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.     

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.     

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.     

Nonlinear Green's function method for the Tzitzéica and related equations

In this short note we apply the nonlinear Green's function method for the solution of the Tzitzéica type equation hierarchies arising in nonlinear science. Using the travelling wave ansatz, we first transform the nonlinear partial differential equations to nonlinear ordinary differential equations. Then, we establish a general representation formula for nonlinear Green's function of these equations. Eventually, using Frasca's short time expansion, we obtain the exact solution to these equations. Numerical analysis shows that the obtained Green's function solution is sufficiently close to the numerical solution obtained by the well-known method of lines. Finally, we involve the inverse transform and study the full nature of the Tzitzéica equation.     

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.     

A time domain FEM approach based on implicit Green's functions for non‐linear dynamic analysis

The present paper describes an unconditionally stable algorithm to integrate the equations of motion in time. The standard FEM displacement model is employed to perform space discretization, and the time‐marching process is carried out through an algorithm based on the Green's function of the mechanical system in nodal co‐ordinates. In the present ‘implicit Green's function approach’ (ImGA), mechanical system Green's functions are not explicitly computed; rather they are implicitly considered through an iterative pseudo‐forces process. Under certain simplifying hypothesis, iterations are not necessary and the ImGA becomes cheaper than standard Newmark/Newton–Raphson algorithm. At the end of the paper numerical examples are presented in order to illustrate the accuracy of the present approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Three‐dimensional time‐harmonic Green's functions for a triclinic full‐space using a symbolic computation system

Time‐harmonic Green's functions for a triclinic anisotropic full‐space are evaluated through the use of a symbolic computation system.This procedure allows evaluation of the Green's functions for the most general anisotropic materials. The proposed computational algorithms are programmed in a MATLAB environment by incorporating symbolic calculations performed using Maple Computer Algebra System. Extensive testing of the numerical results has been performed for both displacement and stress fields. The tests demonstrate the accuracy of the proposed algorithm in evaluating the Green's functions. Copyright © 2001 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.     

Numerical evaluation of harmonic Green's functions for triclinic half‐space with embedded sources—Part II: a 3D model

The displacement and stress Green's functions for a 3D triclinic half‐space with embedded harmonic point load is considered. The resulting displacement and stress fields are expressed in terms of triple Fourier integrals. The first integral was evaluated using contour integration and the 3D Green's functions were obtained as a superposition of 2D results over the azimuthal angle. The resulting algorithm developed for evaluation of the Green's functions avoids repeated calculations of the same quantities and it utilizes the vectorized manipulation within MATLAB environment. The algorithm places no restriction on material properties, frequency and location of source and observation points. Extensive testing of the numerical results was performed for both displacement and stress. The tests confirm the accuracy of the numerical results. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical instabilities of the integral approach to the interior boundary-value problem for the two-dimensional Helmholtz equation

The integral equations arising from the Green's formula, applied to the two-dimensional Helmholtz equation defined in a limited domain, are considered and the presence of instabilities in their numerical solution, when a real Green's function is adopted, is pointed out. A complete study has been carried out for a circular domain and the conditions under which such instabilities can occur in a domain of arbitrary geometry have been investigated. In particular, it has been shown that in every case the use of a complex Green's function is able to avoid their presence.     

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.     

Another kind of numerical instabilities of the integral approach to the interior boundary-value problem for the two-dimensional Helmholtz equation

In a previous paper it has been proved by the author that the integral equations arising from the application of Green's formula to the Helmholtz equation in a limited domain can show a certain type of numerical instability, if a real Green's function is used. It has been also proved that such instabilities cannot arise if a complex Green's function is employed. However, it has been found in this latter case also that numerical instabilities can occur. This has been proved and thoroughly analysed for a circular domain, and a technique of avoiding these instabilities has been devised. Furthermore, when this technique is followed, very accurate results can be obtained, regardless of wavenumber used. Thus, only three or four segments are sufficient to describe a wavelength, contrary to what until now has been obtained, i.e. that at least six segments are always necessary. This last result has been shown to be valid also for geometries other than the circular one.     

On the method of functional equations and the performance of desingularized boundary element methods

A correspondence is made between the reciprocal relation for linear elliptic partial differential equations and the Riesz integral representation. The former relates the boundary distributions and appropriate normal fluxes of two arbitrary solutions, and the latter expresses a continuous linear functional in terms of an integral involving a representing function. When sufficient regularity conditions are met, the representing function is identified with the unknown boundary distribution. In principle, the representing function may be expressed in terms of the images of a complete set of orthonormal basis functions with known normal fluxes, as suggested by Kupradze [Kupradze VD. On the approximate solution of problems in mathematical physics. Russ Math Surv 1967; 22: 59–107]; in practice, the representing function is computed by solving integral equations using boundary element methods. The basic procedure involves expressing the representing function in terms of finite-element or other basis functions, and requiring the satisfaction of the reciprocal relationship with a suitable set of test functions such as Green's functions and their dipoles. When the singular points are placed at the boundary, we obtain the standard boundary integral equation method. When the singular points are placed outside the domain of solution, we obtain a system of functional equations and associated class of desingularized boundary integral methods. When sufficient regularity conditions are met and the test functions comprise a complete set, then in the limit of infinite discretization the numerical solution converges to the unknown boundary distribution. An overview of formulations is presented with reference to Laplace's equation in two dimensions. Numerical experimentation shows that, in general, the solution obtained by desingularized methods becomes increasingly less accurate as the singular points of Green's functions move farther away from the boundary, but the loss of accuracy is significant only when the exact solution shows pronounced variations. Exceptions occur when the integral equation does not have a unique solution. In contrast, and in agreement with previous findings, the condition number of the linear system increases rapidly with the distance of the singular points from the boundary, to the extent that a dependable solution cannot be obtained when the singularities are located even a moderate distance away from the boundary. The desingularized formulation based on Green's function dipoles is superior in accuracy and reliability to the one that uses Green functions. The implementation of the method to the equations of elastostatics and Stokes flow are also discussed.     

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.     

Application of hierarchical matrices to boundary element methods for elastodynamics based on Green's functions for a horizontally layered halfspace

This paper presents the application of hierarchical matrices to boundary element methods for elastodynamics based on Green's functions for a horizontally layered halfspace. These Green's functions are computed by means of the direct stiffness method; their application avoids meshing of the free surface and the layer interfaces. The effectiveness of the methodology is demonstrated through numerical examples, indicating that a significant reduction of memory and CPU time can be achieved with respect to the classical boundary element method. This allows increasing the problem size by one order of magnitude. The proposed methodology therefore offers perspectives to study large scale problems involving three-dimensional elastodynamic wave propagation in a layered halfspace, with possible applications in seismology and dynamic soil–structure interaction.     

浅水环境中基于ISODATA的多目标定位

由于不同距离目标到接收阵之间的格林函数图像存在的明显差异性,利用其特征进行分类可以实现对不同距离目标格林函数图像的区分。文章将聚类分析算法中的迭代自组织数据分析算法(Iterative Selforganizing Data Analysis Techniques Algorithm,ISODATA)应用到传统多目标定位过程中,经过连续干扰消除过程(Successive Interference Cancellation,SIC)的反复迭代,对提取出的多个格林函数进行分类。为解决传统多目标定位过程中需要人工判断格林函数异同的问题提供了一种可行途径,并进一步将分类出的格林函数通过阵不变量方法解算出不同目标的距离信息。     

Boundary effects on sound propagation in superfluids

The equations of strong coupling superconductivity in disordered transition metal alloys have been derived by means of “irreducible” Green's functions and on the basis of the alloy version of the Bari?i?-Labbé-Friedel model for electron-ion interaction. The configurational averaging has been performed by means of the coherent potential approximation. Making some approximations, we have obtained the formulas for the transition temperatureT _c and the electron-phonon coupling constant λ. These depend on the alloy component and total densities of states, the phonon Green's function, and the parameters of the model.     

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.     

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.     

Three-dimensional extended displacement discontinuity method for vertical cracks in transversely isotropic piezoelectric media

The conventional displacement discontinuity method is extended to study a vertical crack under electrically impermeable condition, running parallel to the poling direction and normal to the plane of isotropy in three-dimensional transversely isotropic piezoelectric media. The extended Green's functions specifically for extended point displacement discontinuities are derived based on the Green's functions of extended point forces and the Somigliana identity. The hyper-singular displacement discontinuity boundary integral equations are also derived. The asymptotical behavior near the crack tips along the crack front is studied and the ordinary 1/2 singularity is obtained at the tips. The extended field intensity factors are expressed in terms of the extended displacement discontinuity on crack faces. Numerical results on the extended field intensity factors for a vertical square crack are presented using the proposed extended displacement discontinuity method.