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.     

A boundary integral equation method for plate flexure with conditions inside the domain

A method for solving boundary value problems for thin plate flexure as described by Kirchhoff's theory is proposed. An integral formulation leads to a system of boundary integral equations involving values of deflection, slope, bending moment and transverse shear force along the edge. A discretization leading to a matrix formulation is proposed. To solve problems with inner conditions in the plate domain, an elimination of boundary unknowns proves successful. The degenerate case where the boundary is free (which leads to a non-invertible matrix) is investigated. Three examples illustrate the efficiency of the method.     

An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis

An efficient algorithm is employed to evaluated hyper and super singular integral equations encountered in boundary integral equations analysis of engineering problems. The algorithm is based on multiple subtractions and additions to separate singular and regular integral terms in the polar transformation domain, primarily established in Refs. (Guiggiani M, Krishnasamy G, Rudolphi TJ, Rizzo FJ. A general algorithm for the numerical solution of hypersingular boundary integral equations. Trans ASME 1992;59:604–614; Guiggiani M, Casalini P. Direct computation of Cauchy principal value integral in advanced boundary element. Int J Numer Meth Engng 1987;24:1711–1720. Guiggiani M, Gigante A. A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. J Appl Mech Trans ASME 1990;57:906–915). It can be proved that the regular terms have finite analytical solutions in the range of integration, and the singular terms will be replaced by special periodic kernels in the integral equations. The subtractions involve to multiple derivatives of analytical kernels and the additions require some manipulation to separate the remaining regular terms from singular ones. The regular terms are computed numerically. Three examples on numerical evaluation of singular boundary integrals are presented to show the efficiency and accuracy of the algorithm. In this respect, strongly singular and hypersingular integrals of potential flow problems are considered, followed by a supersingular integral which is extracted from the partial differentiation of a hypersingular integral with respect to the source point.     

Computational-experimental method for solving temperature problems with coordinate- and time-dependent boundary conditions

A method for finding the temperature fields and for identifying the conditions of heat exchange for constructions with coordinate- and time-dependent boundary conditions is presented. Inverse problems are reduced to systems of convolution-type integral equations.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 61, No. 3, pp. 479–484, September, 1991.     

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.     

Method of singular integral equations in linear and elastoplastic problems of fracture mechanics

A brief survey of investigations carried out at the Karpenko Physicomechanical Institute of the Ukrainian National Academy of Sciences and devoted to the application of the method of singular integral equations to the solution of two-dimensional problems of fracture mechanics is presented. Special attention is given to the integral equations defined on piecewise smooth closed or open contours appearing in the boundary-value problems of the theory of elasticity for angular domains. We propose a new method aimed at the solution of dynamic problems by using finite differences with respect to time and singular integral equations on the boundary contours. Integral equations also appear in the elastoplastic problems of fracture mechanics solved by using the model of plastic strips and in the general case of continual plastic zones.Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 3, pp. 38–50, May–June, 2004.Lecture delivered at the Third International Conference Fracture Mechanics of Materials and Strength of Structures in Lviv on 22.06.2004.     

A boundary element method for solving 3D static gradient elastic problems with surface energy

 A boundary element methodology is developed for the static analysis of three-dimensional bodies exhibiting a linear elastic material behavior coupled with microstructural effects. These microstructural effects are taken into account with the aid of a simple strain gradient elastic theory with surface energy. A variational statement is established to determine all possible classical and non-classical (due to gradient with surface energy terms) boundary conditions of the general boundary value problem. The gradient elastic fundamental solution with surface energy is explicitly derived and used to construct the boundary integral equations of the problem with the aid of the reciprocal theorem valid for the case of gradient elasticity with surface energy. It turns out that for a well posed boundary value problem, in addition to a boundary integral representation for the displacement, a second boundary integral representation for its normal derivative is also necessary. All the kernels in the integral equations are explicitly provided. The numerical implementation and solution procedure are provided. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. Advanced algorithms are presented for the accurate and efficient numerical computation of the singular integrals involved. Two numerical examples are presented to illustrate the method and demonstrate its merits. Received: 9 November 2001 / Accepted: 20 June 2002 The first and third authors gratefully acknowledge the support of the Karatheodory program for basic research offered by the University of Patras.     

Integral equations of axisymmetric heat-conduction problems of homogeneous bodies with cracks

The fundamental heat conduction problems for axisymmetric bodies with cracks are considered. Integral representations of temperature functions are constructed in terms of temperature jumps and heat flows in axisymmetric surfaces and cuts. By means of these representations, boundary-value problems for a space with cuts at whose edges are specified either a temperature, heat flow, or heat conduction conditions are reduced to singular integral equations. Integral equations are also found for boundary-value problems for a conical body of revolution with cavities and cracks. Numerical results are presented for a space with simply or doubly connected nonplanar cracks.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 27, No. 2, pp. 75–80, March–April, 1991.     

Dual boundary element analysis using complex variables for potential problems with or without a degenerate boundary

The dual boundary element method in the real domain proposed by Hong and Chen in 1988 is extended to the complex variable dual boundary element method. This novel method can simplify the calculation for a hypersingular integral, and an exact integration for the influence coefficients is obtained. In addition, the Hadamard integral formula is obtained by taking the derivative of the Cauchy integral formula. The two equations (the Cauchy and Hadamard integral formula) constitute the basis for the complex variable dual boundary integral equations. After discretizing the two equations, the complex variable dual boundary element method is implemented. In determining the influence coefficients, the residue for a single-order pole in the Cauchy formula is extended to one of higher order in the Hadamard formula. In addition, the use of a simple solution and equilibrium condition is employed to check the influence matrices. To extract the finite part in the Hadamard formula, the extended residue theorem is employed. The role of the Hadamard integral formula is examined for the boundary value problems with a degenerate boundary. Finally, some numerical examples, including the potential flow with a sheet pile and the torsion problem for a cracked bar, are considered to verify the validity of the proposed formulation. The results are compared with those of real dual BEM and analytical solutions where available. A good agreement is obtained.     

Application of boundary integral method to elastoplastic analysis of V-notched beams

The boundary integral equation method was applied in the solution of the plane elastoplastic problems. The use of this method was illustrated by obtaining stress and strain distributions for a number of specimens with a single edge notch and subjected to pure bending. The boundary integral equation method reduced the non-homogeneous biharmonic equation to two coupled Fredholm-type integral equations. These integral equations were replaced by a system of simultaneous algebraic equations and solved numerically in conjunction with the method of successive elastic solutions.     

Three-dimensional crack problem analysis using boundary element method with finite-part integrals

A set of hypersingular integral equations of a three-dimensional finite elastic solid with an embedded planar crack subjected to arbitrary loads is derived. Then a new numerical method for these equations is proposed by using the boundary element method combined with the finite-part integral method. According to the analytical theory of the hypersingular integral equations of planar crack problems, the square root models of the displacement discontinuities in elements near the crack front are applied, and thus the stress intensity factors can be directly calculated from these. Finally, the stress intensity factor solutions to several typical planar crack problems in a finite body are evaluated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Nonsteady diffusion with variable coefficients in the boundary conditions

A method of generalized integral transformations is used to solve the problem of nonsteady diffusion with time-dependent coefficients in the boundary conditions. Such an approach does not require a solution of an integral equation for the surface potential or of a time-dependent eigenvalue problem. A formal solution is obtained on the basis of an infinite system of ordinary differential equations. An example is considered and numerical results are discussed.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 61, No. 5, pp. 829–837, November, 1991.     

Source point isolation boundary element method for solving general anisotropic potential and elastic problems with varying material properties

This paper presents a new robust boundary element method, based on a source point isolation technique, for solving general anisotropic potential and elastic problems with varying coefficients. Different types of fundamental solutions can be used to derive the basic integral equations for specific anisotropic problems, although fundamental solutions corresponding to isotropic problems are recommended and adopted in the paper. The use of isotropic fundamental solutions for anisotropic and/or varying material property problems results in domain integrals in the basic integral equations. The radial integration method is employed to transform the domain integrals into boundary integrals, resulting in a pure boundary element analysis algorithm that does not need any internal cells. Numerical examples for 2D and 3D potential and elastic problems are given to demonstrate the correctness and robustness of the proposed method.     

On the implementation of 3D Galerkin boundary integral equations

In this article, a reverse contribution technique is proposed to accelerate the construction of the dense influence matrices associated with a Galerkin approximation of hypersingular boundary integral equations of mixed-type in potential theory. In addition, a general-purpose sparse preconditioner for boundary element methods has also been developed to successfully deal with ill-conditioned linear systems arising from the discretization of mixed boundary-value problems on non-smooth surfaces. The proposed preconditioner, which originates from the precorrected-FFT method, is sparse, easy to generate and apply in a Krylov subspace iterative solution of discretized boundary integral equations. Moreover, an approximate inverse of the preconditioner is implicitly built by employing an incomplete LU factorization. Numerical experiments involving mixed boundary-value problems for the Laplace equation are included to illustrate the performance and validity of the proposed techniques.     

Multivalued boundary integral equations for inequality problems. The convex case

Summary In the present paper the boundary integral equation method is extended to inequality problems. Here the case of convex superpotentials is considered. Using saddle-point techniques we derive certain variational inequalities on the boundary of the body with respect to the unknown boundary forces or displacements which are equivalent to multivalued boundary integral equations. The theory is illustrated by numerical examples indicating the connection of the developed method with the classical boundary element method.With 2 FiguresTo the memory of Professor Aris Phillips (Aristoteles Philippides).     

Axial symmetric stationary heat conduction analysis of non-homogeneous materials by triple-reciprocity boundary element method

The heat conduction problems in homogeneous media can be easily solved by the boundary element method. The spatial variations of heat sources as well as material coefficients gives rise to domain integrals in integral formulations for solution of boundary value problems in functionally gradient materials (FGM), since the fundamental solutions are not available for partial differential equations with variable coefficients, in general. In this paper, we present the development of the triple reciprocity method for solution of axial symmetric stationary heat conduction problems in continuously non-homogeneous media with eliminating the domain integrals. In this method, the spatial variations of domain “sources” are approximated by introducing new potential fields and using higher order fundamental solutions of the Laplace operator.     

Einführung in moderne Galerkin-Randeiementmethoden mit einer Anwendung aus dem Maschinenbau

The Galerkin-type boundary element method (BEM) is an discretization procedure for integral equations, represents itself however compared with classical integral equation methods as an universal tool for the solution of practical engineering problems and can be coupled very easily with finite element substructures. The BEM, whose main advantage lies in the fact that only a surface mesh must be generated, is superior to FEM in special applications, i.e. in elastostatics (notch problems) and fracture mechanics. In this paper the individual steps to solving an elliptical boundary value problem of 3-D linear elasticity theory by way of an equivalent system of boundary integral equations will be explained. For the mathematical investigation of elliptical differential equations and integral equations, the theory of Sobolev spaces has proved to be especially suitable. Basic terms to Sobolev spaces will be introduced so that the reader does not have to refer to textbooks for new terms. The transformation of elliptical boundary value problems to systems of singular and hypersingular integral equations will be explained with help of a Calderón projector, which is defined by using fundamental solutions. The discretization of the obtained integral equations with the Galerkin-type BEM will be presented. Finally the approximation of non-linear problems by using the Galerkin-type BEM will be shown. A numerical test for a strength problem will be discussed shortly.     

Matched interface and boundary (MIB) for the implementation of boundary conditions in high‐order central finite differences

High‐order central finite difference schemes encounter great difficulties in implementing complex boundary conditions. This paper introduces the matched interface and boundary (MIB) method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high‐order central finite difference schemes. To attain arbitrarily high order, the MIB method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions. The proposed approach is extensively validated via boundary value problems, initial‐boundary value problems, eigenvalue problems, and high‐order differential equations. Successful implementations are given to not only Dirichlet, Neumann, and Robin boundary conditions, but also more general ones, such as multiple boundary conditions in high‐order differential equations and time‐dependent boundary conditions in evolution equations. Detailed stability analysis of the MIB method is carried out. The MIB method is shown to be able to deliver high‐order accuracy, while maintaining the same or similar stability conditions of the standard high‐order central difference approximations. The application of the proposed MIB method to the boundary treatment of other non‐standard high‐order methods is also considered. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical solution of elastic inclusion problem using complex variable boundary integral equation

Based on a complex variable boundary integral equation (CVBIE) suggested previously, this paper provides a numerical solution for the elastic inclusion problem using CVBIE. A dissimilar elastic inclusion is embedded in the infinite matrix. The original problem is decomposed into two problems. One is an interior boundary value problem (BVP) for the elastic inclusion, while the other is an exterior BVP for the matrix with notch. Both problems are connected by conventional boundary integral equations (BIEs) in complex variables. After performing discretization for the coupled BIEs, the inverse matrix technique is suggested to solve the relevant algebraic equations. Based on the properties of some integral operators, three ways for the inverse matrix technique are suggested. Several numerical examples are carried out to prove the efficiency of the suggested method.     

The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems

This paper deals with an application of the boundary element method to the analysis of nonlinear sloshing problems, namely nonlinear oscillations of a liquid in a container subjected to forced oscillations. First, the problem is formulated mathematically as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions, assuming the fluid to be inviscid and incompressible and the flow to be irrotational. Next, the governing equation (Laplace equation) and boundary conditions, except the dynamic boundary condition on the free surface, are transformed into an integral equation by employing the Galerkin method. Two dynamic boundary condition is reduced to a weighted residual equation by employing the Galerkin method. Two equations thus obtained are discretized by the use of the finite element method spacewise and the finite difference method timewise. Collocation method is employed for the discretization of the integral equation. Due to the nonlinearity of the problem, the incremental method is used for the numerical analysis. Numerical results obtained by the present boundary element method are compared with those obtained by the conventional finite element method and also with existing analytical solutions of the nonlinear theory. Good agreements are obtained, and this indicates the availability of the boundary element method as a numerical technique for nonlinear free surface fluid problems.