Evaluation of Green's function derivatives for exponentially graded elasticity

Effective formulas for computing Green's function of an exponentially graded three‐dimensional material have been derived in previous work. The expansion approach for evaluating Green's function has been extended to develop corresponding algorithms for its first‐ and second‐order derivatives. The resulting formulas are again obtained as a relatively simple analytic term plus a single double integral, the integrand involving only elementary functions. A primary benefit of the expansion procedure is the ability to compute the second‐order derivatives needed for fracture analysis. Moreover, as all singular terms in this hypersingular kernel are contained in the analytic expression, these expressions are readily implemented in a boundary integral equation calculation. The computational formulas for the first derivative are tested by comparing with results of finite difference approximations involving Green's function. In turn, the second derivatives are then validated by comparing with finite difference quotients using the first derivatives. Published in 2010 by John Wiley & Sons, Ltd.     

Approximate Green's functions in boundary integral analysis

It is well known that employing a Green's function which satisfies the prescribed conditions on a part of the boundary is advantageous for boundary integral calculations. In this paper, it is shown that an approximate Green's function, one in which the known data is nearly reproduced, can also be highly beneficial in implementations of the boundary-element method. This approximate Green's function approach is developed herein for solving the Laplace equation, and applied to the modeling of void dynamics under electromigration conditions in metallic thin-film interconnects used in integrated circuits.     

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.     

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.     

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.     

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.     

Computability of series representations for Green's functions in a rectangle

Different series forms of Green's functions are analyzed for various boundary value problems stated for Laplace and Klein–Gordon equation in a rectangular region. The classical double-series representation for the Dirichlet problem for Laplace equation is converted into a single-series form, and a computational experiment is conducted to compare practical convergence of the two forms. By a partial summation of the single-series representation, the singular component of the Green's function is expressed in analytic form radically accelerating convergence of the remaining series for the regular component. Readily computable series forms are obtained for Green's functions of some mixed boundary value problems.     

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.     

Boundary integral equation solution of three‐dimensional elastostatic problems in transversely isotropic solids using closed‐form displacement fundamental solutions

The boundary integral equation method is used for the solution of three‐dimensional elastostatic problems in transversely isotropic solids using closed‐form fundamental solutions. The previously published point force solutions for such solids were modified and are presented in a convenient form, especially suitable for use in the boundary integral equation method. The new presentations are used as a basis for accurate numerical computations of all Green's functions necessary in the BEM process without inaccuracy and redundant computations. The validity of the new presentation is shown through three numerical examples. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

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.     

A hyper-singular numerical Green's function generation for BEM applied to dynamic SIF problems

This paper introduces the extension of the numerical Green's function approach for elastodynamic fracture mechanics problems. The formulation uses the hyper-singular boundary integral equation to obtain the fundamental solution for the cracked unbounded medium. The procedure is general and can be applied to multiple crack problems of general geometry. Applications to time harmonic and transient (through inverse numerical Fourier and Laplace transforms) stress intensity factor (SIF) computations are presented and compared with other numerical and analytical results, showing the good accuracy of the present strategy for these kinds of problems.     

A novel acceleration method for the variational boundary element approach based on multipole expansion

The acoustic radiation of general structures with Neumann's boundary condition using Variational Boundary Element Method (VBEM) is considered. The classical numerical implementation of the VBEM suffers from the computation cost associated with double surface integration. To alleviate this limitation, a novel acceleration method is proposed. The method is based on the expansion of the cross influence matrices in terms of multipoles using the expansion of the Green's function in terms of spherical Bessel functions. Since the resulting multipoles are not dependent on the elements locations, large computation time savings are achieved. Moreover, it is shown that by accounting for the monopole, dipole and quadrupole terms in the multipole expansion, the classical convergence criteria usually used in boundary element guarantee convergence of the proposed method. Several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method. © 1998 John Wiley & Sons, Ltd.     

Dynamic response of tunnels in stochastic soils by the boundary element method

A stochastic boundary element method (SBEM) is developed in this work for evaluating the dynamic response of underground openings excited by seismically induced, horizontally polarized shear waves under steady-state conditions. The surrounding geological medium is viewed as an elastic continuum exhibiting large randomness in its mechanical properties, which implies that the wave number of the propagating signal is a function of a random variable. Suitable Green's functions are proposed and used within the context of the SBEM formulation. More specifically, a series expansion for the Green's functions is employed, where the basis functions are orthogonal polynomials of a random argument (polynomial chaos). These are subsequently incorporated in the SBEM formulation, which employs the usual quadratic, isoparametric line elements for modeling the surfaces of the problem in question. Finally, this formulation is used for the solution of a few problems of engineering interest involving buried cavities (tunnels). We note that the present approach departs from earlier boundary element derivations based on perturbations, which are valid for ‘small’ amounts of randomness in the elastic continuum.     

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 numerical Green's function and dual reciprocity BEM method to solve elastodynamic crack problems

A computational model based on the numerical Green's function (NGF) and the dual reciprocity boundary element method (DR-BEM) is presented for the study of elastodynamic fracture mechanics problems. The numerical Green's function, corresponding to an embedded crack within the infinite medium, is introduced into a boundary element formulation, as the fundamental solution, to calculate the unknown external boundary displacements and tractions and in post-processing determine the crack opening displacements (COD). The domain inertial integral present in the elastodynamic equation is transformed into a boundary integral one by the use of the dual reciprocity technique. The dynamic stress intensity factors (SIF), computed through crack opening displacement values, are obtained for several numerical examples, indicating a good agreement with existing solutions.     

The singular function boundary integral method for a two-dimensional fracture problem

The singular function boundary integral method (SFBIM) originally developed for Laplacian problems with boundary singularities is extended for solving two-dimensional fracture problems formulated in terms of the Airy stress function. Our goal is the accurate, direct computation of the associated stress intensity factors, which appear as coefficients in the asymptotic expansion of the solution near the crack tip. In the SFBIM, the leading terms of the asymptotic solution are used to approximate the solution and to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of Green's theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The numerical results on a model problem show that the method converges extremely fast and yields accurate estimates of the leading stress intensity factors.     

On the Theory of Direct and Inverse Problems in the Elastic Rectangle: Antiplane Case

In this article, we study the antiplane deformation of the boundary surface of a rectangular domain in the presence of a void and a shear force on the outer boundary surface. For a formulated inverse problem, we develop some analytical results and use them to solve the problem numerically for various elliptic geometrical configurations. The analytical method allows us to give an efficient representation for Green's function in the rectangular domain. Then we derive the same Green's function by an alternative method based on Fourier series expansions. Finally, for a number of configurations, we demonstrate the comparison between real and reconstructed defects.     

Efficient computation of the Green's function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis

An alternative scheme to compute the Green's function and its derivatives for three dimensional generally anisotropic elastic solids is presented in this paper. These items are essential in the formulation of the boundary element method (BEM); their evaluation has remained a subject of interest because of the mathematical complexity. The Green's function considered here is the one introduced by Ting and Lee [Q. J. Mech. Appl. Math. 1997; 50: 407–26] which is of real-variable, explicit form expressed in terms of Stroh's eigenvalues. It has received attention in BEM only quite recently. By taking advantage of the periodic nature of the spherical angles when it is expressed in the spherical coordinate system, it is proposed that this Green's function be represented by a double Fourier series. The Fourier coefficients are determined numerically only once for a given anisotropic material; this is independent of the number of field points in the BEM analysis. Derivatives of the Green's function can be performed by direct spatial differentiation of the Fourier series. The resulting formulations are more concise and simpler than those derived analytically in closed form in previous studies. Numerical examples are presented to demonstrate the veracity and superior efficiency of the scheme, particularly when the number of field points is very large, as is typically the case when analyzing practical three dimensional engineering problems.     

Inverse source identification by Green's function

Based on the use of Green's function, we propose in this paper a new approach for solving specific classes of inverse source identification problems. Effective numerical algorithms are developed to recover both the intensities and locations of unknown point sources from scattered boundary measurements. For numerical verification, several boundary value problems defined on both bounded and unbounded regions of regular shape are given. Due to the use of closed analytic form of Green's function, the efficiency and accuracy of the proposed method can be guaranteed.