element computation of Green's functions

Green's functions are important mathematical tools in mechanics and in other parts of physics. For instance, the boundary element method needs to know the Green's function of the problem to compute its numerical solution. However, Green's functions are only known in a limited number of cases, often under the form of complex analytical expressions. In this article, a new method is proposed to calculate Green's functions for any linear homogeneous medium from a simple finite element model. The method relies on the theory of wave propagation in periodic media and requires the knowledge of the finite element dynamic stiffness matrix of only one period. Several examples are given to check the accuracy and the efficiency of the proposed numerical Green's function.     

Heritage and early history of the boundary element method

This article explores the rich heritage of the boundary element method (BEM) by examining its mathematical foundation from the potential theory, boundary value problems, Green's functions, Green's identities, to Fredholm integral equations. The 18th to 20th century mathematicians, whose contributions were key to the theoretical development, are honored with short biographies. The origin of the numerical implementation of boundary integral equations can be traced to the 1960s, when the electronic computers had become available. The full emergence of the numerical technique known as the boundary element method occurred in the late 1970s. This article reviews the early history of the boundary element method up to the late 1970s.     

Maurice Jaswon and boundary element methods

In the direct boundary integral equation method, boundary-value problems are reduced to integral equations by an application of Green's theorem to the unknown function and a fundamental solution (Green's function). Discretization of the integral equation then leads to a boundary element method. This approach was pioneered by Jaswon and his students in the early 1960s. Jaswon's work is reviewed together with his influence on later workers.     

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.     

A numerical green's function approach for boundary elements applied to fracture mechanics

The most accurate boundary element formulation to deal with fracture mechanics problems is obtained with the implementation of the associated Green's function acting as the fundamental solution. Consequently, the range of applications of this formulation is dependent on the availability of the appropriate Green's function for actual crack geometry. Analytical Green's functions have been presented for a few single crack configurations in 2-D applications and require complex variable theory. This work extends the applicability of the formulation through the introduction of efficient numerical means of computing the Green's function components for single or multiple crack problems, of general geometry, including the implementation to 3-D problems as a future development. Also, the approach uses real variables only and well-established boundary integral equations.     

2D Green's functions of defective magnetoelectroelastic solids under thermal loading

Thermomagnetoelectroelastic problems for various defects embedded in an infinite matrix are considered in this paper. Using Stroh's formalism, conformal mapping, and perturbation technique, Green's functions are obtained in closed form for a defect in an infinite magnetoelectroelastic solid induced by the thermal analog of a line temperature discontinuity and a line heat source. The defect may be of an elliptic hole or a Griffith crack, a half-plane boundary, a bimaterial interface, or a rigid inclusion. These Green's functions satisfy the relevant boundary or interface conditions. The proposed Green's functions can be used to establish boundary element formulation and to analyzing fracture behaviour due to the defects mentioned above.     

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.     

An algorithm based on the boundary element method for problems in engineering mechanics

The solution to a two-dimensional problem using the boundary element method requires the evaluation of a line integral over the boundary. The integrand ot this line integral is a product of a known Green's function and an unknown function. A large number of Green's functions for two-dimensional problems can be represented by a linear combination of four singular functions. By approximating the unknown function by a linear combination of known polynomials, integrals are generated whose integrand is a product of the polynomiais and one of the four singular functions. To evaluate these integrals analytically, the boundary is approximated by a sum of straight-line segments. Recursive formulae are established which reduce the generality and the complexity of the integrands to simple expressions. Analytical forms for these simple expressions are found and are used for initiating the algorithm.     

Comparison of boundary and finite element methods for moving-boundary problems governed by a potential

Three formulations of the boundary element method (BEM) and one of the Galerkin finite element method (FEM) are compared according to accuracy and efficiency for the spatial discretization of two-dimensional, moving-boundary problems based on Laplace's equation. The same Euler-predictor, trapezoid-corrector scheme for time integration is used for all four methods. The model problems are on either a bounded or a semi-infinite strip and are formulated so that closed-form solutions are known. Infinite elements are used with both the BEM and FEM techniques for the unbounded domain. For problems with the bounded region, the BEM using the free-space Green's function and piecewise quadratic interpolating functions (QBEM) is more accurate and efficient than the BEM with linear interpolation. However, the FEM with biquadratic basis functions is more efficient for a given accuracy requirement than the QBEM, except when very high accuracy is demanded. For the unbounded domain, the preferred method is the BEM based on a Green's function that satisfies the lateral symmetry conditions and which leads to discretization of the potential only along the moving surface. This last formulation is the only one that reliably satisfies the far-field boundary condition.     

Extended displacement discontinuity Green's functions for three-dimensional transversely isotropic magneto-electro-elastic media and applications

A versatile method is presented to derive the extended displacement discontinuity Green's functions or fundamental solutions by using the integral equation method and the Green's functions of the extended point forces. In particular, the three-dimensional (3D) transversely isotropic magneto-electro-elastic problem is used to demonstrate the method. On this condition, the extended displacement discontinuities include the elastic displacement discontinuities, the electric potential discontinuity and the magnetic potential discontinuity, while the extended forces include the point forces, the point electric charge and the point electric current. Based on the obtained Green's functions, the extended Crouch fundamental solutions are derived and an extended displacement discontinuity method is developed for analysis of cracks in 3D magneto-electro-elastic media. The extended intensity factors of two coplanar and parallel rectangular cracks are calculated under impermeable boundary condition to illustrate the application, accuracy and efficiency of the proposed method.     

3-D and 2-D Dynamic Green's functions and time-domain BIEs for piezoelectric solids

Dynamic Green's functions for linear piezoelectric solids are derived by using Radon transform. Time-harmonic and Laplace transformed dynamic Green's functions are obtained subsequently by applying the Fourier and the Laplace transform to the time-domain Green's functions. Time-domain boundary integral equation formulations are presented for transient dynamic analysis of linear piezoelectric solids. In particular, hypersingular and non-hypersingular time-domain traction BIEs are derived by two different ways. Their potential application in transient dynamic crack analysis of three-dimensional and two-dimensional piezoelectric solids is discussed.     

Numerical Green's functions for some electroelastic crack problems

A plane electroelastic problem involving planar cracks in a piezoelectric body is considered. The deformation of the body is assumed to be independent of time and one of the Cartesian coordinates. The cracks are traction free and are electrically either permeable or impermeable. Numerical Green's functions which satisfy the boundary conditions on the cracks are derived using the hypersingular integral approach and applied to obtain a boundary integral solution for the electroelastic crack problem considered here. As the conditions on the cracks are built into the Green's functions, the boundary integral solution does not contain integrals over the cracks. It is used to derive a boundary element procedure for computing the crack tip stress and electrical displacement intensity factors.     

A general regularization of the hypersingular integrals in the symmetric Galerkin boundary element method

The symmetric Galerkin boundary element method is used to solve boundary value problems by keeping the symmetric nature of the matrix obtained after discretization. The matrix elements are obtained from a double integral involving the double derivative of Green's operator, which is highly singular. The paper presents a regularization of the hypersingular integrals which depend only on the properties of Green's tensor. The method is presented in the case of Laplace's operator, with an example of application. The case of elasticity is finally addressed theoretically, showing an easy extension to any case of anisotropy. Copyright © 2009 John Wiley & Sons, Ltd.     

STRESS INTENSITY FACTORS FOR CRACKS IN NOTCHED FINITE PLATES SUBJECTED TO BIAXIAL LOADING

Stress intensity factors were calculated, based on Bueckner's principle for cracks in both infinite and finite plates with notches subjected to biaxial loading. Approximate Green's functions have been obtained by modifying two existing Green's functions, originally for unnotched plates. Values of stress intensity factors calculated using Bueckner's principle with the approximate Green's functions are in good agreement with published stress intensity factors for cracks in both infinite and finite plates containing a circular notch or an elliptical notch, previously found by the method of boundary collocation.     

Efficient evaluation of three-dimensional Green's functions in anisotropic elastostatic multilayered composites

The three-dimensional Green's functions in anisotropic elastostatic multilayered composites (MLCs) obtained within the framework of generalized Stroh formalism are expressed as two-dimensional integrals of Fourier inverse transform over an infinite plane. Their numerical evaluations involve tremendous computational efforts in particular in the presence of various singularities and near-singularities due to the presence of material mismatches across interfaces. The present paper derives the complete set of the Green's functions including displacement, stress and their derivatives with respect to source coordinates using a novel and computationally efficient approach. It is proposed for the first time that the Green's functions in the MLCs are expressed as a sum of a special solution and a general-part solution, with the former consisting of the first few terms of the trimaterial expansion solution around a source load. Since the zero-order term contains the singularity corresponding to the homogeneous full-space solution and can be evaluated analytically, and the other higher-order terms contain most of the near-singular behaviors and can be reduced to a line integral over a finite interval, the general-part solution becomes regular and the Green's functions overall can be evaluated efficiently. As an example, the Green's functions in a five-layered orthortropic plate are evaluated to demonstrate the efficiency of the proposed approach. Also, the detailed characteristics of these Green's functions are examined in both the transform- and physical-domains. These Green's functions are essential in developing the boundary-integral-equation formulation and numerical boundary element method for composite laminate problems involving regular and cracked geometries.     

Boundary element analysis for viscoelastic solids containing interfaces/holes/cracks/inclusions

With the aid of the elastic–viscoelastic correspondence principle, the boundary element developed for the linear anisotropic elastic solids can be applied directly to the linear anisotropic viscoelastic solids in the Laplace domain. Green's functions for the problems of two-dimensional linear anisotropic elastic solids containing holes, cracks, inclusions, or interfaces have been obtained analytically using Stroh's complex variable formalism. Through the use of these Green's functions and the correspondence principle, special boundary elements in the Laplace domain for viscoelastic solids containing holes, cracks, inclusions, or interfaces are developed in this paper. Subregion technique is employed when multiple holes, cracks, inclusions, and interfaces exist simultaneously. After obtaining the physical responses in Laplace domain, their associated values in time domain are calculated by the numerical inversion of Laplace transform. The main feature of this proposed boundary element is that no meshes are needed along the boundary of holes, cracks, inclusions and interfaces whose boundary conditions are satisfied exactly. To show this special feature by comparison with the other numerical methods, several examples are solved for the linear isotropic viscoelastic materials under plane strain condition. The results show that the present BEM is really more efficient and accurate for the problems of viscoelastic solids containing interfaces, holes, cracks, and/or inclusions.     

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.