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As the Boundary Element Method develops into a tool of engineering analysis more effort is dedicated to studying new applications and solving different problems. This book contains chapters on the basic principles of the technique, time dependent problems, fluid mechanics, hydraulics, geomechanics and plate bending. The number of non-linear and time dependent problems which have become amenable to solution using boundary elements have induced many researchers to investigate in depth the basis of the method. Chapter 0 of this book presents an ap­ proach based on weighted residual and error approximations, which permits easy construction of the governing boundary integral equations. Chapter I reviews the theoretical aspects of integral equation formulations with emphasis in their mathematical aspects. The analysis of time dependent problems is presented in Chap. 2 which describes the time and space dependent integral formulation of heat conduction problems and then proposes a numerical procedure and time marching algorithm. Chapter 3 reviews the application of boundary elements for fracture mechanics analysis in the presence of thermal stresses. The chapter presents numerical results and the considerations on numerical accuracy are of interest to analysts as well as practising engineers.



Chapter 0. Boundary Integral Formulations

An operator is a process which applied to a function or a set of functions produces another function, i.e.,
$$l(u) = b$$
where l(u) is the operator which applied to u produces b; u and b may be scalars or vectors: l() may be an ordinary differential operator such as
$$l() = {a_0}\frac{{{d^2}()}}{{d{x^2}}} + {a_1}\frac{{d()}}{{dx}} + {a_2}()$$
a partial differential operator such as
$$l() = \frac{\partial }{{\partial x}}\left( {k\frac{{\partial ()}}{{\partial x}}} \right) + \frac{\partial }{{\partial y}}(k\frac{{\partial ()}}{{\partial y}})$$
or an integro operator
$$l() = \int\limits_0^t {\upsilon (t - \tau )} ()d\tau $$
which can also be written in terms of convolution notation as
$$L() = \upsilon (t)*()$$
Most of the operators encountered in engineering formulations are differential but for some problem areas, such as visco-elasticity and transient response, one needs to consider integro or integro-differential operators.
C. Brebbia, J. J. Connor

Chapter 1. A Review of the Theory

The modern theory of boundary integral equations began with Fredholm [1], who established the existence of solutions on the basis of his limiting discretisation procedure. It was not envisaged by Fredholm or his immediate successors that solutions could actually be constructed in this way. However the advent of fast digital computers, some 50 years later, opened up the possibility of implementing the discretisation process arithmetically and so enabled numerical solutions of tolerable accuracy to be attempted. This possibility in turn gave a considerable impetus to the development of new and improved boundary integral formulations. In 1962, Hess and Smith [2, 3] formulated a Fredholm integral equation of the second kind for the distribution of simple sources over a surface of revolution. By solving this equation numerically, they were able to compute the perturbation of a uniform potential flow by the surface. In 1963, Jaswon and Ponter [4] threw the torsion problem on to the boundary by formulating an integral equation of the second kind for the warping function, which was solved numerically as a means of computing the torsional rigidity and boundary shear stress for cross-sections inaccessible to other methods of attack. This was one of the first published papers which effectively exploited Green’s formula on the boundary, by emphasising its role as a functional relation between the boundary values and normal derivatives of an arbitrary harmonic function. Also in 1963, Jaswon [5] formulated the electrostatic capacitance problem in terms of a Fredholm integral equation of the first kind for the charge distribution, a formulation which had been noted and discarded by Volterra [6] because of apparent difficulties with the two-dimensional theory.
M. A. Jaswon

Chapter 2. Applications in Transient Heat Conduction

The use of singularities to represent instantaneous sources or sinks of heat for solving time dependent heat conduction problems is described in reference [1, ch.X] where Kelvin is credited with having made systematic use of this method to obtain analytical solutions. The integral representation to be derived below in Section 2 appears in reference [2], but without any numerical treatment.
H. L. G. Pina, J. L. M. Fernandes

Chapter 3. Fracture Mechanics Application in Thermoelastic States

Thermal stress analysis is one of the most important subjects in engineering and technology. Several attempts have been so far made for analysis of thermoelastic problems in steady heat conduction states by the boundary element method (Rizzo and Shippy, 1978; Danson, 1981, for example). However, as far as the authors are aware, there are few investigations for thermal stress analysis in transient heat conduction states, although the transient heat conduction problems are often studied by means of the boundary element method (Wrobel and Brebbia, 1981). This is partly because the boundary integral formulation includes inevitably domain integrals which arise due to the temperature distributions in space and time, and in the transient heat conduction states they can not be transformed into the corresponding boundary integrals. Therefore, we have to innovate a more efficient numerical implementation for such problems.
M. Tanaka, H. Togoh, M. Kikuta

Chapter 4. Applications of Boundary Element Methods to Fluid Mechanics

In this chapter we make an effort to review the applications of boundary methods to fluid mechanics. At the outset we wish to give our definition of boundary methods. First, we exclude such techniques from the general class of finite element methods. Although some of the language and a few of the numerical techniques of the two methods have merged, they are historically quite separate.
J. A. Liggett, P. L.-F. Liu

Chapter 5. Water Waves Analysis

Over the last decades the study of water waves diffraction and radiation problems, especially in offshore structures has interested many researchers. The problem is sometimes solved by using the finite element method as shown by Newton [1]; Berkhoff [2]; Bai [3]; Yue, Chen and Mei [4], and more recently Zienkiewicz and Bettess [5]. Alternatively boundary integral equations have been used as illustrated by the work of Garrison [6]; Hogben and Standing [7]; Eatock-Taylor [8] and Isaacson [9].
M. C. Au, C. A. Brebbia

Chapter 6. Interelement Continuity in the Boundary Element Method

It is well known that homogeneous elliptic field problems may be alternatively posed as infinite systems of boundary integral equations obtained by using a suitable family of kernel functions and integration by parts [1], [2]. A determinate system of non-homogeneous linear algebraic equations is obtained therefrom by discretizing the system, using elements defined after the manner of finite elements and a finite member of Kernel functions [1]. The result is a practical and powerful numerical method for the solution of elliptic field problems which may easily be generalized to rival the finite element method in its range of applicability.
C. Patterson, M. A. Sheikh

Chapter 7. Applications in Geomechanics

The boundary element method is already well established as an efficient numerical technique to solve a variety of continuum mechanics problems [1, 2]. Solutions exist for problems as complex as plasticity [3], viscoelasticity [4], viscoplasticity [5] and other complex time-dependent and non-linear problems. The literature is, nowadays, very extensive and the interested reader is referred to [2].
W. S. Venturini, C. A. Brebbia

Chapter 8. Applications in Mining

Mining is the term used for the extraction of mineral from the earth’s crust. The problems of analysis which arise are thus almost exclusively concerned with the excavation of ore from the “host material” and the stresses and displacements induced by the consequent mining excavations. The determination of these stresses and displacements then allows predictions to be made about the stability of the mine working and may affect the design of the mine layout. There are basically two ways in which ore is deposited, namely,
Tabular: In seams, lenses or strata of large extent in two dimensions but with relatively small thickness in the third dimension.
Massive: A three dimensional body of complicated shape.
G. Beer, J. L. Meek

Chapter 9. Finite Deflection of Plates

The Boundary Element Method (BEM) is now an effective tool for the numerical analysis of nonlinear as well as linear problems. Some of those problems are physical or material nonlinear, such as elastoplasticity and creep in solid mechanics [1]. Mother type of nonlinear problem are those concerned with geometrical nonlinearities, such as finite deformations. Among a variety of geometrically nonlinear behaviours in solid mechanics, the finite deflection of flat plates or shells is one of the most important problems from the standpoint of engineering practice.
N. Kamiya, Y. Sawaki

Chapter 10. Trefftz Method

In recent years, by a boundary method, it is usually understood a numerical procedure in which a subregion or the entire region, is left out of the numerical treatment, by use of available analytical solutions (or, more generally, previously computed solutions). Boundary methods reduce the dimensions involved in the problem leading to considerable economy in the numerical work and constitute a very convenient manner of treating adequately unbounded regions by numerical means. Generally, the dimensionality of the problem is reduced by one, but even when part of the region is treated by finite elements, the size of the discretized domain is reduced [1–2].
Ismael Herrera


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