8.1 Introduction

In 1977, for the first time the term boundary element method (BEM) was coined in some publications and the first book on the BEM appeared in 1992 [1]. In the last 20 years the BEM has become one of the important numerical computational techniques along with the FDM (Chapter 5) and FEM (Chapter 6). It is used to solve certain classes of differential equations by formulating as integral equations 2,3]. It is worth mentioning that for certain classes of differential equations, integral equations' reformulation may not always be possible [ 3 ,4]. The advantage in using the BEM is that only the boundaries of the considered domain of the differential equations require subdivision to develop a boundary mesh [5,6]. Whereas, in other methods like FDM or FEM the whole domain of the differential equation requires discretization. In this chapter, we are presenting a brief introduction of the BEM along with a simple example problem for easy understanding of the method.

8.2 Boundary Representation and Background Theory of BEM

Detailed BEM formulation for handling boundary value problem (BVP) in a field or domain may be found in Refs. [ 6 ,7]. By using known data on the boundary S, one can get the rest of information on the boundary S as well as in the domain D in the BEM. Due to various difficulties [ 7 ,8], BVPs do not always have closed‐form solutions or analytic solutions. In general, this may be done by the facility of defining the boundary as a set of panels which have the same characteristic [8] . For example, a closed two‐dimensional boundary may be represented by a set of straight lines (panels), as shown Figure 8.1.

Let S be the original boundary and (for j = 1, 2, …, n) be known as the panels that represent an approximation to S. If is the surface described by the complete set of panels, then is the approximation to . The representation of the boundary in this way is the first step to discretize the integral operators that occur in the boundary integral formulations [ 8 ,9].

Next, we address linear differential operator, fundamental solution, and Green's integral theorems which play an important role in the BEM.

8.2.1 Linear Differential Operator

A linear differential operator, denoted by L, is a function [ 8 , 9 ] such that

• In one dimension:
  8.1
• In two dimensions:
  8.2

All linear operators have the property that L(αu + βv) = αL(u) + βL(v), where α and β are constants and u and v are functions over the domain. The adjoint operator of L represented as L^* is the function defined as

8.3

and also due to the self‐adjoint property we have L = L^*.

To use the BEM for solving BVPs, we must transform the problem into an equivalent boundary integral equation (BIE) problem. In this regard, the fundamental solution and Green's integral theorems are very useful tools. Accordingly, we discuss next the fundamental solution, Green's function, and integral theorems.

8.2.2 The Fundamental Solution

Before discussing the fundamental solution, a brief review of the Heaviside function (H) and Dirac delta function δ in R² is needed.

8.2.2.1 Heaviside Function

The function H(x − c) is said to be the Heaviside function [10] and is defined as (Chapter 2)

Heaviside functions are also called as step functions.

8.2.2.2 Dirac Delta Function

Let q(x, y) and p(x₁, y₁) represent moving and fixed points [ 10 ,11] in the domain D as presented in Figure 8.2. The set of functions denoted by δ_ε(p, q) for ε > 0 is defined as

where ε is the area of the small circle (Figure 8.2 ).

Now, by taking the limit of δ_ε(p, q) as ε → 0 the Dirac delta function δ may be defined as

Further, the Dirac delta function can be viewed as the derivative of the Heaviside step function

and the Dirac delta function has the fundamental property

8.2.2.3 Finding the Fundamental Solution

The fundamental solution of the Laplace equation 9,12] will be described using the Dirac delta function, where the Laplace equation in two dimension is

8.4

The Dirac delta function will be used to derive the fundamental solution of Eq. (8.4), then it should satisfy

8.5

In polar coordinates the Laplacian operator can easily be written and due to the radial symmetric property [11] we have

8.6

where

Furthermore, the Dirac delta in polar coordinates [ 12 ,13] may be obtained as

8.7

Hence, from Eqs. (8.5)–(8.7) we have

8.8

By applying multiple integration on both sides of Eq. (8.8) over the domain D, one may obtain the fundamental solution as

8.9

where c₁ and c₂ are integral constants. By considering c₁ = c₂ = 0, we may have

8.10

8.2.3 Green's Function

Green's function G(x, s) is described as integral kernel that can be used to solve differential equations [12]. Here, the Green's function G(x, s) may be considered as a fundamental solution for the differential equation

8.11

So, if G(x, s) is a fundamental solution of Eq. (8.11), then it should satisfy L[G(x, s)] =  − δ(x − s) and the solution u(x) of Eq. ( 8.11 ) can be expressed in the following form [ 12 , 13 ]:

8.12

From Eqs. ( 8.11 ) and (8.12), one can get

8.13

Since L is a linear operator and L[G(x, s)] =  − δ(x − s), from Eq. (8.13) we have

8.14

Further, by using the known property for any c, d > 0, one can rewrite Eq. (8.14) as

8.15

8.2.3.1 Green's Integral Formula

Green's integral formula is a very useful tool [14] for finding the derivation of integral equations formed in the BEM. Suppose u and v are continuous functions in a domain D ⊂ R² with continuous first and second derivatives in D. Then, functions u and v satisfy the Green's formula

8.16

where S is the boundary of the domain D and and are the outward normal derivatives of u and v, respectively, on the boundary of domain D.

8.3 Derivation of the Boundary Element Method

The BEM can be derived for differential equations for which we can find first a fundamental solution. Accordingly, here a Laplace equation is handled by using the BEM [15,16]

8.17

In the BEM the fundamental solution will be used as the weight function in the integral equation. To find a weak fundamental solution, we multiply Eq. (8.17) by a weight function (fundamental solution) w and integrating over the domain D we have

8.18

Using the Green–Gauss theorem [17] and Eq. (8.18), one may obtain

8.19

Integrating Eq. (8.19) by parts along with Green's formula, we have

8.20

It may be noted that in the FEM we chose simple piecewise polynomials as our weighting (test) functions. In the BEM we choose the fundamental solution (Eq. 8.5 ) so that the last term of Eq. (8.20) becomes

8.21

Assume that (x₁, y₁) ∈ D and not on the boundary. From Eqs. ( 8.18 ), ( 8.19 ), and (8.21), one may obtain

8.22

References

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