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 matrices for the 2-D Lame system in regions of irregular shape

A method is proposed for the construction of Green's matrices for mixed boundary value problems in regions of irregular shape for the displacement formulation of the plane problem in theory of elasticity. The method is based on the boundary integral equation approach where a kernel matrix B satisfies the 2-D homogeneous Lame system inside the region. This leads to a regular boundary integral equation where the compensating load is applied to the boundary. The Green's matrix is consequently expressed in terms of the kernel matrix B, the fundamental solution matrix of the homogeneous Lame system and a kernel matrix of the inverse regular integral operator. To calculate the stress components, the kernel matrices are differentiated under the integral sign. The proposed method appears highly effective in computing both displacements and stresses.     

On hypersingular surface integrals in the symmetric Galerkin boundary element method: application to heat conduction in exponentially graded materials

A symmetric Galerkin formulation and implementation for heat conduction in a three‐dimensional functionally graded material is presented. The Green's function of the graded problem, in which the thermal conductivity varies exponentially in one co‐ordinate, is used to develop a boundary‐only formulation without any domain discretization. The main task is the evaluation of hypersingular and singular integrals, which is carried out using a direct ‘limit to the boundary’ approach. However, due to complexity of the Green's function for graded materials, the usual direct limit procedures have to be modified, incorporating Taylor expansions to obtain expressions that can be integrated analytically. Several test examples are provided to verify the numerical implementation. The results of test calculations are in good agreement with exact solutions and corresponding finite element method simulations. Copyright © 2004 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.     

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.     

A new boundary element technique without domain integrals for elastoplastic solids

A simple idea is proposed to solve boundary value problems for elastoplastic solids via boundary elements, namely, to use the Green's functions corresponding to both the loading and unloading branches of the tangent constitutive operator to solve for plastic and elastic regions, respectively. In this way, domain integrals are completely avoided in the boundary integral equations. Though a discretization of the region where plastic flow occurs still remains necessary to account for the inhomogeneity of plastic deformation, the elastoplastic analysis reduces, in essence, to a straightforward adaptation of techniques valid for anisotropic linear elastic constitutive equations (the loading branch of the elastoplastic constitutive operator may be viewed formally as a type of anisotropic elastic law). Numerical examples, using J₂‐flow theory with linear hardening, demonstrate that the proposed method retains all the advantages related to boundary element formulations, is stable and performs well. The method presented is for simplicity developed for the associative flow rule; however, a full derivation of Green's function and boundary integral equations is also given for the general case of non‐associative flow rule. It is shown that in the non‐associative case, a domain integral unavoidably arises in the formulation. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

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.     

Numerical simulation of substation grounding grids buried in both horizontal and vertical multilayer earth model

In the simulation of quasi‐static electromagnetic fields produced by a point source located in both horizontal and vertical multilayer media, the conventional image method requires an infinite number of images to get an accurate solution, while the method of quasi‐static complex images needs only a few ones. Based on the method of quasi‐static complex images, the closed form of Green's function of a point source in both horizontal and vertical multilayer earth model is derived through matrix pencil (MP) approach. The fast convergent Galerkin's type of boundary element method (BEM) is taken to simulate and analyse a grounding system including floating electrode with any complicated structure, which can be located anywhere in horizontal or vertical multilayer earth model. Copyright © 2006 John Wiley & Sons, Ltd.     

Green's function expansion for exponentially graded elasticity

New computational forms are derived for Green's function of an exponentially graded elastic material in three dimensions. By suitably expanding a term in the defining inverse Fourier integral, the displacement tensor can be written as a relatively simple analytic term, plus a single double integral that must be evaluated numerically. The integration is over a fixed finite domain, the integrand involves only elementary functions, and only low‐order Gauss quadrature is required for an accurate answer. Moreover, it is expected that this approach will allow a far simpler procedure for obtaining the first and second‐order derivatives needed in a boundary integral analysis. The new Green's function expressions have been tested by comparing with results from an earlier algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

A 3D BEM model for liquid sloshing in baffled tanks

The present work aims at developing a boundary element method to determine the natural frequencies and mode shapes of liquid sloshing in 3D baffled tanks with arbitrary geometries. Green's theorem is used with the governing equation of potential flow and the walls and free surface boundary conditions are applied. A zoning method is introduced to model arbitrary arrangements of baffles. By discretizing the flow boundaries to quadrilateral elements, the boundary integral equation is formulated into a general matrix eigenvalue problem. The governing equations are then reduced to a more efficient form that is merely represented in terms of the potential values of the free surface nodes, which reduces the size of the computational matrices considerably. The results obtained using the proposed model are verified in comparison with the literature and very good agreement is achieved. Finally, a number of example tanks having common configurations are used to investigate the effect of baffle on sloshing frequencies and some conclusions are outlined. Copyright © 2008 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.     

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.     

A successive boundary element model for investigation of sloshing frequencies in axisymmetric multi baffled containers

This study presents a developed successive Boundary Element Method to determine the symmetric and antisymmetric sloshing natural frequencies and mode shapes for multi baffled axisymmetric containers with arbitrary geometries. The developed fluid model is based on the Laplace equation and Green's theorem. The governing equations of fluid dynamic and free surface boundary condition are also applied to proposed model. A zoning method is presented to model arbitrary arrangement of baffles in multi baffled axisymmetric tanks. The influence of each zone on neighboring zones is applied by introducing interface influence matrix which correlates the velocity potential of interfaces to their flux. By discretizing the flow boundaries, the integral equation governed on the boundary is formulated into a general matrix eigenvalue problem. The proposed method has a considerable effect on decreasing computational cost and a good accuracy in determining the sloshing natural frequencies. The obtained results for different types of container based on the application of the presented study are validated in comparison with the literature and very good agreement is achieved. Finally, the effect of baffle parameters on the sloshing natural frequencies was investigated and some conclusions are outlined.     

Numerical solution and utilization of Green's functions

A numerical method for obtaining the Green's functions for Laplace's, Poisson's, and the transient heat diffusion equations is presented. The Green's functions thus obtained are then employed to rapidly obtain numerical solutions of the above equations by matrix multiplication, with subsequent considerable savings in machine time.     

APPLICATION OF THE ORDINARY LEAST-SQUARES APPROACH FOR SOLUTION OF COMPLEX VARIABLE BOUNDARY ELEMENT PROBLEMS

Cauchy's theorem is used to generate a Complex Variable Boundary Element Method (CVBEM) formulation for steady, two-dimensional potential problems. CVBEM uses the complex potential, w=ϕ+iψ, to combine the potential function, ϕ, with the stream function, ψ. The CVBEM formulation, using Cauchy's theorem, is shown to be mathematically equivalent to Real Variable BEM which employs Green's second identity and the respective fundamental solution. CVBEM yields an overdetermined system of equations that are commonly solved using implicit and explicit methods that reduce the overdetermined matrix to a square matrix by selectively excluding equations. Alternatively, Ordinary Least Squares (OLS) can be used to minimize the Euclidean norm square of the residual vector that arises due to the approximation of boundary potentials and geometries. OLS uses all equations to form a square matrix that is symmetric, positive definite and diagonally dominant. OLS is more accurate than existing methods and can estimate the approximation error at boundary nodes. The approximation error can be used to determine the adequacy of boundary discretization schemes. CVBEM/OLS provides greater flexibility for boundary conditions by allowing simultaneous specification of both fluid potentials and stream functions, or their derivatives, along boundary elements. © 1997 by 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.     

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.     

Green's function of a thin circular plate with elastically supported edge

The Green's function that is suitable for immediate computations, is obtained for a thin circular Poisson–Kirchhoff plate of a uniform thickness. The plate's edge is elastically supported so that the boundary values of the radial bending moment equal zero, while the shear force is directly proportional to the deflection function on the boundary. The extended version of the classical method of eigenfunction expansion is used with partial summation of the resultant Fourier expansion. The ‘singular’ component of the Green's function is analytically split off. This makes the representation accessible for engineering computations.     

Computation of singular and hypersingular boundary integrals by Green identity and application to boundary value problems

The problem of computing singular and hypersingular integrals involved in a large class of boundary value problems is considered. The method is based on Green's theorem for calculating the diagonal elements of the resulting discretized matrix using the Nyström discretization method. The method is successfully applied to classical boundary value problems. Convergence of the method is also discussed.