Lecture Notes in Engineering Edited by C. A. Brebbia and S. A. Orszag

35

WTang

Transforming Domain into Boundary Integrals in BEM A Generalized Approach

Spri nger-Verlag Berlin Heidelberg New York London Paris Tokyo

Series Editors C. A. Brebbia . S. A. Orszag

Consulting Editors J. Argyris . K.-J. Bathe' A. S. Cakmak . J. Connor' R. McCrory C. S. Desai' K.-P. Holz . F. A. Leckie' G. Pinder' A. R. S. Pont J. H. Seinfeld . P. Silvester' P. Spanos· W. Wunderlich' S. Yip

Author Prof. Weifeng Tang East China University of Chemical Technology 130 Mei-Iong Road Shanghai 200237 PRChina

ISBN-13:978-3-540-19217-6 e-ISBN-13:978-3-642-83465-3

001: 10.1007/978-3-642-83465-3

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin, Heidelberg 1988

The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

216113020-543210

ABSTRACT

In this work a new and general approach to transform

domain integral terms into boundary integral terms in BEM

formulations is presented. The technique can be used

for both potential and elasticity problems. The method

is based on expanding the integrands in domain integrals

into Fourier series, which ensures convergence of results.

The corresponding Fourier coefficients can be calculated

analytically or numerically. The mathematical implementa­

tion and corresponding programming are described in this

thesis. Numerical applications using the present approach

validate and illustrate how the method can be used in

engineering practice, including the application for

elasto-plastic analysis. The present approach is a

general and reliable transformation technique.

CONTENTS

CHAPTER 1 GENERAL INTRODUCTION

1-1 Numerical Methods

1-2 Domain Methods 2

1-3 Boundary Element Method 3

1-4 The Main Procedures and Features of BEM 5

1-5 The Subject of this Work 7

1-6 Contents of the Present Work 9

1-7 The Cartesian Tensor Notation 11

CHAPTER 2 POTENTIAL PROBLEMS

2-1 Introduction 12

2-2 The Boundary Integral Formulation 13 for Potential Problems

2-3 The Boundary Element Method for 22 Potential Problems

2-4 Motivation and General Ideas 27

2-5 Fourier Analysis 30

2-6 Basic Formulations for Transforming 35 the Domain Integrals into the Boundary for 2-D Problems

2-7 Numerical Approaches 42

2-8 Numerical Accuracy of the Transformation 46 Formula

2-9 Some Further Discussions 68

2-10 Examples 73

2-11 The Transformation Formula for 3-D 90 Poisson's Equation

2-12 Applications in Time-dependent Problems 95

2-13 Application in Non-linear Problems 99

CHAPTER

V

Page

3 LINEAR ELASTOSTATICS

3-1 Introduction 101

3-2 Basic Relationships of Elasticity 102

3-3 Fundamental Solution for Elastostatics 105

3-4 Somigliana Identity 108

3-5 The Boundary Integral Equations of 112 Elastostatics

3-6 The Boundary Element Method in Elasticity 116

3-7 Basic Formulations for Transforming 2-D 117 Elasticity Domain Integrals to the Boundary

3-8 Numerical Implementation 132

3-9 Results of Numerical Experiments 140

CHAPTER 4 APPLICATIONS IN ELASTICITY AND ELASTO­PLASTICITY

4-1 Introduction 158

4-2 An Example of Gravitational Loading 160

4-3 An Example with a More General Type 166 of Distributed Loading

4-4 Relationship between Plastic Stresses 169 and Plastic Strains

4-5 The Governing Equations for Elasto- 177 Plasticity

4-6 Numerical Analysis using Finite 181 Fourier Series

4-7 Application to Elasto-plastic Problems 185

CHAPTER 5 PROGRAMMING

5-1 Potential Problems

5-2 Elasticity Problems

5-3 Elasto-Plasticity Problems

CHAPTER 6 GENERAL DISCUSSION AND CONCLUSIONS

REFERENCES

194

196

198

201

204

CHAPTER 1 GENERAL INTRODUCTION

1-1 NUMERICAL METHODS

For the last two or three decades, scientists and

engineers have used numerical methods as an important

tool in many different areas. This significant fact

has its inexorable historical trend and it is the

inevitable outcome of the recent developments in science,

technology and industry.

Analytical methods have been developed for a long

period and have produced a great amount of successful

results, but they failed to solve most practical engineering

problems with complicated boundary conditions or irregular

geometry. It is also very difficult to solve non-linear

or time-dependent problems using analytical approaches,

even if they are very simple. On the other hand, research

on analytical methods has provided a solid foundation for

different types of numerical methods.

Because of the rapid developments of science and

technology it is now necessary to solve complicated

problems using more efficient and accurate approaches

than before. Not only problems with complicated boundary

conditions or irregular configurations require solutions

but also non-linear or time-dependent problems must be

solved.

Computer hardware and software have developed at an

unexpected high speed. During the last thirty years, ithaz

become possible for scientists and engineers to use

numerical methods with computers easily. This has

2

stimulated scientists and engineers to improve some

classical numerical methods (such as finite difference

method) and to establish new numerical methods (such as

the finite element method and boundary element method).

For all these reasons, numerical methods have

rapidly developed in the areas of mechanics and engineering.

Furthermore, a new discipline, computational mechanics

has already emerged and become a very important and

active branch of mechanics.

The development of computational mechanics provided

excellent tools for different engineering areas. The

numerical solutions can now be obtained accurately and

efficiently even for very complicated boundary conditions,

non-linear problems and time-dependent problems. Moreover

engineers can nowadays use many kinds of software packages,

which are easily available in the software-markets and

can solve problems in their own specialities.

1-2 DOMAIN METHODS

Currently, there are some important domain methods

in the area of computational mechanics, such as finite

difference method (FDM), finite element method (FEM)

and weighted residual method (WRM). All these methods

have already proved their efficiency to solve differential

equations corresponding to complex engineering problems.

For instance, during the last two or three decades, the FEM

has been developed to a high degree of sophistication

both in terms of its fundamental theory and of software

packages.

3

In contrast with analytical methods, numerical methods

give no one formula type expression of the results, but

some discretized and approximate results. The domain

of the problem is discretized into some subdomains,

using grids (in FDM) or elements (in FEM), then the

results are calculated at certain points, which are chosen

in advance, such as intersections of subdomains or Gauss

integration points. The governing differential or

integral equation is reduced into a finite set of

linear algebraic equations, which is suitable for com­

puting. In FDM, for example, every order of difference

of functions is required instead of the derivatives.

Based on some variational principles or weighted residuals

in FEM or WRM, for instance, a linear combination of trial

functions replaces the unknown function. FEM has more

flexibility than other methods because the approximate

expression of the trial functions is only valid in one

of the subdomains. Moreover, very complicated geometrical

domains with arbitrary boundary and initial conditions,

as well as non-linear and time-dependent problems can be

treated by means of iterative and incremental procedures.

1-3 BOUNDARY ELEMENT METHOD

Since the publication of the first book called

"Boundary Elements" in 1978 [ 1), BEM has been developed

rapidly and many new applications in engineering have now

been produced. The technique is an important alternative

method in the area of computational mechanics.

4

BEM is based on well established theoretical

foundations, such as boundary integral equations,

fundamental solutions of partial differential equations

and weighted residual methods. These are combined with

numerical techniques, such as discrete method, numerical

integrations, increment methods, iterative technique and

others.

Early in this century, Fredholm established the

theory of integral equations, then the integral equations

and the boundary integral methods were applied in the

area of mechanics. Some Russian scientists made significant

contributions, particularly Mikhlin, Muskhelishvili and

Kupradze [2,3,41. At that time, integral equations were

considered to be a different and powerful type of

analytical method. Hence, because of the difficulty to

obtain the solutions analytically, it was hard to apply

them in engineering. More recently computer techniques

and numerical methods have provided excellent foundations

for using integral equations in the area of mechanics.

In 1963, Symm and Jaswon presented a numerical method to

solve boundary integral equations for potential problems

using Dirichlet, Neumann or Cauchy boundary conditions

[6,71. For elasticity problems, Cruse and Rizzo presented

the direct integral formulation, in which all variables

are original physical quantities [8-101. It is important

to point out that the direct BEM is more suitable for

solving engineering problems. This direct method is now

more widely used in the integral equation approach.

5

Many books [1,11,1~] and proceedings [7,10,13-21) have been

published and international conferences are held

periodically. Scientists of mechanics and engineers

have recently worked on BEM in the areas of time-dependent

and non-linear problems.

1-4 THE MAIN PROCEDURES AND FEATURES OF BEM

Generally speaking the main procedures of BEM are

the following:

A. Transform the differential equation with boundary

and initial conditions into the corresponding

boundary integral equation:

Using weighted residual method and integration by

parts (or using other equivalent methods, such as

Green's second identity or Somigliana identity), the

governing differential equations can then be made

equivalent to integral equations. After considering

the fundamental solutions, which satisfy the same

operators as the governing equations and are solved

by applying a Dirac delta-function, hence the domain

integrals are replaced by boundary ones. However,

if there are some other functions on the right-hand

side of the governing equation, there remains another

kind of domain integral term in the integral equation.

B. Solve the boundary integral equation numerically:

After expressing the variables by means of inter­

polation functions and discretizing the intagral

equations, one can calculate all the boundary integrals

as summations of the values on all boundary elements.

6

Hence the final formulations can be reduced to a

set of linear algebraic equations. The set of

equations can then be solved to obtain all boundary

values.

C. Find the results at internal points of the domain

Afterwards, if values at internal points are required,

they can be calculated from the boundary solutions.

The developments and applications of BEM are recent,

but the method has shown some remarkable advantages over

domain methods.

These advantages can be summarised as follows:

A. BEM requires only the discretization of the boundary

of the domain. So that the mesh generation and the

preparation of initial input-data are considerably

simplified. For the same reason, the total degrees

of freedom of the problem are reduced considerably.

For these reasons BEM systems are easier to use and

to interface with other CAD systems.

B. The application of fundamental solutions in BEM not

only improves the accuracy of the results, but also

permits to represent boundary conditions at infinity.

Hence, BEM avoids taking a large mesh to present

infinite domain problems, while FEM requires a very

large number of elements to solve the same problems.

C. Both the unknown functions (displacement in elasticity

or value of field in the potential problems) and their

derivatives (traction in elasticity or flux in the

potential problems) are present in the BEM formulationj

hence, it is a mixed type formulation which gives the

7

same degree of accuracy for the derivatives as for

their functions using BEM. This makes the BEM more

suitable for cases of stress concentration and high

temperature gradient.

There are some other advantages in BEM, such as the

possibility of using discontinuous elements and simple

mesh refinements.

It is important to realise, however, that the BEM

also has some disadvantages over the FEM. For inGtance,

BEM formulations are usually more difficult than FEM ones

and the BEM systems of equations are fully populated, while

the FEM are not.

These disadvantages can be solved by using available

software systems developed by experts (such as BEASY [22]),

efficient numerical methods and large computers. Regarding

the last point, it is worth mentioning that the formulations

of BEM are especially suitable for the type of supercomputers,

which are rapidly becoming available for solving engineering

problems.

1-5 THE SUBJECT OF THIS WORK

As already mentioned, one of the obvious advantages

of using BEM is that it requires discretization only on

the boundary. Unfortunately this feature is generally

lost when source terms are present in the governing

differential equations. For example, when internal

distributed body forces (in elasticity) or internal

distributed sources (in potential problems) exist in

8

the domain, also in time-dependent and in non-linear

problems, domain inte-gral terms appear in the formulations

of the boundary integrals. Usually there are two

approaches to compute the domain integral terms in these

cases, first, in some special cases the body force (or

source) term can be transformed into boundary integral

terms. For instance, in the case of potential problems,

if the source function is constant or a harmonic: in

elasticity problems, if the body force is a gravitational,

centrifugal or therm& load [25-28]. The second approach

consists in dividing the domain into a certain number of

cells and then computing the domain integral numerically

as a summation of values over the cells. For this purpose,

quadrature techniques using Monte Carlo method and an

adaptive quadrature algorithm were presented in referencest24.

29-31]. These algorithms, however, consume a great

amount of CPU time and sometimes fail to get the results

within the specified accuracy.

The disadvantage of the second approach is that it is

necessary to define internal mesh points and cells over

the domain, so that a part of the main advantage of BEM

is lost [23]. In addition, the approach can not be used

for the case of infinite or semi-infinite domain, as it is

impossible to divide these domains into a finite number of

cells.

In order to transform the domain into boundary

integrals, a method called dual reciprocity (DRM) has

been developed by Brebbia and Nardini [32-35]. This method

can be used both in potential and elastic problems and some

9

accurate results have been obtained. Another advantage of

this method is that it introduces no new influence matrices

in the formulations, so it is computationally very efficient.

Its main drawback is that the body force functions are

expressed using an incomplete set of functions, and the

expressions are not sufficiently accurate. This can

cause considerable errors in some results.

In this thesis, a generalized method of transforming

the domain integrals into boundary ones is presented both

for potential and elasticity problems. Although some

expressions are similar to those presented in the

DRM , the idea and the derivations of formulations in

this thesis are independent work and their numerical

implementation is different. In addition, the approximate

functions used here are a complete set which is a.:comp1ete1y

different type from the ones used in DRM.

1-6 CONTENTS OF THE PRESENT WORK

In Chapter 2 a short review of BEM formulations in

potential problems is presented. Then, the method of

transforming the domain into boundary integrals is presented

and the formulations are derived for 2-D and 3-D potential

problems. Afterwards, the numerical approach for computing

the source integrals is introduced. In order to examine

the numerical accuracy of these formulations, tests have

been carried out, comparing results with those obtained

using the original domain integral. It is possible to

conclude from these results that the present formulation

is accurate. Some Poisson's equation examples are presented

10

and discussed. Finally it is shown how to generalize this

method for time-dependent and non-linear problems.

Chapter 3 is concerned with elasticity problems. All

the contents are almost parallel to potential problems.

Although the basic formulation is similar to the one for

potential cases, the problem is now more complex due to

the relative complexity of the elasticity equations. The

numerical approach is also more complex. Some problems,

which do not occur in potential problems, are discussed

in numerical implementation.

Applications in elasticity are presented in Chapter 4.

As has been mentioned above, body force integral terms

can be due to non-linearities. This is the case in

plasticity, even in the absence of body forces due to the

existence of the plastic strains. Using incremental methods,

the plastic strains can be obtained from the previous step

and a domain integral formulated. Then this integral can

be transformed to the boundary using the approach previously

described.

Chapter 5 describes the programs developed throughout

this work, with special emphasis to the programming of the

transformation formulae.

Chapter 6 presents a general critical discussion of

the topics investigated in the previous chapters and gives

the conclusions and suggestions for further work.

11

1-7 THE CARTESIAN TENSOR NOTATION

Throughout this work the Cartesian tensor notation

is used. This notation not only permits expressions to

be written in compact forms, but also is useful in deriva-

tion and proof of theorems. Such notation makes use of

subscript indices (1,2,3) to represent the spatial

coordinates (Xl, X2, X3).

These are some rules:

A. A repeated index in a term implies a summation with

respect to this index over its range.

B. The Kronecker delta symbol 0ij and the permutation

symbol e ijk are used throughout.

The definitions of 0ij and e ijk are as follows:

0"" = jl 1.] 0

when i j

when i ;. j

when any two indices are the same

when i,j,k are an even permutation of 1,2,3

otherwise

C. The spatial derivatives are indicated by a comma

and the index corresponding to that derivative, for

example, u" stands for au/ax1." • ,1.

CHAPTER 2 POTENTIAL PROBLEMS

2-1 INTRODUCTION

Many phenomena of mechanics and physics can be

reduced to the solutions of potential problems, such as

heat conduction, diffusion, flow of ideal fluid, flow

in porous media, torsion, electrostatics and others.

The Poisson's equation is discussed, first, in this

chapter. It is shown how a problem governed by a partial

differential equation and with prescribed boundary con­

ditions can be recast into a boundary integral equation

form. Because there is a source term on the right-hand

side of the Poisson's equation, the boundary integral

equation involves a domain integral term too. Even

though the integrand is a known function in order to

compute the numerical value of this integral the domain

of the problem under consideration can be divided into

some cells {subdomainsl and these cells must be numbered

for numerical procedures. However in order to avoid

dividing the domain into subdomains, the source function

in the Poisson's equation can be expanded as a Fourier

series. Since each function in this series has some

particular features, the domain integral can be transformed

into a summation of boundary integrals. The error of the

integral values depends on how accurately the source function

can be expressed by this series. After deriving the

formulations, the numerical approach and the results for

examining are presented. Some examples are then presented

13

to describe the validity of the approach and finally

applications in time-dependent and non-linear problems

are described.

2-2 THE BOUNDARY INTEGRAL FORMULATION FOR POTENTIAL PROBLEMS

For steady potential problems the governing equation

can be expressed as Poisson's equation, assuming that

there exist sources b(x) inside the domain Q. These

sources can be due, for instance, to internal heat genera-

tion for heat conduction problems. The potential function

u is governed by Poisson's equation:

b(x)

and the following boundary conditions

A. Dirichlet type boundary condition:

u(x) u(x)

B. Neumann type boundary condition:

q(x) = q(x)

where V2 is the Laplace operator:

V is the Nabla operator:

and x is a spatial point with its coordinates xi

e; is unit vector along the axis x • i'

au q an

(2-2-1)

(2-2-2)

(2-2-3)

14

n is the unit outward normal to the boundary r,

u and q are prescribed values of the function u and

its normal derivative over the boundary. The total

boundary is given by r = r1 + r2 (Fig. 2-2-1).

Multiplying Eq.(2-2-1) by a function u* and

integrating it over all the boundary, one can express

Eq. (2-2-1) as an integral equation as follows:

J u*(E,;,x) [V 2 u(x) - b(x)]dn(x) = 0

n (2-2-4)

where u* is the fundamental solution of Laplace's equation,

i.e. u* satisfies the following equation

V2 U *(E,;,X) -t,(E,;,x) (2-2-5)

where E,; is an arbitrary point in the space. ~(E,;,x) is

the Dirac delta-function which has the following

properties:

and

~(E,;,x) {: for E,; " x

for E,; = x

J u(x)~(E,;,x)dn(x) u(E,;)

n

u*

u*

1 21T

1 41ir

1 In(r) for 2-D

for 3-D

(2-2-6)

15

Figure 2-2-1 Notation

16

Considering Green's second identity

J (u av - v aU)df an an (2-2-7)

f

and letting v = u*

one obtains the following equation from Eq. (2-2-4)

J [ * au au* )df - J u an - u an-r n

Using the notations

q au an and q* au*

an-

u* b(x)dn

and considering the integral feature of the Dirac delta-

function, the previous equation becomes

J u*q df - J q*u dr -

f f

Taking into consideration

(2-2-3), one obtains

J u*q dr J u*q df +

r fl

and

J q*u dr J q*u df + r f1

J u* b(x)dn

n u (t;) (2-2-8)

the boundary conditions Eqs (2-2-2)and

J u*q dr

f2

J q*u df

f2

Eq. (2-2-8) is valid for any point ~ inside the domain n.

But in order for this equation to be valid for any value

17

of ~ including those on the boundary r, the point ~ needs

to be taken to the boundary r. After accounting for the

jump in the second integral on the left -hand side of

(2-2-8)

J = J q*u dQ Q

the boundary integral equation (2-2-8) becomes

c(~)u(O J u*q dr - J q*u dr - J u*b(x)dQ (2-2-9) Q r r

Eq. (2-2-9) is the basic integral equation for BEM

potential problems.

The value of c(~) depends on the position of point

lj as follows:

c = 1 for an internal point ~

c = 0 for an external point ~ (2-2-10)

c = 1 + lim J q* dr for ~ on the boundary r E+O r E

In order to c"a1cu1ate the value of c (~), assume that the

domain under consideration can be augmented by a small

region r E ' which is a part of a sphere (3-D case) or

a circle (2-D case) centred at point ~ with radius E

(Fig.2-2-2). So the integral J can be separated into

two parts

J J q*u r-r

E

(2-2-11)

18

{}

Figure 2-2-2 Boundary f augmented by f£

19

The local polar (r,e) coordinate is introduced for

the 2-D case (Fig. 2-2-3). Considering the direction of

n along the radius, r = E and dr = Ede, one can write

q* 1 - 2nE

substituting this relationship into J 2 of Eq. (2-2-11), one

obtains

J q*u dr

r E

For the 3-D case, one can use spheroidal coordinates

(r,e,$) as shown in Fig.2-2-4. Therefore, relationships

are:

q* 1 - 4nE3

substituting them into J 2 of Eq. (2-2-11), one has

q*u dr

21T n/2

1 E2

- u!~) J I cos$ d~ de

o ~1

_ u(~) (1-sin~ ) 2 1

The geometric meaning of I r E

i dr (for 2-D) and J r E

Figure 2-2-3

Figure 2-2-4

20

----. ---

Evaluating c(~) for two-dimensional

Evaluating c(~) for three-dimensional

21

(for 3-D) is the plane and the solid angle respectively

which is spanned on r£ with centre ~ (Fig.2-2-3). As £ tends to zero, the values of these angles are equal to

the external plane angle (for 2-D) and external solid

angle (for 3-D) at point ~ on r. The values c(~) equal the

internal plane angle divided by 2n(for 2-D) and internal

solid angle divided by 4n(for 3-D) at point on ~ •

The fact of c(~) = I when ~ is located inside the

domain Q and c (~) = 0 when ~ is located outside ·the domain

Q is clear by looking at the geometric meaning of J 2 •

After taking into account the jump on the boundary

and expressing c as Eq. (2-2-10), the integral J is

considered as a Cauchy principal value integral because

when £ tends to zero in Eq.(2-2-11) J 1 is a Cauchy

principal integral. The existence of the integral J 2

can be proved if u(x) satisfies a Holder condition [45] at

point ~ as follows

lu(x) - u(~) I ~ B rcx

where Band cx are positive constants.

There is also a domain integral term on the right-hand

side of Eq.(2-2-9) as follows

I(~) = I u*b(x)dQ Q

(2-2-12)

For some particular cases, such as b(x) equals a con-

nt8.nt: or when b (x) is a harmonic function in Q, this

domain integral can be transformed into equivalent boundary

integrals, but for an arbitrary function b(x) numerical

22

quadrature is usually applied on each cell.

Besides the Poisson's equation, domain integral

terms also exist for the diffusion equation, time-dependent

problems and non-linear problems.

2-3 THE BOUNDARY ELEMENT METHOD FOR POTENTIAL PROBLEMS

This section presents the numerical implementation

of Eq. (2-2-9) for 2-D problems using only linear continuous

elements. In addition, the source function b(x) can be

assumed identically zero for simplicity as later on the

study will be extended to include the case of a more

general b(x) function.

In order to solve Eq. (2-2-9) numerically, one needs

to compute the integrals in Eq. (2-2-9) numerically

therefore, two important concepts need to be introduced.

A. BOUNDARY ELEMENT DISCRETIZATION

The boundary integrals, f, are computed as a r

summation of a certain number of numerical integrals, each

of which is carried out on a segment of the boundary.

This kind of segment is regarded as a boundary element

(Fig.2-3-1). Thus the integrals in Eq.(2-2-9) can be

expressed as

f NE

f u*q dr L u*q dr]

r j=1 rj

(2-3-1)

f NE

f q*u dr L q*u dr]

r j=l r. J

where NE is the total number of elements.

23

Figure 2-3-1 Discretization of boundary elements

24

B. INTERPOLATION FUNCTION

Using interpolation functions, only some values of

variables u and q are needed. The points, at which these

values are considered are, called nodes. The variables

elsewhere can be expressed in terms of these values at the

nodes. Using interpolation functions for continuous linear

elements, for instance, both ends of each element are

taken as nodes and the value of variables at every point

between the nodes can be expressed as a linear combination

of the values at the nodes (Fig.2-3-2).

(2-3-2)

where u jl and u j2 represent the potential values of the

nodes. qjl and qj2 are fluxes at the nodes. ~1 and ~2

are interpolation functions, i.e.

~1 J, (1-n) J, (1+n) (2-3-3)

where n is the local coordinate of the boundary element.

Notice that the length of every element in the local

coordinate is 2, Le. n e [-1,+1].

as:

After that, one can express the integral over element j

I u*q dr

rj

where

J q*u dr J q*[4>l 4>21

r. r. J J

gj1 J u* 4>1 dr

r. J

gj2 J u* 4>2 dr

r. J

hj1 J q* 4>1 dr

r j

hj2 J q* 4>2 dr r.

J

26

t'l f u j 11 J rr=[h j1 hj21

u j2 u. 2 J l J

(2-3-4)

(2-3-5)

Note, because all u*, q* and 4>1' 4>2 are known functions,

the results of hj1' hj2' gj1 and gj2 can be evaluated

by Eqs(2-3-5). The integrals in (2-3-5) are usually

computed by means of numerical quadrature, which is

described in Section 2-6. However, when the source point

~ is located on the element j, Eqs(2-3-5) can be calculated

analytically [1,121.

Substitution of Eqs(2-3-1) and (2-3-4) into Eq. (2-2-9)

leads to the following matrix equation:

H U G Q (2-3-6)

27

where matrix H is assembled with the elements h j1 , hj2 and c

and matrix G is assembled by gj1' gj2 (j = 1,NE). U and

Q are vector, the components of which are potentials and

fluxes at the nodes, i.e.

(2-3-7)

Q

After replacement of certain components in U and Q

by their prescribed values, given by the boundary conditions,

and rearrangement, Eq.(2-3-6) becomes a set of linear

algebraic equations, from which all unknown values of

potential and flux at the nodes can be solved. Furthermore,

the potential at the internal points can be computed from

Eq. (2-2-9) numerically.

2-4 MOTIVATION AND GENERAL IDEAS

When the function b(x) of Eq. (2-2-9) happens to be

a particular case, such as constant or sine function, the

domain integral on the right-hand side of Eq. (2-2-9)

1(0 J u*(~,x)b(x)dn(x) n

can be transformed into a boundary integral.

Let us suppose the function b(x) satisfies

IPb (x) = 6b (x)

(2-4-1)

(2-4-2)

28

where B is a non-zero constant.

After substituting Eq. (2-4-2) into (2-4-1), one

obtains

1(1;;) i J u* V' 2 b(x)Ml

n

and then considering Green's second identity (2-2-7)

and letting u

1(1;;) 1 B

u*, v = b, one has

J u* ab J an dr - q* b(x)dr + r r

J b(x)V'2 u *drl J rl

Considering u* is a fundamental function of Laplace

equations (Eq. (2-2-5», the last integral of previous

equation can be rewritten as

- J b(x) V' 2 u* (I;;,x)drl = b(l;;)

rl

Hence the previous equation now becomes

1(1;;) 1 B J u*

r ab J an dr - q*b dr - b (0 ]

r (2-4-3)

When the point I;; is on the boundary r, one calculates

the second integral on the right-hand side of Eq. (2-4-3)

as a Cauchy principal value integral. Then one can

rewrite Eq. (2-4-3) as follows:

I ( 1;;) 1 B J u*

r

ab dr an J q*b dr - c(l;;)b(I;;)]

r (2-4-4)

29

where c is as given in Eq. (2-2-10).

Hence, if the function b(x) in the domain integral

(2-4-1) satisfies Eq. (2-4-2), then this domain integral

(2-4-1) can be transformed into a boundary integral as

shown in Eq. (2-4-4).

An arbitrary function b(x) generally does not satisfy

Eq.(2-4-2) but based on the above idea, one can expect

that, if

A. Here choose a set of functions {bi (x) I i=l, ••. , oo} ,

and each function among this set satisfies Eq. (2-4-2),

B. Every arbitrary function b(x) can be expanded as a

c.

series of this set of functions, i.e.

b(x) = L kibi(x) i

(2-4-5)

When the set of functions {b.} is a complete set,then a 1.

finite part of series (2-4-5) instead of the infinite

series can express the original function b(x) with

sufficient accuracy. Thus in all the following

sections, the index i of the summation in the series

be only taken from 1 to a certain number N. In fact,

in this case, the meaning of 'equal to' in the

formulae will be 'approximately equal to'.

After substituting Eq. (2-4-5) into Eq. (2-4-1), it

becomes

I(O u*b. drt 1.

(2-4-6)

Then, using Eq. (2-4-4), the domain integral of Eq. (2-4-6)

can be expressed as follows

30

I(O J obi J u* az:l dr -r r

(2-4-7)

In order to satisfy the above conditions, one can use

trigr>nometric function as a set and expand b (x) as a

Fourier series. Because trigonometric functions

constitute a complete set, when one takes enough

terms in Eq. (2-4-5), this finite part series can

express the funcion b(x) accurately enough. This idea

allows us to transform any general domain integral into

a boundary one in potential problems.

2-5 FOURIER ANALYSIS

In this section a short review of Fourier analysis

is presented [36,67].

First, one defines a one dimensional function fIx)

as periodic if, and only if, there exists a positive

number 2a, such that for every x in the domain of

function f

fIx + 2a) = f(x) (2-5-1)

The number 2a is called a period of fIx).

Consider a function fIx) which is periodic with

period 2n, and suppose that it can be represented in the

interval [-n, n] by the following infinite trigonomp.tric

series

fIx) ko +I (k~ cos nx + k~ sin nx) n

(2-5-2)

31

where ko ' kl and k 2 (for n=l, ... ) are constants. In n n order to determine these unknown constants, one can use

the orthogonality of harmonics, i.e.

11

f sin(nx)dx 0

-11 11

J cos(nx)dx 0

-11 11

J sin(nx) sin(rnx)dx= 11 a (2-5-3) nm -11 11

J cos (nx) cos (mx>dx= 11 a nm -11

11

J sin(nx) cos (mx)dK= 0

-11

To find ko,one can integrate series (2-5-2) term by term

over the interval [-11, 111. Considering the properties

(2-5-3), one obtains

1 211

11

f -11

f(x)dx (2-5-4)

since f(x) is a known function which is assumed to be

integrable.

In a similar way to the above, to find kl or k 2 (for n n

n = 1,2, ... ) one multiplies each side of series (2-5-2)

by con(nx) or sin(nx) and then integrates from -11 to 11,

assuming again that term-by-term integration is justified.

using the orthogonality of sines and cosines and reducing,

32

one obtains

1T kl I

J fIx) sin(nx)dx n 1T -1T

1T

k 2 I

J fIx) cos (nx)dx (2-5-5) -n 1T -1T

Formulae (2-5-4) and (2-5-5) are regarded as Euler

formulae and the constants ko' k~ and k~ defined above

are called the Fourier coefficients of function fIx).

In this case, the series (2-5-2) is called the Fourier

series representation of function fIx).

Next, if the periodic function fIx) has a period

of 2a in the interval [-a, a), one can transform it into

a periodic function of y with period 21T, i.e.

y 1T a x (2-5-6)

Therefore, the function f(ay/1T) has a Fourier series

representation as given by Eqs(2-5-2) and (2-5-3).

substituting (2-5-6) in these equations, one obtains the

Fourier series expressed in terms of the variable x with

period 2a, Le.

fIx) . (n1TX)] s~n --a

(2-5-7)

where

33

a

ko 1 I f(x)dx 2a

-a a

kl 1 I fIx) (n 1TX) (2-5-8) n ""a cos -a dx -a

a

k 2 1 I fIx) sin (n;x) dx -n a

-a

Up to now, the function fIx) with period 2a is expanded as

a series (2-5-7) only formally. A theorem which gives

the conditions under which a Fourier series converges

to this function is called a Fourier theorem. One of such

theorems is referred to as Dirichlet theorem, i.e.

If fIx) is a bounded periodic function which in

any period has at most a finite number of discontinuous

points then the Fourier series of function fIx) converges

to fIx) at all points where fIx) is continuous and

converges to the average of the right- and left-hand limits

of fIx) at each point where fIx) is discontinuous.

The conditions of the above theorem usually are called

the Dirichlet conditions. Obviously,the Dirichlet conditions

are Valid for piecewise continuous functions. Although the

Dirichlet theorem is a sufficient condition for conver-

gence, almost all functions of practical engineering can

satisfy these conditions. Therefore one supposes ~ll the

functions used throughout this thesis satisfy the Dirichlet

conditions.

The following theorem gives the asymptotic behaviour

of the Fourier coefficients of a periodic function

fIx) [67].

34

THEOREM: As n becomes infinite, the coefficients

k~ and k~ in the Fourier expansion of a period function

satisfying the Dirichlet conditions always approach zero

at least as rapidly as c/n where c is a constant independent

on n. If the function has one or more points of discon­

tinuity, then either k~ or k~, and in general both, can

decrease no faster than this. In general, if a function

f(x) and its first m-l derivatives satisfy the Dirichlet

conditions and are everywhere continuous, then as n becomes

infinite, the coefficients k~ and k~ in the Fourier series

f f () d 1 . dl / m+l o x ten to zero at east as rap1 y as c n • If

in addition, the mth derivative of f(x) is not everywhere

continuous, then either k~ or k~ , and in general both,

m+l can tend to zero no faster than c/n •

From the above theorem, generally speaking, the

smoother the function, the faster its Fourier series

expansion converges.

Finally, when f(x) possesses certain symmetry prop-

erties, the coefficients in its Fourier expansion become

especially simple. Suppose that f(x) is an even function,

i.e. f(-x) = f(x). The coefficients in the Fourier series

of f(x) (2-5-8) become

a

ko 1 I f(x)dx a

0

a kl 2 I (mrx) (2-5-9) n a f(x) cos a dx

0

k 2 0 n

35

Conversely, if fIx) is an odd periodic function, i.e.

f(-x)= -fIx), the coefficients in the Fourier series of

fIx) (2-5-8) become

k kl 0 0 n

(2-5-10) a

k~ 2

f fIx) . (mrx) dx s~n --n a a 0

The properties of Fourier analyses of two or three

dimensional functions are similar to these for one

dimension~ function and these will be used in this thesis

directly.

2-6 BASIC FORMULATIONS FOR TRANSFORMING THE DOMAIN INTEGRALS INTO THE BOUNDARY FOR 2-D PROBLEMS

In this section, one derives the basic formulae,

which are expressed using only boundary integrals. For

simplicity, the domains are assumed to be two dimensional,

so the vector x in all the equations has two components

x 1 and x 2' and ~ = (~ 1 ' ~i . Let us suppose the function b(x) = b(x1 ,x2 ) in Eq. (2-2-1)

is an arbitrary function, which satisfies the Dirichlet

conditions mentioned in the previous section in the domain

n of the problem under consideration. One extends b(x)

from the original domain n to the domain n f , which involves

domain nand is[2a1 X 2a 2 ] (shown in Fig.(2-6-1)). Then one

carries out the periodic extension of b(x) with the period

2a1and 2~ with respect to xl and x 2 directions respectively.

36

-----'----------2a,

Figure 2-6-1 Notation

37

From the above hypothesis, one can expand the function

b(x1,x2) as a convergent Fourier series. Therefore one

takes the finite part summation of Fourier series to

express the function b(x) approximately as follows:

b(x)

where

f 0

fl n

f2 n

f3 n

f4 n

fl nm

f2 nm

f3 nm

f4 nm

1

k f o 0 +

nrrx 1 cos (--) a 1

nrrx 1 sin (--) a 1

nrrx2 cos (--) a 2

nrrx 2 sin (--) a 2

fl. f3 n m

fl. f 4 n m

f2.f3 n m

f2.f4 n m

4 N 4 N M L L L L L

R,=1 n=1 R,=1 n=1 m=1

(2-6-1)

(2-6-2)

38

k 1

J J b(x)drl 0 4a1a 2

rl f

k£ 1 J

( f£(x)drl

2a 1a 2 J b(x) (2-6-3)

n n rlf

k£ = 1 J J b(x)f;m(X)drl nm a 1a 2

rl f

After substituting Eq. (2-6-1) into the domain

integral term (2-4-1), one obtains

1(0

+

J u*b drl

rl

u*f o

4 N M J L L L k£ u*f£ £=1 n=l m=l nm nm

rl

drl (2-6-4)

Consider that for each of the functions f o ' f~(x) or

f~m(x) there exists a corresponding function uo ' u~(x) or

u£ (x), which satisfies one of the corresponding equations nm

as follows

1,2,3,4 n,m 1, 2, ...

(2-6-5)

these functions u can be expressed as

39

1 (x 2-X 20 )2] Uo "4 [(x l -x lO )2 +

! -[:~l' fR. R. 1,2 n R. (2-6-6) u n

_ [:~) 2 fR- R- 3,4 n

where x lO and x 20 are arbitrary constants.

Using Eqs. (2-6-5)one by one and the second Green's

identity and considering the source point ~ can be

located anywhere, one has

- f q*uo dr - cUo(~)

r

f u*f~ f u* V2 UR. f dUR-

dQ dQ u* n dr n an Q Q r

- f R. dr - cu~(~) (2-6-7) q*u n

r

f u*fR. f u* V2 UR. f dUR.

dn dQ u* nm dr nm nm ----an Q Q r

- f R. dr - R-q*u cUnm (~) nm r

40

where c are the same as in Eq. (2-2-10).

Substituting Eqs. (2-6-7) into (2-6-4), one can transform

each term of the domain integral on the right-hand side of

Eq. (2-6-4) into boundary integrals. After defining

a go an Uo

R. Cl fR. (2-6-8) gn an n

R. a fR. gnm= an nm

the Eq. (2-6-4) can be rewritten as a transformation formula

for the domain integral 1(~):

1(0 = I u*b dSl So [ I u*g dr- I q*u dr - c U o (~)] + 0 0

Sl r r

4 N sR. I i I q*f i f~(~)] + + L L u*gn dr - dr - c

.11.=1 n=1 n n r r

4 N M i I i I *f i fi (~)] + L L L Bnm u*gnm dr - q nm dr - c R.=1 n=1 m=1 nm

r r

(2-6-9)

where

1,2

3,4

41

(2-6-10)

The first term of Eq. (2-6-4) can also be transformed

as follows. Let us introduce a function v* defined as

r 2 1 v* = an [In(r) + 1] (2-6-11)

and that function v* satisfies

u* (2-6-12)

The first integral on the right-hand side of Eq. (2-6-4)

can now be expressed as

f u* dn n

= f av* dr an r

(2-6-13)

Now, having substituted Eqs. (2-6-7) and (2-6-13) into

Eq.(2-6-4), we obtain

I (E,;) f u*b(x)dn ao f av* dr + an

n r

4 N a2. f

2-

f *f2. f2. (E,;)] + + L L u*9n dr - dr - c 2.=1 n=l n q n n

r r

4 N M 2. I 2. f *f2. - c f~m(E,;)] + L L L anm [ u*qnm dr - dr

2.=1 n=1 m=1 q nm

r r (2-6-14)

The above Eq. (2-6-9) or (2-6-14) is the final formula

which transforms the domain integral of the body source the

term into boundary integrals. This is calledAtransformation

42

formula for simplicity throughout this thesis. For two

dimensional problems, one has

u*

q*

v*

<lv* <In

1 1 2n In (i)

1 <lr - 2nr an

1 <lr - an [2r(ln r-1) + r1 <In

2-7 NUMERICAL APPROACHES

..f-[2 I (1) 11 <lr 871 n i + an

The function b(x) on the right-hand side of Poisson's

equation (2-2-1) is known, so the coefficients of its

Fourier series can be determined by Eqs. (2-6-3). After

that, all the functions and constants are known in Eq.

(2-6-9) or (2-6-14). In this case, the source integral

term can be computed by means of numerical quadratures

with formula (2-6-9) or (2-6-14). Using numerical quad-

ratures directly instead of using interpolation functions

one can obtain more accurate results. To this purpose,

one discretizes all the boundary r into NE elements and

calculates every boundary integral in Eq.(2-6-9) as a

summation of the values over all these elements:

J f(x)dr

r

NE L

j=l J f(x)dr

rj

When computing the values of the integral over every

element, using standard Gauss quadrature as follows [38,461

43

J f(x)dr r.

(2-7-1)

J

where NP is the number of Gauss integral points chosen.

Sp is the local coordinate of the Gauss integral point p

and wp is the corresponding weight value Both sp

and w can be found in the tables, which are included in p

references [38,12]. Because the values are listed in

those tables in terms of normal local coordinates in the

interval [-1,+1], it is necessary to transform the length

of element, rj,into the length 2. In Eq. (2-7-1) IJI is

the values of the Jacobian of this transformation which

gives

IJ I = r. /2 J

It can be proved, if the integrand is continuous

on the interval [A, B] that the results of Gauss quadrature

converge to the exact value of the integral as the number

of Gaussian points increases. For regular integrals

Gauss quadrature gives good results with few integration

points.

But this standard Gauss quadrature (2-7-1) is not

suitable for the case when the source point s is on the

element, because of singularities of functions u*, q*

and ~*/an. In this case, a special coordinate transform-

ation can improve the accuracy of results significantly

[39] •

Let us consider the singular integral I as follows

I

+1

J f(x)dx

-1

44

(2-7-2)

in which fIx) is singular at a point x in the interval

[-1,11.

Now, one chooses a non-linear transformation as

follows

x (z) az 3 + bz 2 + cz + d (2-7-3)

with the following conditions on the boundary and at the

singular points:

x(l) = 1

x(-l) =-1

dxl = 0 dz -z

~~~ 1_ 0 z

(2-7-4)

where the image point z is the point, which corresponds

to the original singular point x. The third equation

in Eqs. (2-7-4) means that the Jacobian value of this

transformation equals zero at the singular point z and

the fourth equation in Eqs (2-7-4) means the Jacobian is

an extreme value, which can be chosen as extremely small.

So, after using the transformation (2-7-3) the singularity

of integrand can be smoothed by the Jacobian, and Eq. (2-7-2)

becomes

I

+1

J f(x(z))dx/dz dz

-1

45

(2-7-5)

From the constrained conditions (2-7-4) of this

transformation one obtains all these coefficients in

Eq. (2-7-3) as follows

a = 1/Q

b - 3z/Q

d -b

Where z is simply the value of z which satisfies x(z) x,

this value can be evaluated by

z = (xx* + Ix*I)1/3 + (xx* - Ix*I)1/3 + x

and

x* = x:Z - 1

Thus, after transformation the integral (2-7-2) becomes

I (2-7-6)

Using this transformation, the order of accuracy in

numerical integration can be improved up to one to two

orders in comparison with the standard Gauss quadrature.

The value c in Eq. (2-6- 9) was expressed as Eq. (2-2-10)

In regard to the constants x 10 and x 20 in the first

of Eqs (2-6-6), the values of (x 10 ' x 20 ) can be taken as

coordinates of a certain reference point. This reference

point can be chosen at an arbitrary position, such

46

as the origin of coordinates or the middle point of domain

Q. It must be avoided that taking the magnitude of slO or

x 20 is significantly different from that of xl or x 2 '

otherwise it causes some numerical error.

2-8 NUMERICAL ACCURACY OF THE TRANSFORMATION FORMULA

In this section the numerical accuracy of formula

(2-6-9) or (2-6-14) is examined. For this purpose, let

the source function b(x) equal each term in Eq.(2-6-1)

one by one and then compute the value of the integral using

formula (2-6-9). Notice that all the integrals are

implemented only on the boundary. In order to compare the

results the same domain integrals are computed also by

means of the original domain integral method, the procedures

of which are as follows:

A. Divide the domain Q into some subdomains.

B. Compute the values of this integral over these subdomains.

C. Add all the values of the subdomain integrals to obtain

the original domain integral.

When the above step B is carried out, the standard Gauss

quadrature is applied over the subdomain. If the source

point is located outside this subdomain, one has

f NP1 NP2

Is(~) f(x)dQ L L f(~ , ~ )w iJi Q P1=1 P2=1 PI P2 P1P l

s (2-8-1)

where iJi is the Jacobian of transformation for Q s into

the standard 2-D Gauss quadrature region [2 x 21, i.e.

47

IJI = A /4 s (2-8-2)

where As represents the area of the subdomain Os.

When the source point ~ is within the subdomain Qs'

the same transformation as shown in Eq. (2-7-3) can be used

in both the xl and x 2 directions.

In order to compare the results for the two different

methods, one selects the same subdivision of the domain and

boundary (shown in Fig. 2-8-1). The number of boundary

elements is NE = 2(NX + NY), where NX and NY are the numbers

of divisions in xl and x2 directions. The numbers of

Gaussian points NP, NP1 and NP2 for standard Gauss quadrature

are 4 or 6, which depend on the distances between the source

point and the middle point of the boundary element or the

subdomain. If the distance is less than 1.5 times the

length of the boundary element or the side of the subdomain,

NP (or NP1, NP2) equals 6 otherwise it equals 4. For the

singular boundary element or subdomain, in which the trans-

formation (2-7-3) is carried out, the number of integral

points is 8. Furthermore, without losing generality, the

area of Fouri~r expansion, Qf iS2a1 X2a2 and it involves

the integral domain O. The source point ~, when located

on the boundary r, is notated ~l or ~2. If the source

point is located outside or within the domain, it is notated

~3 or ~4· The locations of these source points are shown

in Fig. '2-8-1. The results of domain integrals, which are

defined as

I(~) J u*b(x) dQ(x)

°

48

20

15

• nesh points 10

5

2 7 10

Figure 2-8-1 Geometry and mesh

49

are computed with Eq. (2-6-9) and the original domain

integral method respectively, the results of which are

noted as II and 12 separately in Tables 2-8-1 - 2-8-10.

In these tables, the wave numbers of harmonics (2-6-2) are

nand m in the xl and x 2 directions respectively.

From these tables, some interesting facts can be drawn:

A. If the wave number n or m is less than 6, II and 12

coincide with each other exactly in the numerical sense.

It means, in this case, the Eq.(2-6-9) is correct both

theoretically and numerically.

B. While the wave number n or m is increasing, the errors

of these results increase as well, so the differences

between II and 12 become more pronounced. This kind of

numerical error is also reasonable, because in every

boundary element or every side of a subdomain there are

many wave numbers, when compared with the number of

integral points NP, NPI or NP2. For example: when n = 24

in every boundary element along xl direction there are 2

waves of harmonics, it means that the signs of these

harmonics alternate four times along one element, but the

number of integral points, NP, is only four per regular

element. Obviously it causes unacceptable errors. The

same case happens for the results of the subdomain integrals

1 2 •

C. Even if there are as many as four waves of harmonics

in three boundary elements (that is nor m equals sixteen),

the results still have four or five digit accuracy.

50

Table 2-8-1 Comparison of boundary inte,gral II

with domain integral I2

I = I u*.1 d!1

!1

NX I1 I2 E:. NY

3 -0. 8272959X102 -0.8272960X102

6 -0.8272958X102 -0.8272957X102

E:.! 9 -0.S272958X102 -0.8272958X102

12 -0. 8272959X1 02 -0. 8272965X102

15 ":'0. 8272959X102 -0.8272972X102

3 -0.7534999X102 -0. 753500OX102

6 -0. 7535000X102 -0. 7535000X102

E:.2 9 -0. 7535000X102 -0. 7535007X102

12 -0. 7534999X102 -0.7535001X102

15 -0. 7534999X102 -0.7535009Xl02

51

Table 2-8-2 Comparison of boundary integral 11

with domain integral 12

~ n

1

2

3

4

5

~l 6

7

10

15

25

1

2

3

4

~2 5

6

7

10

15

25

I J u* cos(n;;) dQ Q

11

-0.7622810 X101

0.5229649 X102

0.5056158 X101

-0.3344568 X101

0.1191101 X102

-0.1082302 X102

-0.8292571 X101

0.2712600 X100

-0.4247177 X101

0.2479330 X101

-0.8491453 X101

0.4558578 X102

0.5711345 X101

0.1174665 X100

0.1297357 X102

-0.9624903 X101

-0.9006425 X101

0.1432590 X100

-0.3697073 X101

0.2233753 X101

12

-0.7622812 X101

0.5229648 X102

0.5056158 X101

-0.3344566 X101

0.1191102 X102

-0.1082301 X102

-0.8292586 XlI)1

0.2712435 X100

-0.4247263 X101

0.2420987 X101

-0.8491419 X101

0.4558580 X102

0.5711358 X101

0.1174508 X100

0.1297358 X102

-0.9624891 X101

-0.9006439 X101

0.1432513 X100

-0.3697100 X101

0.2183748 X101

52

Table 2-8-3 Comparison of boundary integral II

with domain integral 12

I J * 'n(~) dn u S1 10 n

E; n 11 12

1 -0.7424272 X102 -0.7424271 X102

2 -0.9916166 x101 -0.9916165 X101

3 0.2555545 X10 2 0.2555545 Xl02

4 -0.4099361 X101 -0.4099360 X101

5 0.8766171 X10 1 0.8766171 X101

6 0.1361520 X102 0.1361522 X102

E; 1 7 -0.7063324 X101 -0.7063317 X101

10 -0.1041555 X102 -0.1041555 X102

15 0.2715321 X101 0.2715315 X101

25 0.1675875 X101 0.1604687 X101

1 -0.6699394 X102 -0.6699390 X102

2 -0.1106592 X102 -0.1106590 X102

3 0.2007587 X102 0.2007588 X102

4 -0.4384451 X101 -0.4384450 X10

5 0.9765131 X101 0.9765128 X101 E;2

0.1482711 X102 0.1482712 X102 6

7 -0.4656162 X101 -0.4656166 X101

10 -0.1085651 X102 -0.1085651 X102

15 0.3093648 X101 0.3093693 X10'

25 0.1968400 X101 0.1919084 X101

Table 2-8-4

I; n

1

2

3

4

5

I; 1 6

7

10

15

25

1

2

3

4

1;2 5

6

7

10

15

25

53

Comparison of boundary integral II

with domain integral I2

I f u* cos (n27Tl) dS"l S"l

I1

0.1025681 X102

0.5387757 X102

-0.1768966 X102

-0.2668609 X101

0.5235638 X101

-0.1596658 X102

0.6657169 X101

o 9456897 X101

0.2848718 X1~1

-0.1464234 X101

0.1295744 X102

o 5021597 X102

-0.2207117 X102

-0 4883914 X101

0.5886055 X101

-0.1325916 X102

0.9007068 Xl01

0.7555257 X101

0.3923257 X101

-0.1937091 X101

I2

0.1025684 X102

0.5387759 X102

-0.1768966 X102

-0.2668607 X101

0.5235623 X101

-0.1596659 X102

o 6657169 Xf01

o 9456928 X101

0.2849329 X101

-0.1491409 X101

0.1295761 X102

o 5021592 X102

-0.2207116 X102

-0.4883922 X101

0.5886029 X101

-0.1325917 X102

0.9007059 X101

0.7555306 X101

0.3922994 x101

-0.1962734 X101

54

Table 2-8-5 Comparison of boundary integral II

with domain integral 12

£;

£; I

£;2

I f u* sin(n;;) dn

n

n 11

1 -0.7482952 X10 2

2 0.1685810 X102

3 0.2697564 X10 2

4 -0.1301431 X102

5 0.1242537 X102

6 -0.2266829 X10 1

7 -0.1027152 X102

10 0.3230910 X100

15 -0.4582665 X101

25 -0.2598Q64 X101

1 -0.6848425 X10..!

2 0.2121402 X102

3 0.2658022 X102

4 -0.1591316 X102

5 0.9124641 X101

6 -0.3630992 X101

7 -0.9268256 X101

10 0.1419310 X101

15 -0.3825251 X101

25 -0.1944320 X101

12

-0.7482947 X102

0.1685812 X102

0.2697564 X102

-0.1301432 X102

0.1242537 X102

-0.2266842 X102

-0.1027155 Xl02

0.8232314 X100

-0.4582321 X101

-0.2510424 X101

-0.6848415 X102

0.2121401 X102

0.2658024 X102

-0.1591315 X102

0.9124645 X101

-0.3630999 X101

-0.9268279 X101

0.1419359 X101

-0.3825188 X101

-0.1874890 X101

55

Table 2-8-6 Comparison of boundary integral II

with domain integral 12

~ n m

1 1

1 2

2 1

1 3

2 2

3 1

1 4

~l 2 3

3 2

4 1

1 1

1 2

2 1

1 3

~2 2 2

3 1

1 4

2 3

3 2

4 1

I J u* cos(n;;) cos(m2~Y) dn n

11 12

0.2918378 ~101 0.2918366 X101

0.5592705 X101 0.5592709 X101

-0.5243825 X101 -0.5243832 X101

-0.4876962 X101 -0.4876965 X101

-0.3356133 X102 -0.3356137 X102

-0.3528005 X101 -0.3528009 X101

-0.1588836 X101 -0.1588832 XH)1

0.9215223 X101 0.9215222 X101

-0.4111199 X10' -0.4111189 X101

-0.1495466 X101 -0.1495464 X101

0.2733293 X101 0.2733322 X101

0.6142702 X101 0.6142701 X101

-0.7863301 X101 -0.7863300 X101

-0.4531018 X101 -0.4531018 X101

-0.3039432 X102 -0.3039432 X102

-0.3782635 X101 -0.3782633 X101

-0.1569167 X101 -0.1569179 X101

0.1338504 X102 0.1338503 X102

-0.4924170 X101 -0.4924174 X101

-0.1025702 X10-1 -0.1025700 X10-1

56

Table 2-8-7 Comparison of boundary integral 11

with domain integral 12

E;, n m

1 1

1 2

2 1

1 3

2 2

E;, I 3 1

1 4

2 3

3 2

4 1

1 1

1 2

2 1

1 3

2 2 E;,2

3 1

1 4

2 3

3 2

4 1

I =J u* cos(nnx) s1'n(mny ) d n 10 20" Q

11 12

-0.7075934 X101 -0.7075955 X101

0.4749635 X101 0.4749632 X101

0.4715874 X102 0.4715874 X102

0.3588468 X101 0.3588460 X101

-0.8670514 X101 -0.8670522 X101

0.4808176 X101 0.4808191 XlO 1

-0.3405131 X101 -0.3405133 X101

-0.1618861 X102 -0.1618862 X102

-0.5755103 X101 -0.5755108 X101

-0.2804108 X101 -0.2804120 X101

-0.7857435 X101 -0.7857417 X101

0.4437354 X101 0.4437364 X101

0.4143626 X102 0.4143625 X102

0.3840045 X101 0.3840088 X101

-0.1287099 X102 -0.1287098 X102

0.5514474 X101 0.5514489 X101

-0.3119807 X101 -0.3119800 X101

-0.1610427 X102 -0.1610427 X102

-0.6095337 X101 -0.6095333 XlO 1

0.1185471 X100 0.1185437 X100

57

Table 2-8-8 Comparison of boundary integral 11

with domain integral 12

I I * . (n 1TX) (m1TY) dSl u I:!l.n 10 cos 20 Sl

~ n m 11 12

1 1 0.8825028 X10 1 0.8825037 X101

1 2 0.4819830 X102 0.4819831 X102

2 1 0.4297675 X101 0.4297669 X101

1 3 -9.1527316 X102 -0.1527316 X102

2 2 0.7400675 X101 0.7400671 X101

3 1 -0.1230353 X101 -0.1230359 X101 ~l

-0.2078539 X101 -0.2078537 X1.0 1 1 4

2 3 -0.7196420 X101 -0.7196427 X101

3 2 -0.1586730 X102 -0.1586731 X102

4 1 -0.1256455 X101 -0.1256454 X101

1 1 0.1152898 X102 0.1152899 X102

1 2 0.4465325 X102 0.4465328 X102

2 1 0.4179062 X101 O. 4~;?9073 X101

. 1 3 -0.1963381 X102 -0.1963380 X102

~2 2 2 0.8256349 X101 0.8256357 X101

·3 1 -0.3480577 X101 -0.3480571 X101

1 4 -0.4353125 X101 -0.4353124 X101

2 3 -0.6880232 X101 -0.6880241 X101

3 2 -0.1338276 X102 -0.1338276 X102

4 1 -0.1998341 X101 -0.1998334 X101

58

Table 2-8-9 Comparison of boundary integral II

with domain integral 12

~ n m

1 1

1 2

2 1

1 3

2 2

3 1

~ 1 1 4

2 3

3 2

4 1

1 1

1 2

2 1

1 3

2 2 ~2

3 1

1 4

2 3

3 2

4 1

I J u* sin(~~X) Sin(~cr) dn

n

11 12

-0.6710900 X102 -0.6710900 X102

0.1452079 X102 0.1452079 X102

-0.9240564 X101 -0.9240568 X101

0.2394382 X10 2 0.2394383 X102 ,

0.6998923 X101 0.6998926 X101

0.2289640 X10 2 0.2289640 X102

-0.1129755 X102 -0.1129755 X102

0.4891803 X101 0.4891800 X10 1

-0.2101770 X101 -0.2101773 X101

-0.3598833 X101 -0.3598852 X101

-0.6089127 X102 -0.6089122 X10 2

0.1887393 X10 2 0.1887393 X102

-0.1031246 X102 -0.1031246 X102

0.2364258 X102 0.2364259 X102

0.6770080 X101 0.6770090 X101

0.1824759 X102 0.1824760 X102

-0.1415146 X102 -0.1415146 X102

0.5440886 X101 0.5440889 X101

-0.5698493 X101 -0.5698484 X101

-0.3653636 X101 -0.3653646 X101

59

Table 2-8-10 Comparison of boundary integral Il

with domain integral I2

~ n m

1 1

1 2

2 1

1 3

2 2

~3 3 1

1 4

2 3

3 2

4 1

1 1

1 2

2 1

1 3

2 2 ~4

3 1

1 4

2 3

3 2

4 1

I Jf u* . (nnx) . (mny) dn s~n --ro s~n 20

n

Il I2

-0.4055582 Xl02 -0.4055582 Xl02

0.1226002 Xl02 0.1226003 Xl02

-0.2847070 Xl01 -0.2847061 Xl01

0.6456060 Xl01 0.6456055 Xl01

0.4988304 Xl01 0.4988313 X101

0.1210875 Xl02 0.1210877 Xl02

-0. 133441 0 Xl02 -0.1334412 Xl02

-0.2021419 Xl01 -0.2021427 Xl01

-0.3698058 Xl01 -0.3698051 Xl01

-0.7289629 Xl01 -0.7289616 Xl01

-0.9541643 Xl02 -0.9541643 Xl02

0.7483069 Xl01 0.7483064 Xl01

-0.1921332 Xl02 -0.1921332 Xl02

0.3226102 Xl02 0.3226102 Xl02

0.2354544 X101 0.2354541 X10 1

0.2946567 X102 0.2946566 Xl02

-0.5857213 Xl01 -0.5857200 XlO 1

0.6655581 Xl01 0.6655580 Xl01

-0.1936955 X101 -0.1936950 Xl01

-0.1324903 Xl01 -0.1324900 Xl01

60

To decrease the numerical errors in B, one needs

only to take more integral points or divide the boundary

using finer meshes.

Table 2-8-11 presents the results obtained using the

finer meshes. It can be found that when NE equals 24

(that means there is one wave along every boundary element),

the relative error of the result decreases to only 4 x 10- 6

with Eq. (2-6-9). By contrast, for the same division as

in the original subdomain integral method, the results do

not improve as much as those for Eq. (2-6-9). This can be

explained by noticing that the domain integral has one

more dimension and here it causes more errors than the

boundary integrals.

In addition, formula (2-6-14) has the same accuracy

as formula (2-6-9).

All these mean that formula (2-6-9) or (2-6-14) gives

reliable results for the values of the domain integrals.

After testing the transformation formulae term by

term in series (2-6-1), some further examples are examined.

The exact results of following examples are using the

refined mesh, i.e. they are in the numerical sense.

EXAMPLE 2-8-1:

Let us assume that the source function b(x) is as

follows

b(x) (2-8-3)

The Fourier [:;erien e::presnio;'l of thin function is

Table 2-8-11

I

NX

~ NY

3

6

9

~2 12

15

18

21

exact solution:

61

Comparison of boundary integral II

with domain integral 12

J u* sin(nl~x)dn n

11

0.1968400 X10 1

0.1969215 X101

0.1969207 X101

0.1969207 X101

0.1969207 X101

0.1969207 X101

0.1969207 X101

.0.1969207 X101

n = 25

12

0.1919084 X101

0.2019204 X101

0.1968971 X101

0.1969194 X101

0.1969230 X101

0.1969200 X101

0.1969203 XHl1

62

Table 2-8-12

The results of Example 2-8-1

2 x 5

Relative errors of (%)

N I I function

2 -0.6785131 x 10 2 1.11 2.72

~1 5 -0.6713483 x 10 2 0.039 0.75

10 -0.6710841 x 10 2 -0.00082 0.19

exact. -0.6710896 x 10 2

2 -0.6153646 x 10 2 -1. 06 2.79

~2 5 -0.6090678 x 10 2 -0.026 0.76

10 -0.6088849 x 10 2 0.0046 0.19

exact. -0.6089122 x 10 2

63

(2-8-4)

The geometry and meshes are shown in Fig. 2-8-2

and the results are listed in Table 2-8-12. In this table

the relative errors of Fourier series (2-6-1) and the

results of the integrals are listed as well. From these

relative errors, it is clear that the convergence of ~e

results is faster than that of Fourier series. For example,

when N increases from 2 to 5, the relative errors of the the

Fourier expression of A function are improved from 2.72%

to 0.19%, but the relative errors of integral results are

improved rapidly from 1.11% to -0.00082% at point ~1.

This is because from relationships (2-6-10) one can find

that the coefficients of B tend to zero faster than the

coefficients of Fourier series as n is increasing.

EXAMPLE 2-8-2:

Let us suppose the source function is in the form of

This is a periodic function with a period 4a1 . Obviously,

if one takes the expansion domain nf as 4a l in the xl

direction, one can expect the solutions to have the same

accuracy as in Table 2-8-2 with n = 1. Let us see what

happens if one expands it as a periodic function with a

64

a 2 ------,

15

10

5

o ~~~ ___ ~ __ _L __ ~

2

Figure 2-8-2

7

Geometry and meshes for Examples 2-8-1, 2-8-2 and 2-8-3

period 2a1 , i.e.

b(x) 1 + 1 'IT 'IT

65

(2-8-5)

The results are listed in the Table 2-8-13 with the

same meshes in Fig. 2-8-2 as in example 2-8-1. From

Table 2-8-13 one can find that the relative errors of

integrals are much smaller than those of Fourier series

ofb(x).

EXAMPLE 2-8-3:

In this example one supposes function b(x) is equal

to unity. In this case, the expression of Fourier series

only has the constant term ko = 1. It means that one

expands b(x) as an even function. After periodic extension

of b(x), this function equals unity everywhere. The results

for the same mesh shown in Fig. 2-8-1 are listed in Table

2-8-1. The results have six or seven digit accuracy. Here

also to test the discontinuous function, one carried out

the period extension for b(x) as an odd function. Then

one can express it as a Fourier series in the form of

b(x) 1 4 'IT

y __ 1 __ sin (2n-1) nX1 n=l 2n-1 a 1

(2-8-6)

Using the Fourier series (2-8-6), one computes the

integral (2-6-14). The results are listed in Table 2-8-14,

from which one can also find the convergence of integrals

is faster than that of Fourier series. Obviously, the

accuracies of results in Table 2-8-14 are worse than those

shown in Table 2-8-1. Because the periodic extension is as

66

Table 2-8-13

The results of Example 2-8-2

2 x 5

Relative error of (%)

N I I function

5 -0.5993685 x 10 2 -0.027 0.91

~l 10 -0.5994077 x 10 2 -0.031 0.17

20 -0.5991896 x 10 2 0.0052 0.0498

exact. -0.5992106 x 10 2

5 -0.5507037 x 10 2 -0.016 0.98

~2 10 -0.5507957 x 10 2 -0.032 0.18

20 -0.5505929 x 10 2 0.0038 0.051

exact. -0.5506117 x 10 2

67

Table 2-8-14

The results of EXqmple 2-8-3

2 x 5

Relative errors of

~ N I I ( %) function(%)

2 -0.836827 x 10:1 -1.15 13.59

5 -0.827162 x 10:1 0.016 7.1

~1 10 -0.827076 x 10:1 0.027 3.69

Eg. (2-8-6) 15 -0.827288 x .• l0:l 0.00097 2.40

2 -0.767789 x 10:1 -1. 90 13.01

5 -0.752942 x 10:1 , 0.074 6.46 ~2

10 -0.753207 x 10:1 +0.039 3.29

15 -0.753461 x 10:1 0.0052 2.19

2 -0.847423 x loa -2.43 21. 85

5. -0.835158 x loa -0.95 7.14

~1 10 -0.827229 x loa 0.008 4.62

Eg. 20 -0.827246 x loa 0.006 2.44 (2-8-7)

2 -0.773523 x loa -2.66 21.15

5 -0.759431 x loa -0.79 6.47 ~2

10 -0.753249 loa 0.033 4.48 x

20 -0.753457 x loa 0.0057 2.19

~ = ~1 -0.827296 x loa exact.

~ = f,;2 -0.753500 x 10:3

68

an odd function for b(x), the values of b(x) have some

discontinuities on the lines as xl = 0, aI' 2a l , •••.

In contrast, b(x) is constant as an even function and has

continuous derivatives. According to the theorem in section

2-5, the above results are reasonable. If one considers

the period extension for b(x) as an odd function both in

Xl and x 2 directions, the function can be represented as

b(x) 1 16 1f2

N M 1 L L ...,-:::--~...,...,,----~

n=l m=l (2n-l) (2m-I)

. ( 2 n -1) 1TX 1 s l.n.------=-

. ( 2m-I) 1TX 2 s l.n.------=- (2-8-7)

Integrating with the same mesh, one lists the results also

in Table 2-8-14. It is clear the accuracy of the results

is much worse than before, because now the discontinuity

exists not only in the xl ' but also in the x 2 direction.

2-9 SOME FURTHER DISCUSSIONS

Based on the results discussed in the previous section,

if the source function can be expressed as a convergent

Fourier series, then sufficiently accurate solutions of

the formula (2-6-9) or (2-6-14) can be expected with enough

terms in the Fourier series. Usually only very

few terms are sufficient for the continuous function.

Let us consider that the distributed source function

b(x) is defined over all regions of the problem under

consideration, that is, the domain nb equals the domain n.

69

In this case, the boundary rb coincides with r. However,

if the source function b(x) is defined only on a part of

the domain n of the problem, i.e. ~ is a subdomain of nand

one can now apply two different approaches.

One of them is that the source function b(x) can be

extended to all the domain n, for example, a new function

b(x) is defined over the whole domain n as follows

J b(x) b(x) = 1 0 (2-9-1)

Then, one can compute the values of the integral (2-4-1)

with the transformation formula (2-6-9) and integrate along

the boundary r of the domain n.

The other approach is to compute the value along the

boundary rb of thesubdomain nb , noticing that

I (~) J u*b(x) dn

n (2-9-2)

Usually, the second approach is more efficient and

accurate than the first one, i.e. computing the integral

along the boundary r b • In addition, sometimes it simplifies

the expansion in Fourier series. For example, in Fig.

(2-9-1), b(x) only exists over the subdomain nb , which

is involved by n and the function b(x) is

b(x) 1: (2-9-3) otherwise

70

Figure 2-9-1 Notation

71

The extension of the function b(x) can be carried out

as follows

b(x) = 1 everywhere (2-9-4)

After making the expansion domain nf coincide with nb ,

in this case, only a constant term exists in the Fourier

series of b(x). The value of domain integral is computed

by the integral on the right-hand side in Eg. (2-9-2).

Obviously, it is an efficient and accurate approach.

However, how to choose the integral boundary Pb and

the expansion domain nf strongly depends on the features

of the function b(x). The following section shows some

examples with numerical results.

When function b(x) is a piecewise function, the

Fourier theorem ensures that the Fourier series converges

to the average of the right-and left-side limits. Con­

sider, for instance, that the function b(x) as shown in

Fig.2-9-2 (a) is expanded as a Fourier series for periodic

functions of period a l • Since b(O) ~ b(a l ),

the Fourier series converges to the value [b(O) + b(a l )]/2

= b(a l )/2 at the points xl 0 and a l (Fig. 2-9-2 (b».

This kind of discontinuity in Fourier series does not

produce any problems when using Eg. (2-6-9) or (2-9-14),

even in the case that the integral boundary Pb coincides

with the discontinuous lines of Fourier series. However,

in this case, more terms of Fourier series must be taken

for obtaining a sufficiently accurate expansion. For

this reason, it is better to avoid discontinuity when

choosing the expansion domain nf and the integrating

boundary Pb . In this example, the same function b(x)

72

(a) (b)

Figure 2-9-2 Periodic extensions of function b(x)

73

can be extended as a periodic function of period 2a l

on nf and computed on the boundary of fb with interval

[0, all (shown in Fig.2-9-2 (c». This way is more

efficient and accurate than the approach shown in Fig.

2-9-2 (b).

2-10 EXAMPLES

After testing the transformation formulae (2-6-9)

and (2-6-14) in the previous section, one presents here

some examples applying this transformation technique to

Poisson's equation. In these examples, only continuous

linear elements are used and all the integrals in the

form of (2-4-1) are computed with formula (2-6-9).

Besides the results and relative errors listed

later, some results are given related to the analyses

of errors, how to choose the extension of function b(x) ,

how to choose the expansion domain nf etc.

Comparing with the original subdomain integral,

one finds that the transformation formula is an efficient

and accurate approach to compute the integral (2-4-1).

EXAMPLE 2-10-1:

In this example, one assumes that the source

function b(x) in Poisson's equation (2-2-1) is the form

of

b(x) x " n (2-10-1)

74

and can easily find out the theoretical solution as

follows

l:iafxf 1 x~ u = - 12 1

(2-10-2)

and

dU x~

a1xf 1

dn - 3a1 (2-10-3)

One considers that the geometry of domain ~ (Fig.2-10-1) is a

circle centred at the origin of coordinates with radius

a 1 = 10. The prescribed boundary conditions are

Dirichlet type over all the boundary of the problem.

The given values for boundary conditions at all the mesh

points are obtained from expression (2-10-2).

The source function then can be expressed by

Fourier series as

b(x)

Results at boundary points and internal points

are listed in Table 2-10-1. From this table, one can

find:

A. The errors, which correspond to mesh A and N=5, can

be reduced using the finer mesh B with the same

number of terms N=5 in the Fourier series expression.

B. With the same mesh B, the results are not apparently

different between N=5 and N=10.

C. Some results of integral (2-4-1), which are computed

with formula (2-6-9), are listed in Table 2-10-2.

From them, it is clear then when the mesh is

refined from A to B (for there N=5), the results

change about 3%,but using the same mesh B

75

• internal points

x Itesh points

4·( 4

Figure 2-10-1 Geometry and meshes in Example 2-10-1

76

Table 2-10-1 The Results of Example 2-10-1

Fluxes at the boundary points

Mesh N 1 2 3 4

A 5 675.41 571.33 222.45 -22.29 1.31% 1. 57% -2.94% -

B 5 668.85 564.83 226.40 -8.30 0.33% 0.41% -1.21% -

B 10 668.64 564.50 226.67 -8.52 0.30% 0.36% -1. 09% -

Exact 667.67 562.50 229.17 0

Potential values at the internal points

Mesh N 5 6 7 8 9 10

A 5 -224.93 918.42 1974.17 103.80 351. 31 746.11 5.44% 1.13% 0.67% 33.20% 13.60% 8.66%

B 5 240.48 914.18 1964.85 87.71 321.13 701.33 3.41% 0.66% 0.19% 12.56% 3.84% 2.14%

B 10 240.24 914.16 1965.01 87.49 321.10 701. 55 3.31% 0.65% 0.20% 12.28% 3.80% 2.17%

Exact 232.54 908.20 1861. 06 77.92 309.25 686.65

77

Table 2-10-2 The Results of Integral (2-4-1) in

Example 2-10-1

At boundary points

Mesh N 1

A 5 -53684.75

B 5 -55072.12

B 10 -55098.05

* At internal points

Mesh N 5

2

-53100.58

-54435.98

-54461.98

6 7

A 5 -0.398901 -0.429945 -0.476575

B 5 - 0.411500 -0.442669 -0.489507

B 10 -0.411777 -0.442952 -0.489796

3 4

-51936.06 -51354.02

-53162.21 -52526.11

-53194.01 -52560.28

8 9 10

-0.396652 -0.433191 -0.463712

-0.409169 -0.434585 -0.475912

-0.409450 -0.434878 -0.476236

* All the figures have to be multiplied by 10 5

78

and increasing the number of terms from N=5 to

N=10, the results are improved only about 0.05%.

D. From above A, Band C, one can see that the errors

are mainly due to use of too coarse meshes.

EXAMPLE 2-10-2:

In order to compare results, let us set the potential

function u(x) in the form of

2 2 lIx1 u(x) = (2.) cos(-)

11 2a x c: n (2-10-5)

This function satisfies Poisson's equation with the

source function b(x) as

(2-10-6)

so that, the flux function on the boundary is

x <:: f (2-10-7)

where n 1 is the first component of direction cosine

on the boundary f, i.e. n 1 = cos(n,x1 ) and n is outward

unit normal of f •

. The representation in Fourier series of function

(2-10-6) is

b (x) 2 11

1 N (_l)n nTIx 1 11 L 1 cos (-a-)

n=l '4 - n 2

(2-10-8)

The configuration of domain n of the problem under

consideration, mesh points and internal points are shown

79

in Fig.2-l0-2. The expansion domain for the Fourier

series, nf is [-a l ~ xl ~ aI' -a2 ~ x 2 ~ a21 and a l = 10,

a 2 = 20.

The boundary conditions at all mesh points are

prescribed as potential values from Eq. (2-10-5). The

results of potential at internal points are listed in

Table 2-10-3. One can see accuracy of results is

sufficiently good even for only N=2. They improve

rapidly from N=2 to N=5 especially at poi~ 3~

The values of integral (2-4-1), which are computed

by formula (2-6-9), are listed in Table 2-10-3 as well.

Comparing the accuracies of potential and integrals

in Table 2-10-3, one can find that the errors in potentials

at internal points are caused by the coarse meshes. The

relative errors of the values of integral (2-4-1) are

improved by more than thirty times from N=2 to N=5.

But the relative erors of the results of potential at the

internal points are only improved two or three times

from N=2 to N=5 at internal points I' and 2'.

EXAMPLE 2-10-3:

This example is for the case for which the source

function b(x) exists only partly in the domain n of the

problem under consideration. One supposes that:

b(x) = 1: (2-10-9) elsewhere

and the boundary conditions are homogeneous, i.e.

u o on r

80

15

3~ • internal points

10 l' 2'

x zresh points . .

5 2 3

0~~2~------~7----------

Figure 2-10-2 Geometry and mesh in Example 2-10-2

81

Table 2-10-3 Results of Example 2-10-2

The Values of Potential at the Internal Points

Internal Point N l' 2' 3'

2 34.020 27.185 30.858 0.088% 0.245% 0.130%

5 33.9BO 27.15B 30.B16 -0.029% +0.146% -0.006%

Exact 33.990 27.119 30.B1B

The Values of Integral (2-4-1) at the Mesh Points

Mesh Point

N 1 2 3

2 60.3416 55.4421 57.0107 0.7018% 0.6929% 0.9480%

5 59.9369 55.0704 56.5014 0.0265% +0.0162% 0.0462%

Exact 59.9211 55.0612 56.4753

82

The geometry of the domain and mesh points are

shown in Fig. (2-10-3).

As mentioned in section 2-9, one can extend the

function b(x) as an even function, so that the expression

of Fourier series of b(x) can be represented as follows

b(x) = 1 everywhere

Hence, in this case, the integral (2-4-1) is only

integrated over the domain Qb as shown on the right­

hand side in Eq. (2-9-2) and the boundary of integration

in formula (2-6-9) is r b • Dividing rb into boundary

elements, one can obtain the results of (2-6-9) for

certain source points. In the numerical practice one

finds that only four boundary elements, each of which

is one side of the rectangular domain Qb' are sufficient

to obtain the exact results in the numerical sense. The

values of integral (2-4-1) for various mesh points are

listed in column 'exact'in Table 2-10-4. The values of

potential, which are approximated numerically, are listed

in the same column as 'exact' too. But the errors of

potential values are due to computing the first and

second integrals on the right-hand side of Eq. (2-2-9)

numerically, and are not due to computing the third

integral in Eq.(2-2-9) with the transformation formula

(2-6-9) •

One can also expand function b(x) as a Fourier

series in the domain Q, i.e. the expansion domain Qf

coincides with Q.

83

I I I I I I : I I

14 __ ~ __ fL-

o r. I b b I

--- /; ---f' 11--I r I I I I I I 0 I

-- _fL_f~-F-I I I I I I

• internal points

x rresh points

I I I

Figure 2-10-3 Geometry and mesh in Example 2-10-3

84

Table 2-10-4 Results of Example 2-10-3

The results of integral (2-4-1) at the boundary points

Number N = 2 N = 5 Exact of points

1,4 -6.8559508 -7.1768198 -7.0897632

-3.30% 1. 23%

2,3 -6.6487422 -6.9796948 -6.8836842

-3.41% 1. 39%

5 -7.1871481 -7.5382462 -7.4431710

-3.44% 1. 28%

6,9 -6.0212121 -6.3417339 -6.2561951

-3.76% 1. 37%

7,8 -4.9700890 -5.2666259 -5.1861911

-4.17% 1. 55%

10,13,16 -5.4122548 -5.5643382 -5.5176592

-1.91% 0.85%

11,12 -4.9976969 -4.9998822 -4.9908090

0.14% 0.18%

14,15 -3.5861070 -3.6087379 -3.5955679

-0.26% 0.37%

The results of potential at the internal points

Number N = 2 N = 5 Exact of points

17 -0.0823950 -0.1218647 -0.1121903

-26.56% 8.62%

18 -0.1125917 -0.1588545 -0.1474590

-23.65% 7.73%

19 -0.0697340 -0.1012834 -0.0936967

-25.57% 8.10%

20 -0.2401351 -0.2758389 -0.2670178

-10.07% 3.30%

21 -0.2331309 -0.2518989 -0.2470396

-5.63% 1.97%

85

In this case, one obtains more complex Fourier

coefficients as follows

ko

kl n

k 2 n

k 3 n

k~ n

1 d 3 ) -- (d - d ) (d -a 1a 2 2 1 4

(d4-d3) ~ nTTd 2 sin (--) nTTd 1 ]

- sin(--)

-

-

2nTTa 2

(d4-d3 )

2nTTa2

(d2-d1 )

2nTTa1

(d2-d1 )

2nna 1

1 n2nm

1 - n2nm

1

a 1 a 1 .

G nTTd 2 cos (--) a 1

nTTd 1 ] - cos(--) a 1

sin ( __ 4) -~ nTTd a 2

nTTd 3 ] sin (--) a 2

~ nnd4 nnd 3 J cos (--) - cos (--) a 2 a 2

mnd 3 ] - sin(--) a 2

mnd 3 ] - cos(--) a 2

mTTd 3 J - sinC·-) a 2

= -- ~ nnd2 cos (--) -a 1

nTTd 1 l r mTTd 4 cos (-a-) ICOS (-a-) TTnm 1 .. ' 2

mTTd 3 ] - cos(--) a 2 .

(2-10-10)

where a 1 6, a 2 = 10

d 1 x(2), d 2 = x(3)

d 3 y(15), d 4 = y(14)

86

The results obtained integrating along the boundary

r with formula (2-6-9), are shown in Table 2-10-4.

By comparison, the first approach, which is integration

along rb , is much more efficient and accurate than the

latter one, obviously.

EXAMPLE 2-10-4:

This example analyses the torsion of a shaft with

keys as shown in Fig. 2-10-4. The governing equation of

this problem is Poisson's equation. By using a dimension-

less torque function, u, the value of b(x) can be set

to two. The shear stresses in a section are expressed

by

au T ~ zy

(2-10-11)

au T -ay zx

Because of symmetry, only a quarter of the shaft

is calculated.

Obviously, the simple&way is to extend b(x) as

an even function and then to expand b(x) with only ko

equal to two in the Fourier coefficients, the others

are all zero. The results are shown as case A in Table

2-10-5.

.I

I I

I I

D

/ I/,'"

I /'".'" / /'/

I ~/ I~

MESH2

B A

\ I \ I

\ 1 'I

l

MESH 3

\ I \1 \I \I

87

14

mash points

Figure 2-10-4 Geometry and meshes in Example 2-10-4

88

Table 2-10-5 The Shear Stresses in Example 2-10-4

Item Mesh N A B B' C D

BEASY I a zx 110.0 301. 0 180.1 78.4 95.2

a 0 0 -310.4 -111. 8 -55.1 zy

II a zx 112.8 253.6 195.3 75.2 88.3

a 0 0 -338.2 -107.4 -51. 0 A zy

III a 104.8 zx

204.8 156.7 63.6 91.8

a 0 0 zy -271. 4 -90.8 -53.0

III 5 a 102.1 zx 198.9 152.3 61.8 87.6

a 0 0 -263.7 -88.2 -50.6 zy

B III 10 a 103.6 200.3 153.3 62.2 89.9 zx a 0 0 -265.5 -88.8 -51. 9 zy

III 20 a 104.5 202.4 154.9 62.7 90.8 zx a 0 0 -268.3 -89.6 -52.4 zy

III 5 a 98.9 zx 174.2 113.2 53.4 80.8

a 0 0 zy -196.1 -76.3 -46.3

c III 10 a zx 99.4 191. 6 146.6 59.5 87.4

a 0 0 zy -254.0 -85.0 -50.5

III 20 a 102.0 zx 197.0 151. 3 61.3 89.5

a 0 0 zy -262.1 -87.6 -51.7

89

By comparison, b(x) is extended as an odd function

as well. There are two extensions for function b(x)

as follows:

One of them is

x > 0 b(x) (2-10-12)

x < 0

Then after expanding the Fourier coefficients are

k 0 0

kR. 0 n

j 8 k 2 nlT n

0

kR. 0 nm

Another is

b (x) -j' - 2

R. 1,3,4

n 1,3,5, ••• , (2N+1)

n 2,4, •••

x.y > 0

x.y < 0 (2-10-13)

and the corresponding Fourier coefficients are

k 0 0

kR. 0 n kR. 0 R. nm 1,2,3

j" k4 ~mlT2 nm

n, m = 1, 3, ••• , (2N+ 1)

n or m = 2,4, •••

90

In the first one, b(x) is discontinuous on the

axis y in Eq. (2-10-12), but in Eq. (2-10-13) b(x) is

discontinuous on both axes x and y. The corresponding

results are listed as cases Band C in Table 2-10-5.

The results in Table 2-10-5 are compared with

solutions found using the BEASY code [22] with quadratic

boundary elements. Obviously, from the shear stresses

at the singular points Band B', one can find the results

of case A are excellent. In contrast, in cases Band C

one can not expect very accurate solutions. They also

require much more CPU times than that in case A because

of the discontinuity of b(x) in cases Band C.

2.11 THE TRANSFORMATION FORMULA FOR 3-D POISSON'S

EQUATION

In order to derive the formulations, which trans-

form the domain integral into boundary ones in 3-D,

a similar procedure as in section 2-6 can be carried

out.

Suppose there the function b(x) in Eq. (2-4-1) is

a piecewise continuous function in a 3-D region

After periodic

extension of the function b (x) , this can be expanded

as a convergent Fourier series:

b(x) b(xl'x2,x3)

6 kR.fR. +

12 kR. fR. kofo + L L L L L +

R.=l n n R.=l nm nm n n m

8 + L L L L kR. fR.

i=ln m p nmp nmp (2-11-1)

91

where f 1 a

f I nlfx 1 nlfx 1

cos (-a-) f;! sin (--) n 1 n a 1

nlfx 2 f4

nlfx 2 f3 cos (-a-) sin (--) n 2 n a 2

fS nlfx 3

f6 nlfx 3 cos (--) sin (--) n a 3 n a 3

fl flf3 f;! flf4 nm n m nm n m

f3 f2f3 f4 f;!f4 nm n m nm n m

fS f3fs f6 f3f6 nm n m nm n m

e t4f s f8 f4f6 (2-11-2) nm n m nm n m

f9 fSfl flo f s t2 nm n m nm n m

f I I f6fl f 12 f6f2 nm n m nm n m

fl e f3fS nmp n m p

f;! flf3f6 nmp n m p

f3 flf4fS nmp n m p

e flf4f6 nmp n m p

fS f;!f3fS nmp n m p

f6 f;!f3f6 nmp n m p

e f;!ef s nmp n m p

f6 f;!f4f6 nmp n m p

92

and the Fourier coefficients are as follows:

ko 1 III b(x)dll 8a 1a 2a 3

II

kR. 1 III b (x) f~(X)dll n 4a 1a 2a 3 II

kR. 1 fJI b(x) f~m(X)dll (2-11-3) nm 2a1a 2a 3 II

kR. 1 III b(x) f~mp(X)dll nmp a 1a 2a 3 II

Now we define some functions in the following way:

R. u nmp

a fR. _(-1.)2

nn n a

fR. _(~) 2 nn n

a fR. _ (-2) 2

nn n

fR. nm

n2 Ull.) 2 + a 1

fR. nm

n2 Ull.) 2 + a 2

fR. nm

n2 [(ll.) 2 + a 3

R. 1,2

R. 3,4

R. 5,6

R. 1,2,3,4 (2-11-4) (~) 2J a 2

R. 5,6,7,8 (~) 21 a 3

R. 9,10,11,12 (~)2J a 1

where x10 ' x 20 ' x 30 are constants, which are the coordinates

of reference point xo'

93

f . iii The above unct~omuo,un' unm and u nmp satisfy

the following equations respectively:

\]lU f 0 0

\]lU~ fi n

\]lU i fi = nm nm

(2-11-5)

\]lU i fi nmp nmp

Then, defining

a go an u

0

i a fi gn an n

i a fi gnm = an nm

(2-11-6)

i a fi gnmp = an nmp

and

1,2

3,4

5,6

(2-11-7)

94

k~ nm

~ 1,2,3,4 1T2 [(..!!..) 2 + (~) 21

a 1 a 2 _

~ k~

nm ~ 5,6,7,8 Snm [ n 2 (~) 2] 1T2 (-) +

a 2 a 3

k~ nm

~ 9,10,11,12 [ n 2 (~) 2J 1T2 (-) +

a 3 a 1

k~ nmp

and using the second Green's identity, we obtain the

transformation formula for 3-~ as follows:

I (E,;) fU*bdr.l

r.l

6 + L L

~=1 n

12

df - f q* Uo df - c Uo 1 +

f

f ~

df - f q* f~ dr u*gn - c n f f

f~ n

+ L L I f ~

u*gnm df - f q* f~ df - c ~=1 n m f

nm f

8 ~

f * ~ f f~ + I I I L Snmp u gnmp df - q* df ~=1 n m p nmp

f f

- c f~ nmp

1 +

f~ 1 + nm

(2-11-8)

where c

u*

95

c(~) is shown in Eq.(2-2-10) and

1 411r q* 1 ar

- 411r2 an

The coefficient c(~)depends on the position of the

source point ~.

The numerical techniques for computing formula

(2-11-8) are similar to those mentioned in section 2-7.

2-12 APPLICATIONS IN TIME-DEPENDENT PROBLEMS

Not only does the source function term produce domain

integrals in potential problems, but also time-dependent

or non-linear terms can cause this type of domain integral

in boundary'e1ement formulations. A similar method, which

transforms domain into boundary integrals for the source

term, can be applied to these problems. In this section

time-dependent problems are discussed. After recasting

the partial differential equation into a boundary integral

form the similarities between the previous and the present

problems are pointed out.

Diffusion problems are governed by the following

equation:

V2 u(x,t) a at u(x, t) x (: g

with boundary conditions of the two types:

Dirichlet B.C. u(x,t)

Neumann B.C. q(x,t)

= u(x,t)

a an u(x,t)

(2-12-1)

q(x,t)

(2-12-2)

96

and initial conditions

u(x,O) = uo(x) (2-12-3)

where u, q and Uo are known functions.

To recast Eq. (2-12-1) with boundary and initial

conditions (2-12-2) and (2-12-3) into a boundary integral

equation, it is usual to combine with finite differences.

The derivative of function u with respect to time t,

d at u , can be expressed in finite difference form as

follows

d at u(x,t) 1 ~t [u(x,t + ~t) - u(x,t)] (2-12-4)

where ~t is a sufficiently small time step. Eq. (2-12-1)

can then be rewritten as

1 V2 U (X,t + ~t) - ~t u(x,t + ~t) 1

- ~t u(x,t)

(2-12-5)

Now the fundamental solution of the above Helmholz

equation, u*, is introduced [40,41]:

- ~(cx) (2-12-6)

u* for 2-D

(2-12-7)

where Ki is the modified Bessel function of the second

kind of order i.

97

Applying the procedures described in section 2-2,

the differential equation (2-12-5) can be rewritten as

a boundary integral equation, i.e.

c(t;) u(t;) J u*q dr -

r J q*u dr + It J u*u dn r n

(2-12-8)

where the unknowns u and q are at time step t+~t, and u

is a known function of u(x,t) over all the domain n from

previous time step t.

Using the finite difference method with the time-

independent fundamental solution (2-12-7), Eq. (2-12-8)

can be solved in series of time steps. To obtain suffic-

iently accurate results, small 6t time steps must be used

and after every step the solutions at a sufficient number

of internal points must be computed to find u which can

then be expressed as a Fourier series. For these

reasons this method usually requires large CPU times.

Instead of using the time-independent fundamental

solution, one can apply the time-dependent fundamental

solution u* in the form of [42,43]

u* 1 d/2 exp [- 41r (~~l)l H (t-l) 411 (t-l) ]

(2-12-9)

where d is the number of spatial dimensions of the

problem. H is Heaviside function. The function u*

possesses the following properties:

98

a 1J 2 u*(E;,x,t-T) - at u*(E;,x,t-T) - ll(E;,x)ll(t,T)

(2-12-10)

After applying the fundamental solution (2-12-9)

one can rewrite Eq. (2-12-1) as a boundary.integral

equation, i.e.

c(t,)u(E;)

t

J J u*q dfdT -

o f

t

J J q*u dfdT + o f

(2-12-11)

Note that in the domain integral on the right-hand side

of the above equation there is only the known function

Uo from the initial condition. Comparing with Eq. (2-2-9)

and setting the notation b(x) = -uo(x) , one finds that

Eq. (2-12-11) is of the same type as Eq. (2-2-9) excepting

the different fundamental solutions u*. The Fourier

expansion (2-6-1) can be used now after setting the

fundation b(x) = -uo(x). Because of the different

fundamental solutions, Eqs. (2-6-5) must be changed as

follows

1J 2 u a f -at u 0 0 0

(2-12-12)

1J 2 u a fll. -at u n n n

1J 2 u lI. a lI. fll. nm -at u nm nm

Consequently, the transformation formula (2-6-9)

or (2-6-14) can be used in the same form for the domain

integral in Eq. (2-12-11) .

99

The same procedure can be applied for hyperbolic

time-dependent problems.

2-13 APPLICATION IN NON-LINEAR PROBLEMS

Usually non-linear equations are solved numerically

by applying an iterative procedure: substituting the

results of a previous step into the non-linear term, which

can then be considered as a known function. Thus, for

each step only a linear problem is solved.

Suppose a non-linear equation is given by

L(u) + N(u, .•. ) R(x) (2-13-1)

where L is a linear operator, function N is a non-linear

function of unknown function u and its derivatives, R

is a function of the spatial coordinates. Using

iterative procedures, Eq. (2-13-1) can be expressed as

follows for the s step.

(2-13-2)

where Us and u s _ 1 are the solutions for the sand s-l

steps, the former are unknown and the latter are known

from the previous s-l step. In this case it is possible

to solve Eq.(2-13-2) using boundary element methods

with the fundamental solution u*, which satisfies the

linear operator:

L(u*) + lI(Cx) 0

After having defined b(x) = R-N, one can transform

the differential Eq. (2-13-1) with its boundary conditions

100

into a boundary integral equation with a domain integral

term as

1(0 J u*b(x)dl1 11

J u*[R(x) - Njdl1 11

(2-13-3)

This integral can be transformed into boundary integrals

as explained before.

CHAPTER 3 LINEAR ELASTOSTATICS

3-1 INTRODUCTION

In this chapter the bodies under consideration are

assumed to be homogeneous and isotropic. It is also

assumed that their behaviour can be analysed using linear

theory of elasticity which implies that:

A. Within certain limits the behaviour of the material

can be represented by a linear relationship between

stresses and strains.

B. The relationship between displacements and strains

is also linear, that is second or higher order

derivatives of displacements with respect to spatial

coordinates can be neglected.

A short review of the basic linear theory of elasto­

statics is presented first, including definitions of

stresses and strains, relationship between stresses and

strains, stress equilibrium equations, compatibility

equations and boundary conditions. In order to derive

the basic boundary integral equation of elastostatics,

the Somigliana identity and the fundamental solutions of

elasticity are also presented. Then the fundamental

concepts of BEM in elasticity are described, using the

2-D case for simplicity.

The main concepts are introduced regarding computation

of body forces integrals, following the same idea as

described in the previous chapter, i.e. using the trans­

formation formula, which evaluates the body force domain

integrals in terms of only boundary integrals. The

102

numerical implementation of this technique is discussed

and numerical results presented. From these solutions

it is clear that the transformation formula is reliable

for computing the body force integrals.

3-2 BASIC RELATIONSHIPS OF ELASTICITY

Let us define the state of stress at a point as a

stress tensor

o = (0 .. ) ~J

(3-2-1)

In the absence of body moments, the following relationships

apply for the tensor components

o .. ~J

o .. J~

(3-2-2)

Introducing the distributed internal body force

vector b = (b i ), the following equilibrium equations over

the body can be obtained

0 ... + b). ~J,~

o (3-2-3)

When the traction vector ~ acting at a point on the

boundary of the body is represented by ~ = (Pi)' the

following relationship holds over the boundary

p.=o .. n. ~ J~ )

(3-2-4)

where (n j ) represents the direction cosines of the outward

normal vector on the boundary.

A body is displaced from its original place or

configuration under the action of forces. Let (Xi) denote

the initial position of a point Q, (x.+u.) its deformed ~ ~

position i then (ui ) represents the displacement

103

vector of point Q (Fig.3-2-1). According to the linear

theory, the strains can be represented by the following

Cauchy infinitesimal strain tensor:

Eo 0 = ~(uo 0 + Uo 0)

1J 1,J J,1 (3-2-5)

To solve for displacements from Eq. (3-2-5) uniquely,

the compatibility equations of strains must be satisfied:

Eo 0 k. + Ek • 0 0 - Eo k o. - Eo. 0 k 1J, '" "',1] 1 ,]'" ]",,1

o (3-2-6)

All the above relationships are independent of

material properties. If the material is linear and

isotropic, the linear Hooke's law relating stress and

strain tensors can be stated in the form of

(3-2-7)

or inversely

Eo 0

1] (3-2-8)

where v is Poisson's ratio and G is the shear modulus.

~ and ~ are Lame's constants. Notation E is Young's

modulus.

For an isotropic material only two of these constants

are independent and can be related to one another by the

following relationships

Ev (l+v) (l-2v)

G E

2 (1+v)

2Gv 1-2v

(3-2-9)

104

Figure 3-2-1 Displacement

105

Hence in linear elasticity, the variables are (ui ),

(a .. ) and (E: •. ) and the equations are Eqs. (3-2-3), (3-2-5) ~J ~J

and (3-2-7) (or (3-2-8)). These equations can be further

manipulated and the equilibrium equation can be written

as the Navier equation, in which the variables are the

displacement components, i.e.

G + _G_ b uj,kk 1-2v Uk,kj + j o (3-2-10)

Using Eqs. (3-2-4), (3-2-5) and (3-2-7), one can also

obtain the traction boundary condition on the boundary

as follows

2Gv 1- 2 " uk, k n; + G (u. . + u. .) n. = p.

v ~ ~,J J,~ J ~ (3-2-11)

For the case of elastostatics, the boundary conditions

can be expressed as the fdlowing two types

The displacement B.C. u j on r 1 (3-2-12)

and the traction B.C.

(3-2-13)

where u i and Pi are prescribed displacements and tractions

and the total boundary is r = r 1 + r 2 (Fig.3-2-2).

3-3 FUNDAMENTAL SOLUTION FOR ELASTOSTATICS

The fundamental solutioffi for displacement u~. are ~J

the solutions of the Navier Eqs. (3-2-10) for an unbounded

domain when applying particular concentrated body forces

as follows

106

Figure 3-2-2 Boundary conditions

107

(3-3-1)

In this case, the fundamental solutions Uij satisfy

o (3-3-2)

The body force components b! in Eq. (3-3-1) correspond to

a positive unit load applied at a point ~ in the direction

of the unit vector e i • Here u~.(~,X) represent the ~J ::. .

displacements in j direction at the point x corresponding

to a unit force bi applied at point ~ in the i direction.

After substituting the fundamental solutions of

displacements u*ij into Eq.(3-2-11), the fundamental

traction components prj can be found. Then from the

definition of strain tensor in Eq. (3-2-5) and the

relationship (3-2-7) the corresponding fundamental

solutions of strain (E'!j) and stress (Gij) tensor components

are obtained. The fundamental solut~ons due to Kelvin

can be represented as follows

1 16n(1-v)Gr [(3-4v)oij + r,ir,j]

for 3-D (3-3-3)

1 8n(1-v)G [(3-4v)ln r 0ij - r,ir,j]

1

for 2-D plane strain

(3-3-4)

C3r { [(l-2v) 0 .. + B r . r . h-- -4an(1-v)ra ~J,1 ,J ~n

- (1-2v)(r .n. - r .n.)]} ,1 J ,J 1

(3-3-5)

108

and

[(1-2\1)(r k O" + r .o'k) 8aH(1-\l)Gra , 1J ,J 1

1

-r 'O'k + B r .r .r k1 ,1 J ,1 ,J , (3-3-6)

* O'k'(Cx) J 1 [(1-2\1) (r k O .. + r .ok'

4aH(1-\l)r a , 1J ,J 1 1

- r . 0J'k) + B r .r .r k1 ,1 ,1 ,J ,

(3-3-7)

where r

for 3-D and 2-D plane strain respectively, B a+1.

The notation £jki and 0jki represent the components of

strain £jk and stress 0jk at point x due to a unit point

load applied at s in i direction:

£jk £jki e. 1

(3-3-8)

°jk °jki e. 1

Note, the plane strain expressions are valid for plane

stress only if \I is replaced by \1/(1+\1).

3-4 SOMIGLIANA IDENTITY

In order to derive the boundary integral equation

of linear elasticity, the weighted residual method can

be used [1,401, although one can use the Somigliana

identity as will be shown here.

Consider a body defined by ~ with its boundary r

(Fig. 3-4-1), which is in equilibrium under some given

loads and displacements. This state is represented by

the variables 0 .. , 1J

109

Figure 3-4-1 Elastic body with region n and boundary r

110

In addition, suppose there is another domain Q*

with boundary r*, which contains the body Q+r of the

problem under consideration (Fig. 3-4-2) and with the

same properties as Q. As before, this domain is also

in equilibrium with variables a~" E~, , u~, p~ and b~. ~) ~) 1. ~ 1.

Now, the well known reciprocity principle, which can

be inferred by simple symmetry of the constitutive

equation of stresses and strains, is introduced, i.e.

(3-4-1)

After integrating by parts both sides of Eq. (3-4-1)

and considering the equilibrium equations under the Q

integrals, one obtains

f bk u~ dQ + f Pk u~ dr n r

(3-4-2)

which corresponds to Betti's second reciprocal work

theorem.

Eq. (3-4-2) can be further modified by assuming uj'

a!)" E~" p~ are fundamental solutions which are caused ... .1.) )

by bI = 6(~,x)ei' where ~ and x belong to Q*. Therefore,

if ~ c Q too, the first integral in Eq. (3-4-2) becomes

(3-4-3)

Furthermore, considering the point loads bk are

applied separately in each direction, the corresponding

displacements and tractions can be rewritten in the form

of

111

Figure 3-4-2 General region Q* + r* containing the body Q + r

112

(3-4-4)

p~ = p~. (Cx)e. J 1.J 1.

where Uij(~'x) and pij(~'x) represent the results in

j direction at the point x corresponding to a unit point

force applied at point ~ in the i direction. After that,

substituting Eqs. (3-4-3) and (3-4-4) into Eq. (3-4-2) ,

Eq. (3-4-2) can be rewritten in function of components

J ulj (ex) Pj (x) dr (x)

r - J Pij(~,x)Uj(x)dr(x) +

r

+ J u.*. (~,x)b. (x)d~(x) 1.J J

(3-4-5)

~

The above Eq. (3-4-5) is called Somigliana identity.

3-5 THE BOUNDARY INTEGRAL EQUATIONS OF ELASTOSTATICS

This section considers the derivation of the boundary

integral equation applying the Kelvin fundamental solutions

and Somigliana identity presented in the previous two

sections.

As in the case of potential problems, Eq. (3-4-5) must

be modified for when the point ~ is on the boundary r.

For this purpose, the boundary r of domain ~ can be

augmented by a small region r£ (Fig.3-5-1). Therefore,

the second integral on the right hand side of Eq. (3-4-5) can

be written as

and

113

r

Figure 3-5-1 Augmented boundary

J Pij(~,X)Uj(x)dr r

+ lim J £-+0 r£

lim £-+0

(3-5-1)

p~.(~,x)dr ~J

After that, the first integral on the right-hand side

of Eq.(3-5-1) is evaluated in the Cauchy principal

value sense [44], the existence of which can be proved

if u(x) satisfies a Holder condition [45] at the point

~ in the form

114

where B and a are positive constants.

The first integral on the right-hand side of Eq.

(3-4-5) has no contribution from f£ since

(3-5-2)

Therefore, after defining Cij(~) in the form

(3-5-3)

one obtains the boundary integral equation of linear

elastici ty, i. e.

c .. (Ou.(O 1.) ) J u~. (~,x)p. (x)df(x) -

1.) ) f

- J p~. (I;,x)u. (x)df(x) 1.) )

+ J uij(I;,X)bj(x)dn(x) n f

(3-5-4)

It relates the surface displacements and surface tractions

With the boundary conditions, the unknown displacements

and tractions on the boundary can be solved numerically

by means of BEM.

Once the displacements and tractions on the boundary

f are solved, Eq. (3-5-4) can also be used to find out the

displacements at any internal points I; inside the domain n.

o

115

If the stress state at internal points is required,

they can be obtained by:

A. Taking derivatives of Eq. (3-5-4) with respect to

the coordinates of ~ to obtain the strain tensor.

B. Substituting the strains into Hooke's law (3-2-7).

The expression of stresses at internal points can be

reduced to

J Uijk(~,x)pk(x)dr(x) -

r

- J pijk(~,x)uk(x)dr(x) + J Uijk(~,x)bk(x)dn(x) r n

(3-5-5)

The tensor terms Uijk and pijk corresponding to

Kelvin fundamental solutions are in the form of

- 0* ijk (3-5-6)

G ar --"'----::-B {B an [( 1-2\1)0 .. r k 2a~(1-\l)r ~J ,

+ \I(o;k r . + 0J'k r .) - y r .r .r kl + L,J ,~ ,~ ,J ,

+ B\I(n.r .r k + n.r .r k) + (1-2\1) (Bnkr .r . + ~ ,J, J ,~ , ,~ ,J

where a = 2 and a = 1 for 3-D and 2-D plane strain

respectively. B = a+1 and y = a+3. For 2-D plane

stress problems, as before, it is necessary to use

116

-v v/(1+v) instead of v in all above equations.

3-6 THE BOUNDARY ELEMENT METHOD IN ELASTICITY

The boundary integral equation with boundary con-

ditions can be solved numerically in potential problems

as shown for Eq. (2-2-9). The same approach can be used

in elastostatics.

For the discretization of Eq.(3-5-4), the boundary

r is approximated by a certain number of elements. For

2-D linear elements, the edges of each element are taken

as nodes, at which the values of variables are considered.

In this case, the displacement 0' traction and body force

can be expressed with their components:

u = {::) {::) b

{::) (3-6-1)

and the fundamental solutions can be written as matrices:

p* u*

For linear elements, linear interpolation functions are

used

~1 ~(1-n) (3-6-3)

~2 ~(1+n)

Then, the variables ~ and p can be expressed with the

values at nodes and interpolation functions ~1 ' ~2 :

<P 1u j1 + <P 2Uj2

<P 1Pj1 + <P 2Pj2

117

(3-6-4)

where u j1 ' u j2 and Pj1, Pj2 are displacements and

tractions at nodes 1 and 2 of the element j. After that,­

in a similar way to that in Section 2-3 - by considering

the prescribed boundary conditions one can obtain a set

of linear algebraic equations, which contains all the

unknown variables at the nodes. These unknown variables

can be solved from this set of linear algebraic equations.

Then, if the displacements and stresses at the internal

points are required, they can be computed from Eqs. (3-5-4)

and (3-5-5) numerically.

During the above numerical procedures, domain

integrals, which exist in the last terms in Eqs. (3-5-4) and

(3-5-5), are usually computed by subdividing the original

domain into subdomains or cells and using numerical

quadrature formulae over subdomains. The approach however

is expensive as it requires domain integrations, but the

following section presents instead a transformation

technique which can transform domain into boundary integrals.

3-7 BASIC FORMULATIONS FOR TRANSFORMING 2-D ELASTICITY

DOMAIN INTEGRALS TO THE BOUNDARY

In Eqs.(3-5-4) and (3-5-5) there exist domain

integral terms of the forms

J u~. (I;,x)b. (x)d!1(x) ~J J

(3-7-1)

!1 and

118

J u!jk(~,x)bk(x)dQ(x) Q

(3-7-2)

Using Somigliana identity, one derives in this

section the formulations to transform domain integrals

such as (3-7-1) and (3-7-2) into boundary integrals. The

Somigliana identity describes the relationship between

displacements u j ' tractions Pj and body forces b j

distributed on the boundary and in the body respectively.

Note, that in the Somigliana identity, the displacements

and tractions are present only in the boundary integrals

and the body forces are in the domain integrals. There-

fore if components of displacements and tractions, which

correspond to the body forces in Eq.(3-7-1), can be

found, the domain integral term in Eqs. (3-7-1) can be

transformed into boundary integrals. The general idea

is similar to that represented for potential problems,

for it is difficult to find the solutions u j and Pj

corresponding to arbitrary body force functions b .• J

The series expansion method will again be applied. Only

2-D problems are studied here for simplicity.

In order to expand the components of body forces

as series in integral (3-7-1), let us choose the 2~D

complete set of functions:

{ f f R. fR. I 0 = 1 4 o'n' nm'" , ••• "n,m 1, ••• , oo} (3-7-3)

where f o ' f~ and f~m are shown in Eqs.(2-6-2).

This is not only a complete set, which means that the body

force function can be expressed in a sufficiently precise

119

way, but also the actual displacements and tractions

corresponding to the functions (3-7-3) can be found easily.

Suppose the body force functions b1 (x) and b 2 (x) are

bounded and periodic functions of periods 1a1 and 2a2 in

xl and x 2 directions respectively. The components of

the body force can be expanded as Fourier series in the

region nf :2a1 x2a2 as follows

b. J

4 N M L L L

1=1 n=l m=l k~ fR,

Jnm nm

j = 1,2 (3-7-4)

The coefficients of the Fourier series are

kjO 1

J b. (x)dn 4a1a 2 J nf

k~ 1 J

R,

In 2a1a 2 bj(x)fn(x)dn

nf

(3-7-5)

k~ 1 J b. (x) fR, (x) dn

Jnm a 1a 2 J nm nf

In order to find out the solutions of displacements

and tractions corresponding to each term of the expansion

in Eq. (3-7-4), all the terms in Eq. (3-7-4) are classified

in three groups:

(A) Constants

(3-7-6)

120

(B) The function of sine or cosine

4 N kR, fR, b l I I

R,=I n=1 In n

(3-7-7) 4 N

kR, fR, b 2 I I R,=I n=1 2n n

(e) The multiple harmonics

4 N M L I I

R,=l n=l m=l

(3-7-8)

4 N M I I I

R,=I n=1 m=1

First, one takes body forces in Group (A) and, for

simplicity omit the subscript 0 from k lO and k 20 in the

following derivations. Notice that now kl and k2 are

known constants. Let us assume that the stress components

can be written in the form of

(3-7-9)

where c l ' c 2 ' c 3 and c 4 are unknown constants and x IO '

x 20 are arbitrary reference coordinates.

Substituting stresses (3-7-9) and body forces (3-7-6)

into the equilibrium equation (3-2-3),one obtains

121

(3-7-10)

Considering the relationships (3-2-8), the strains

can be rewritten as follows

l+v [(1-v)c 1 (x 1-x10 ) - VC 2 (X 2-X 20 )] £11 ~

l+v [-vc 1 (x 1-x 10 ) + (I-v) c 2 (x 2-x20 ) 1 (3-7-11) £22 ~

l+v [c 3 (x1-x10 ) + c 4 (x 2-x20 )] £12 £21 ~

It can be found that the above strains satisfy the 2-D

compatibility equation (3-2-6) as follows:

2 £12,12 (3-7-12)

Integrating the first two Eqs. (3-7-11), one finds

the following displacements

(3-7-13)

Fom Eqs. (3-7-13), one can obtain the shear stress

(3-7-14)

122

Comparing Eq. (3-7-14) with the last equation of Eqs.

(3-7-11), one has

(3-7-15)

From Eqs. (3-7-15) and (3-7-10), one can easily find all

the constants c 1 ,c 2 , c 3 and c 4 ' which are involved in the

expressions of stress (3-7-9), strain (3-7-11) and dis-

placement (3-7-13) components, i.e.

2 k1 c 1 - 2-v

2 k2 c 2 - 2-v

(3-7-16) v

k2 c 3 2-v

v k1 c 4 2-v

Furthermore, one can express the tractions as

(3-7-17)

123

where n 1 and n 2 are components of direction cosine of

the unit outward normal vector on the boundary r.

After obtaining displacements (3-7-13) and tractions

(3-7-17) corresponding to the constant body forces (3-7-6)

and considering Somigliana identity (3-4-5), one can

represent the constant domain integrals as boundary

integrals, i.e.

f u!j b j drI = - f u!j Pj df + f p!j u j df + CijU j n r r

(3-7-18)

where u j ' Pj are functions known from (3-7-13) and

(3-7-17). c .. are given in Eqs. (3-5-3). ~J

Let us consider the body forces from group (B),

because in this there are several terms. To find actual

displacements and tractions corresponding to these forces,

one rearranges the functions in Group (B) in the following

four kinds of combinations:

l. b 1 kl e In n

b 2 k~ f~ 2n n

2. b 1 k~ f.:l In n

b 2 k 1 fl 2n n

3. b 1 k 3 f3 In n (3-7-19)

b 2 k4 f4

2n n

4. b 1 k4 f4 In n

b 2 k 3 f3 2n n

124

Let us consider the body forces b l and b 2 from

(3-7-19 1) as an example and find out the corresponding

solutions for displacements and tractions:

(3-7-20)

where n' = nn/a 1 .

Here one uses notations kl and k2 instead of k~n and k~n

for simplicity.

In order to satisfy the equilibrium equation (3-2-3), one

can expect the stresses to be

(3-7-21)

Substituting the expressions for the body forces and

stresses - (3-7-20) and (3-7-21) - into the equilibrium

Eq. (3-2-3), one finds

o (3-7-22)

From Eq. (3-2-8), one can write the strains in the form of

where

d f2 2 n (3-7-23)

125

d l 1 [(H21.1)c l - Ac2l 41.1 (HI.I)

d 2 1

[-Ac 1 + (I..+21.1) c 2l (3-7-24) 41.1 (1..+1.1)

d 3 1

21.1 c 3

or

1:;) ['-' -v 0

] {:;l l+v I-v 0 (3-7-25) ~ -v

0 0 1

They must satisfy the compatibility equation (3-7-12),

hence one has

n '2 d 2 0

therefore

substituting the above relationship into the second

equation in (3-7-24), one obtains the following

relationship

(3-7-26)

Taking into consideration the relationships between

constants C l ' c 2 and c 3 Eqs. (3-7-22) and (3-7-26), one

can solve for these unknown constants provided the

determinant is not equal to zero, i.e.

126

n' 0 0

0 0 -n' + 0

-A 1.+21.1 0

that is n' 2 (;\+21.1) + 0

Hence n' nn,'a l is non-zero, then

A + 21.1 + 0

or

v + ~ and v + I (3-7-27)

The conditions (3-7-27) can be satisfied by every

compressible material. One now obtains the following

results

kl c i - i1'

v kl ( 3-7-28) c 2 - I-v i1'

k2 c 3 i1'

substituting these results into Eq. (3-7-25), one finds

the coefficients in Eqs. (3-7-23) as

(l+v) (1-2v) c i (l-v) E i1'

o (3-7-29)

so that the displacements can be written as

(1+\1) (1-2\1) k1 f I

(1+\1) E fiT 2 n

and tractions

1 fiT

1 -fiT

127

(3-7-30)

(3-7-31)

For the other combinations of body forces in Eqs.

(3-7-19 2. 3. and 4.), one just lists all the solutions

as shown in Table 3-7-1.

Consequently, from the combinations of body forces

b j in the forms of (3-7-7', one can find out the

corresponding displacements u j and tractions Pj' Then the

domain integrals (3-7-1) can be expressed in the form of

Eq. (3-7-18), in which u j and Pj must be found from the

corresponding columns in Table 3-7-1

Similarly from Group (C) in Eq. (3-7-8), one can

obtain stresses, strains, displacements and tractions

corresponding to certai~ combinations of body forces and

these solutions are listed in Table 3-7-2. The existence

condition of all these solutions is the same as for

(3-7-27).

The only reason for choosing the combinations of body

forces as given in Tables 3-7-1 and 3-7-2 is that it is

then easy to find out the solutions of displacements and

tractions corresponding to each combination of body forces.

Tab

le

3-7

-1

b.

b1

~

b2

all

02

2

01

2

o ..

~J

c1

c2

c3

-------

Th

e S

olu

tio

ns

co

rresp

on

din

g to

B

od

y

Fo

rces

in

Eq

s.

(3-7

-19

)

1 2

3 4

k fl

1

n k

f2

1 n

kf3

1

n k

fit

1 n

k f2

2

n k

fl

2 n

k e

2 n

k f3

2

n

c f2

1

n c

fl

1 n

C f

3

1 n

C f

it

1 n

c f2

2

n c

fl

2 n

C f

3

2 n

C f

it

2 n

c fl

3

n c

f2

3 n

C f

it

3 n

C f

3

3 n

-kIln

' k2

/n'

.-..Y..

... kin

"

1-\1

2

-_

\1-

kin

"

1-\1

2

-.-..

Y.....

kin

' 1-

\1

1 1~\I k

2/n

' k2

/n"

-k

2/n

"

k2

/n'

-k2/n

' -k

Iln

"

kin

"

1 ~ -

~

----

I

I'J

(XI

£ ..

~J

Ui

Pj

Tab

le

3-7

-1

(Co

nti

nu

ed

)

1

£1

1

d f2

1

n

£2

2

d f2

2

n

£1

2

d fl

3 n

(l+

v)

(1-2

v)k

1 d

l -

(l-v

)E n

'

d2

0

d3

(l+

v)

k2

E n

'

d u

l _

--.!.

f I

n'

n

2d

3 U

2 --

f2

n'

n

I P

I 11

' [-k

1 f~

nl +k2f~n21

-l~

v kl

f~n2

+k2f

~nl

P2

n

'

wh

ere

n

' n1

T a

1 n

" n1

T a

2

2

d fl

1

n

d fl

2

n

d f2

3

n

(l+

v)(

1-2

v)

kl

(l-v

)E n

'

0

(l+

v)k

2 -

E n

'

d --.!.

f2

n'

n

2d

3 -ri' f

~

kIf~nl-k2f~n2

n'

l~v

klf~n2-~2f~nl

n'

3

d f3

I

n

d f3

2

n

d f'

3 n

0

(l+

v)

(l-2

v)

k2

(l-v

)E

n"

(I+

V)k

l -

E

nil

2d

3 -~ f~

d 2f"

n"

I~v

k2f~

nl-k

If~n

2 n

"

k2f~

n2-k

lf~n

l n

"

4

d f'

I n

d f'

2 n

d f3

3

n

0

(l+

v)

(l-2

v)

k2

-

(l-v

) E

n"

(l+

v)

kl

E

nit

2d

3 fi"

"" f~

d

2 -

fiTt f~

-I~V

~2f~

nl+k

If~n

2 n

" ..

-k2f~n2+klf~nl

n"

,

N c.o

Tab

le

3-7

-2

Th

e S

olu

tio

ns

Co

rresp

on

din

g to

B

od

y

Fo

rces

in

Eg

s. (

3-7

-8)

bi

I

o ..

>

J

E ..

>J

u. J

Pj

wh

ere

n' b

1

b2

all

02

2

01

2

c1

c2

c3 Ell

E2

2

E12

d1

d2

d3

u1

u2

PI

P2

-

n~

a1

1

kl~

k f3

2

nm

clf~

m

c f'

2 nm

c f'

3 nm

n* [-

k1

+m

'c3

]

;.

[-k

2+

n'c

3]

m'e

1k

1+

n'e

2k2

(I-v

) (m

' 2

+n

,:2

) 2

d f'

1

nm

d f'

2 nm

d f'

3 nm

d1

-fiT

f~

d2

-ffiT

f~

m

c1

f:un

nl +c3

f~mn

2

c3fA

mnl+

c2f~

mn2

m'

mn

a2

e1

nr2

\)

-m

l2 (

I-v

) e

2

2 I

3 4

k f3

1

run

k f'

1 nm

k

f'

1 ru

n

k f>

2

nm

k f'

2

nm

k f'

2 nm

clf~

m c

f3

1 nm

c

f'

1 nm

c2f~

m c

f3

2 nm

c

f'

2n

m

c f'

3 nm

c

f'

3 nm

c

f'

3n

m

;,

[k1

+m

'c3

' n1

, [-

k1-m

'c3

' n1

, [k

1-m

'c3

'

ml,

[k2

+n

'c3

J m

l, [k

2-n

'c)J

,;

, [-

k2-

n'c

3,

m'p

. 1k

1+

n'e

2k

2 m

'elk

l-n

'e2

k2

-m

'e1

k1+

n'e

2k

2 (1

-\1

) (m

' 2

+n

ti)

a

(I-v

) (m,2+n'~).2

(1-v

) (m

' ~+n' i

) 2

I d

f'

1 ru

n d

f3

1 nm

d

f'

1 nm

I

d f'

2 nm

d

£3

2

nm

d f'

2 nm

d f'

3 nm

d

f'

3 nm

d

f3

3 nm

(l~vl

[(l-

vlc

1 _

c2

'

(l~Vl

[-c

1 +

(l

-vl

c2

,

l+v

~c3

~ f

3

n'

nm

_ ~ f

' n

' nm

d

1 f'

n'

nm

~ f

' m

' nm

~ f

' m

' nm

_

d2

£'

m'

nm

clfAmnl+c3f~n2

c1 f~mnl +c3f~n2

clf~mnl+c3f3n2

c3f~

mn2+

c2f~

mn2

c3f~mnl+c2f~mn2

c3f~

mnl+

c2f~

rnn2

.-

m,2

v -

n,2

{l-

v)

w

o

131

One can now express the domain integral (3-7-1) as

a summation of boundary ones:

1. (0 = f u~.(t,;,x)b.(x)dn(x) ~ ~] ]

n =-f Uij(Pj)o dr + f pij(uj)o dr + cij(uj)o

r r

£11 nIl [ [ Uij(Pj)~ dr - [ Pij(Uj)~ dr - Cij(Uj)~l

4 N M I I I

£=1 n=l m=l

(3-7-32)

where (u.) , (p.) correspond to Eqs.(3-7-13) and ] 0 ] 0

(3-7-17), (u.)l and (p.)l can be found in Table 3-7-1 ] n ] n

I £ and (uj)nm and (Pj)nm in Table 3-7-2. The superscript

£ is the type of combination of body forces. Note, that

only boundary integrals exist on the right-hand side of

formula (3-7-32). All the results presented above are

valid for plane stresses by transforming v into v = v/(l+v).

As regards the integrals (3-7-2), one can take

derivatives of Eq.(3-7-32) and consider the matrix c .. is ~]

a unit matrix for every internal point t,;. Finally the

transformation formula for integral (3-7-2) can also be

obtained simply by replacing Uij and pij by Uijk and pijk:

132

J Uijk(~,x)bk(x)dn(X) n

=-J Uijk (Pk) 0 df+ J p ~ . k (uk) d f + c (u . ) ~J 0 ~ 0

r f

4 N

J Uijk(Pk)~ J Pijk(Uk)~ R.

- L L [ df - df -c(ui)n1 R.=1 n=l f r

4 N M

J Uijk(Pk)~m J Pijk(uk)~m - L L L df - df R.=1 n=l m=l f f

(3-7-33)

where Uijk and pijk are as shown in Eqs. (3-5-6) and

(3-5-7) . c is equal to 0ij'

3-8 NUMERICAL IMPLEMENTATION

In order to compute the results of Eqs. (3-7-32) and

(3-7-33), one needs to discretize the problem using boundary

elements and applying numerical integration techniques.

As mentioned in potential problems, the integral along

the boundary is divided into several integrals over the

boundary elements:

J F(x)df f

NE L

j=l J F(x)dr 1

fj

(3-8-1)

Because the displacement u j and tractions Pj in the

integrands of integrals (3-7-32) and (3-7-33) are continuous

functions over every element j, the singularity is only due

133

to the fundamental solution components Uij and pij .

There are some kinds of singularities for 2-D problems,

such as In(r) and l/r, in Eqs. (3-3-4) and (3-3-5), there-

fore for the purpose of numerical implementation one can

classify the integrals into three types, i.e.

A. Those that can be evaluated using fue standard Gauss

quadrature for regular functions.

B. Those that require a transformation (2-7-3) for the

functions involving the factor of In(r) in the integrand.

c. Those that can be computed using Kutt quadrature

(finite-part integration) applicable for functions of

the type l/r.

Types A, and B. have already been described in section

2-7, therefore, one needs to discuss here only the case C.

For this case, the finite-part integral technique is used

later. The finite-part integral technique and the corres-

ponding tables can be found in some references [47-49].

In order to use the tables directly, one must first

scale the interval of the integral from [a,b] to [0,1]:

I

b

J fIr) dr r-a

a

1

J f[(b-~)t + a] dt (3-8-2)

o

Then one can find the local coordinates of the integral

points t. and the weights w. in the finite-part integral 1 1

tables and express the integral as

I N L f[(b-a)t i + a]wi

i=l (3-8-3)

134

Although the form of Eq. (3-8-3) is similar to that of

Gauss quadrature, their deduction and tables are completely

different.

Moreover, when one evaluates the kind of integral

(3-8-1), in which p~. exists, they are computed in the 1)

meaning of Cauchy principal value as mentioned in section

3-5. For this reason, if the interval [a,b] involves the

singular point s, the integral I becomes

b

I f 1 f(r)dr r-s

a

s-£ b

lim f f(r) dr + f

f(r) dr ] (3-8-4 ) £~o

r-s r-s a s+£

where f(r) is supposed to satisfy the Holder condition at

point sand f(s) + 0, so that the singularity is actually

of order one. Note that the same value of £ must be taken

in the first and second integrals on the right-hand side

of Eq. (3-8-4) in Cauchy principal integral. Hence, after

transforming Eq. (3-8-2) there is an additional term in the

formulation. Using Eq. (3-8-2), Eq. (3-8-4) can be rewritten

as

I

b

f ~ f(r)dr a

lim £~o

£/Is-al

[ f 1

lim £~o

s-£ b

f f ~) dr + f a s+£

1

f (r) dr ] r

f[(a-s)t+s] dt + t f f[(b-s)t+s] l

t dtJ £/ Ib-s I

135

£/ls-al f[(b-s)t+sl dt +

t lim £-+0

[J 1

1 f[(a-~)t+sl dt + J

£/ls-al

= lim £'-+0

£/ls-al + J f[(b~S)t+sl d~

£/lb-sl

£ '

[ f 1

1

f[(a-~)t+sl dt + f £ '

Ib-sl + f(s) ln -Is-al (3-8-5)

The above integral (3-8-5) can be evaluated numerically

by the finite-part integral as follows

NP NP I -~ f[(a-s)x +slw + p p ~ f [ (b- s ) x + s 1 w

p p p=l p=l

(3-8-6)

Consequently, when the finite-part integrals are used to

compute the terms in Eq.(3-7-32) in the form of

the addition term u(s) ln [I~=:I) must be considered as

shown in Eqs.(3-8-5) and (3-8-6).

The matrix c .. in Eq.(3-7-32) can be considered now. ~)

From Eqs. (3-5-3) and (3-3-5), one has

and

136

J pij (t;,x) dr(x)

r£

( 3-8-7)

1 dr 4n(l-v)r {[(l-2v)oij + 2r,ir,jl an -

- (l-2v) (r .n. + r .n.)} (3-8-8) ,1) ,) 1

when the source point t; is on the boundary r. One supposes

the boundary is augmented by a small boundary rc which is

a part of a circle centred at point; with radius £ (Fig.

3-8-1). If one uses the polar coordinates and considers

r = £ on the boundary r£, one has

£ cos e

X2 = £ sin e

r , 1 cos e

r ,2 sin e

dr 1 'Yii

grad r x !! o (3-8-9)

137

Figure 3-8-1 Boundary r + re:

138

substituting Egs. (3-8-9) into Eg. (3-8-8), one obtains

I .. (i;) = ~J

1

f p~. (i;,x)dr ~J

r £ 92

41f (l-v) f 1. [ (1- 2 v) <5.. + r . r . 1 rd 9 r ~J, ~ , )

(3-8-10)

The components of matrix I .. can be integrated as follows: ~J

(3-8-11 )

Then the matrix c ij can be expressed by Egs. (3-8-11)

as follows

<5 •• + I .. (0 ~J ~J

(3-8-12)

There are constants x lO and x 20 in Egs. (3-7-13)

and (3-7-17), which are the coordinates of a reference

point as mentioned in section 3-7. When one calculates

formula (3-7-32) or (3-7-33), the values of (x 10 ' x 20 )

can be taken as the coordinates of the point near the

middle of domain n or, for simplicity, the origin of

coordinates. Only do not let mag'nitudes xl and x 10 or

x 2 and x 20 be significantly different, which causes some

rounding errors for numerical reasons.

139

Finally, one wants to mention here that in Eqs. (3-7-13)

and Tables 3-7-1 and 3-7-2 the displacements can involve

some other terms, which correspond to the arbitrary rigid

body motion (translated and/or rotated) and pure shear

state. These rigid body motion and pure shear state terms

have no influence on the results of integrals (3-7-1) and

(3-7-2). For testing the rigid body motion, one can assume

the following displacements

(3-8-13)

where w is the rotation angular velocity, (x IO ' x 20 )

are the coordinates of the centre of rotation. u 10 and

u 20 are components of the translated motion. In this case

the tractions corresponding to displacements (3-8-13)

vanish.

For testing the pure shear state, one takes all the

variables as follows

u 1 k 1 y

u 2 k 2x

0 0 11

0 0 22 0 0 E E

12 21 l+V 12 (3-8-14)

E 0 11

E 0 22 E E l.j(k1 + k 2 ) 12 21

PI 0 21 n 2

P2 0 n 1 12

140

Substituting u. and p. in Eqs .• (3-8-13) and (3-8-14) J J

into formulae (3-7-32) and (3-7-33), one obtains results

which are all numerically zero, so that, one has the

following conclusion: in Eqs. (3-7-32) and (3-7-33) the

displacements u. and tractions p. involve arbitrary rigid J J

body motion and pure shear state without any influence

to the results of integrals (3-7-1) and (3-7-2). So

in the Eqs. (3-7-13) and (3-7-17) and the expressions of

u. and p. in Tables 3-7-1 and 3-7-2, one can add some J J

terms, which correspond to the rigid body motion and pure

shear state, without changing the values of integrals

(3-7-32) and (3-7-33).

3-9 RESULTS OF NUMERICAL EXPERIMENTS

This section discusses some results, which are

obtained for examining Eqs. (3-7-32) and (3-7-33) numeric-

ally. In order to test the accuracy of Eqs. (3-7-32) and

(3-7-33) thoroughly, one studies the body forces b 1 (x)

and b 2 (x) term by term for each combination in Eqs. (3-7-6),

(3-7-7) and (3-7-8). Besides using formulae (3-7-32) and

(3-7-33), integrals (3-7-1) and (3-7-2) are computed for

comparison also using the original domain integrals as

well. The domain is subdivided into several cells in

advance and the numerical quadrature formulae and trans-

formations used to compute the integrals (3-7-1) and

(3-7-2) are the same as those mentioned in section 2-8.

Because of this they will not be repeated here.

The results for the two methods mentioned above are

listed in Tables 3-9-1 to 3-9-16. The integral domain n

141

expansion domain nf and discretizations of boundary

elements and subdomains (cells) are the same as shown

in Figure 2-8-1, which is for testing the transformation

formula in potential problems in section 2-8. From these

results, one can also obtain a set of conclusions as

those in section 2-8. Therefore, formulae (3-7-32) and

(3-7-33) can be used for transforming the domain integrals

(3-7-1) and (3-7-2) into boundary integrals provided one

chooses suitable element lengths.

It is important to specify how to choose the boundary

rb for computing the boundary integrals in transformation

formulae and the domain of the Fourier expansion, Qf .

In principle, those are chosen according to the features

of body force functions bi(x). When the periodic extensions

of bi(x) are carried out, the less discontinuous are

the functions and their derivatives the faster will be

the convergence of the solution. Here it is also emphasized

that the convergence for formulae (3-7-32) and (3-7-33)

is faster than for their Fourier series as mentioned in

section 2-8.

~ n

r1

r1

r2

rL

. 1

2 1

2 8

1 -0

.109

710X

10-5

0.

2411

490X

10-7

-0

.;09

810X

10-5

0.

2411

439X

10-7

III

t:r

I-

' (l

)

2 0.

6110

476X

10-5

-0

.814

7151

X10

-6

0.61

1047

3X10

-5

-0.8

1471

59X

10-6

H

.....

w

I

3 0.

8866

362X

10-6

0.

1809

437X

l0-7

O

. 886

6364

X1·

0-6

0.18

0940

7X10

-7

J"'""

I \0

I .....

4 -0

. 222

0403

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0.17

7581

9X10

-6

-0.2

2204

11X

l0-6

0.

1775

824X

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~l

5 0.

1368

458X

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-0

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5229

X1

0 ~

o.13

6845

8X10

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-0. 1

2152

20X

l0-6

:8'_

__

__

~

(")

c .....

0

..... *

r1"

;3

w.

::r

'0

6 -0

. 127

1625

Xl0

-5

0.15

4515

8Xl0

-6

-0.1

2716

23X

10-5

0.

1545

163X

10-6

III

J"

'""I

0..

11

0 .....

7 -0

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6687

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0-5

0.77

4362

1X10

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0466

87X

10

5 0.

7743

695X

10-7

X

;3

ti

l III

0

t:r

.....

::l

w.

::l

8 0.

1800

425X

l0 6

-0

.105

5635

Xl0

-6

0.18

0043

2X10

-6

-0.1

0556

20X

l0-b

0

X

.....

I"tl

::l

.j>

.

1 -0

. 116

8440

X10

-5

-0.2

9297

84X

l0-6

-0

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8423

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-0. 2

9297

42X

1 0-

6 0.

. r1

" t:r

N

:8

(l

) 0

\Q

C

2 0.

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. 505

2277

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0.56

3845

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0522

66X

l0-6

X

11

::l

III

0.

. I-

' III

11

3 0.9637214Xl0-~

0.46

3545

7X10

-6

O. 9

6372

17X

1 0-

6 0.

4635

461X

10-6

H

'<

t:r

.....

.. .....

.....

~2

4 0.

1109

685X

l0-7

0.

4148

128X

l0-6

0;

1109

358X

l0-7

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4148

129X

10~6

::l

r1

" (l

)

5 0.

1408

068X

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. 115

5626

Xl0

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0.11

4080

67X

10-5

0.

1155

627X

10-6

0

\Q

0 11

ti

l III

6 -0

. 118

5283

Xl0

-5

-0.4

7115

70X

l0-8

-0

. 118

5823

X1

0-5

. -0

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1408

Xl0

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f I-

'

o ~

H

X

..... -

~

7 -0

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7829

Xl0

-5

-0.1

0581

38X

10-6

-0

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7831

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-O.1

0581

47X

l0-6

t:r

8 0.

2229

765X

l0-7

-0

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2218

Xl0

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0.2;

2·97

90X

10-7

-0

. 204

2223

X1

0-6

IV

0

E; n

rl

rl

r2

r~

'1

'')

'1

>-3

1 -0

. 880

172

5Xl 0

-5

0.10

6278

2Xl0

-5

-0.8

80

1725

Xl 0

-5

O.1

0627

82X

l0-5

III

0

-H

I-

' ....

CD

2 -0

. 147

7261

Xl0

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0.20

9800

8Xl0

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-0. 1

4772

63X

l0-5

0.

2093

UO

OX

10-7

.....

W

I

3 0.

2863

316X

l0-5

-0

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7780

Xl0

-6

0.28

6331

7Xio

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8877

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1.

0 I N

4 -0

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4693

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-0.3

0839

76X

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0.

7646

866X

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E;

1

:0--

---.

c: .... *

~

()

0.11

4116

9Xl0

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6651

66X

l0-7

0.

1141

168X

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i 5

u.

.... 0

rt

S .....

::T

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195X

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. 124

3190

Xl0

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0.16

6019

6Xl0

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2431

88X

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III

X

0

-11

-

0 ....

7 -0

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0188

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o • 1

6042

65X

l 0-6

-0

.736

0183

Xl0

-6

0.16

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0-

S en

u

. III

0

.... ::l

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::l

8

· 0.

5887

573X

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1284

062X

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5887

579X

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1315

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HI

.... :0

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1 -0

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0 c: ::l

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6635

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0

. III

11

3 0.

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10-6

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' ....

4 -0

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5762

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5750

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9157

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2 0.

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3Xl0

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. 172

0595

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9925

5Xl0

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0.17

2059

6Xl0

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9921

4Xl0

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II ::l

rt

en

CD

....

\Q

::l

11

-f III

I-

'

~~

H

....

I-'

7 -0

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4408

Xl0

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O. 1

5859

37X

l 0-6

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4404

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8 0.

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9325

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·0.5

4542

87X

l0-7

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. 930

9359

Xl0

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0

~ r'

r'

r~

r~ .

I n

1 2

1 2

>-3

0.43

0663

0X10

-7

0.10

0946

6X10

-5

0.43

0665

7X10

-7

0.10

0946

2X10

-5

,

1 i

III

tr

......

(1)

2 -0

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9702

X10

-6

0.54

4470

6X10

-5

-0. 7

8797

02X

1 0

-6

0.54

4470

6X10

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H ...

IN

I

3 :"

0.10

5397

6X10

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7584

42X

1 0

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66X

1 0

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10-5

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\0

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4 0.

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10-b

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. 123

6893

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10-o

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----

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~ 1

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O

. 562

1677

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0-6

0.

1019

903X

10-6

0.

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10-6

~

n s::

... 0

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1-rt

" a

u.

::T

'0

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10-6

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7127

X1o

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0406

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-6

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7171

26X

10-5

III

J'

"t

0.

t1

0 ...

>< a

III

7 -0

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i846

X;0

-7

0.61

4214

9X10

-6

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2511

55X

1o-7

0.

6l42

143X

10-6

II>

0

tr

... ::

l u

. ::

l

8 -0

. 656

5576

X1

0-7

0.

2366

767X

10-7

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0.23

6709

3X10

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1 0.

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607X

10-7

0.

1438

303X

10-5

0.

1904

696X

10-7

.0

. 14 3

8248

X1

0-5

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3-9

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o

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ou

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h

do

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~ n 1 2 3 4 5 6

I--

~3

7 8 9 10

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le

3-9

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om

par

iso

n o

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ou

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ary

in

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ral

I!.

1]

wit

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E,;)

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0

0.78

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0

~.259033 X

l00

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r-:l

~ n 1 2 3 4 5 6 7

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8 9 10

Tab

le

3-9

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C

ompa

r:is

on o

f b

ou

nd

ary

in

teg

ral

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wit

h

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n in

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=

! ul.~'k(~,x)bk(x)dn(x)

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212

X10

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0.60

2097

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20

2 111

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9035

6 X

10-1

0.

3806

05 X

100

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2533

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1

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10'

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9422

1 X

100

0.47

5541

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100

0.60

2399

X10

0

U1

W

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n m

1 1

1 2

2 1

1 3

2 2

3 1

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1 4

2 3

3 2

4 1

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le

3-9

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C

om

par

iso

n o

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ou

nd

ary

in

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ral

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wit

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ral

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0

J Uijk(~,x)bk(x)dQ(x)

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37

83

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2806

63 X

10-2

0.49

5116

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0 -0

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3757

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4515

2 X

100

0.22

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0.33

5846

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0 0.

1067

66 X

100

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3308

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10-1

0.86

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24 X

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. 0.

1363

81

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0

0.19

2607

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0 0

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0.78

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1959

4 X

100

0.13

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0 0.

8689

72 X

10-2

0.58

7276

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(n1T

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(m1T

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0

co

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3781

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0.86

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2329

17 X

100

0.19

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0 0.

4815

65 X

10-1

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1958

8 X

l00

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4107

Xl0

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7260

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0

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9423

2 X

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4816

96 X

10-1

0.29

6978

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0 -0

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386

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2

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8020

7 X

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5314

9 X

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3433

5 X

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6355

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0

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8142

6 X

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0.86

8969

X10

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8825

4 X

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6825

5 X

10-1

0.43

3799

X10

-1

Ul

.j:>.

~ n 1 1 2 1

£;3

2 3 1 2 3 4

Tab

le

3-9

-14

C

om

pari

son

o

f b

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nd

ary

in

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ral

I~.

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n in

teg

ral

I~.

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Iij (

0

J Ui

jk(~

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k(x)

dQ(x

) Q

b1

sin

(n

;;)

co

s (m

;l)

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1 11

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1786

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3271

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1786

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32

70

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0 0.

1028

31

X10

1 0.

1039

38 X

100

1 0.

9309

65 X

10-1

-0

.70

41

80

X10

-2

0.21

9268

X10

-1

0.93

0984

X10

-1

-0.7

05

05

6 X

10-2

2 122

-0.1

35

55

2 X

10-1

-0.6

8947

3 X

10-1

-0.9

2793

9 X

10-3

0.50

6530

X10

-1

0.11

3977

X10

0

-0.2

6402

9 X

10-1

-0.3

35

53

5 X

10-1

-0.2

14

28

9 X

10-1

0.20

4675

X10

0

-0.2

19

25

0 X

10-1

U1

U1

E;, n 1 1 2 1

E;,s

2 3 1 2 3 4

Tab

le

3-9

-15

C

om

par

iso

n o

f b

ou

nd

ary

in

teg

ral

I~. ~J

m

1 2 1 3- 2 1 4 3 2 1

wit

h

do

mai

n in

teg

ral

I~.

~J

Iij (

E;)

J

Uij

k(T

I,x

lbk

(xld

Q(x

l Q

bI

0

1 I11

1 I12

1 I22

0.35

4809

X10

-1 -

0.60

0064

X10

-1

0.25

8646

X10

0

0.36

8907

X10

-1

0.30

7772

X10

0 -0

.265

865

X10

0

-0.8

8294

2 X

10

-2 -

0.10

4416

X10

0 0.

5966

09 X

100

-0.6

3763

2 X

10-1

0.

1212

75 X

100

0.49

0090

X10

0

-0.6

0969

1 X

10-2

0.

5351

02 X

100

0.60

3650

X10

0 -

-0.8

2513

0 X

10-1

0.

6610

94 X

10

0.40

9206

X10

0

-0.3

0770

5 X

10-1

-0.

7567

19 X

10-1

0.

4429

97 X

100

0.88

2097

X10

-2

0.20

8585

X10

0 o

1113

28 X

1 01

-0.8

6429

3 X

10-1

-0.

3992

75 X

100

0.42

2950

X10

0

-0.5

4096

5 X

10-2

0.

1795

57 X

100

0.83

2501

X

10-1

--

------

b2

cos(~~)cos(m;l)

2 I11

2 I12

0.35

4871

X

10-1

-0

.600

000

X10

-i

0.36

8973

X10

-1

0.30

7747

X10

0

-0.8

8406

9 X

10-2

-0

.104

403

X10

0

-0.6

3777

0 X

10-1

0.

1212

54 X

100

-0.6

1087

7 X

10-2

0.

5350

57 X

100

-0.8

2527

5 X

10-1

0.

6610

13 X

10-1

-0.3

0793

3 X

10-1

-0

.756

570

X10

-1

0.88

4556

X10

-2

0.20

8548

X10

0

-0.8

6444

9 X

10-1

-0

.399

249

X10

0

-0.5

4107

6 X

10-2

0.

1795

23 X

'-OO

--

----

---

2 I22

-0.2

5865

8 X

100

-0.2

6587

8 X

100

~

0.59

6631

X

100

0.49

0119

X10

0

0.60

3673

X10

0

0.40

9237

X10

0

0.44

3044

X10

0

-0.1

1133

3 X1

01

0.42

2981

X

100

0.83

2798

X10

-1

E; n

m

1 1

1 2

2 1

1 3

·

E;3

2 2

3 1

1 4

2 3

3 2

4 1

Tab

le

3-9

-16

C

om

pari

son

o

f b

ou

nd

ary

in

teg

ral

1~, ~J

wit

h

do

main

in

teg

ral

1~,

~J

1ij

(t;)

J u!

jk(~

,x)b

k(x)

dn(x

) n

bl

1 111

1 112

1 122

0.11

8509

X10

-1

0.52

4739

X10

-1

0.87

3266

X10

0

0.78

7263

X10

-2 -

0.3

55

48

8 X

100

1-0.

8835

90 X

100

0.64

8160

X10

-1 -

0.8

71

41

7 X

10-1

t-0

.411

675

X10

0

-0.1

05

34

0 X

10-1

-0

.93

33

88

X10

-1

0.16

2865

X1

01

0.67

5941

X

10-1

0.

4644

55 X

100

0.42

3902

X10

0

-0.2

98

94

3 X

10-2

-0.

1569

71

X10

0 0.

2627

49 X

100

0.11

6968

X10

0 0.

5805

95 X

10-1

0.

1161

08 X

101

-0.1

17

74

4 X

100

0.17

4248

X10

0 0.

7828

64 X

100

-0.1

71

69

2 X

10-2

v.

697'

:!54

X10

0 0.

2658

64 X

100

-0.8

48

86

2 X

10-1

0.

6541

03 X

10-2

0.

2794

99 X

100

o b

2 ,

(n 1T

X)

(m1T

Y)

s~n 1

0 c

os

20

2 111

2 112

0.11

8662

X10

-1

0.52

4693

X10

-1

0.78

8871

X

10-2

-0

.35

54

73

X10

0

0.64

8275

X10

-1

-0.8

71

30

4 X

10-1

-0.1

0566

9 X

10-1

-0

.93

32

67

X10

-1

0.67

4416

X10

-1

0.46

4416

X10

0

-0.2

9946

5 X

10-2

-0

.15

69

47

X10

0

0.11

6913

X10

0 0.

5805

08 X

10-1

-0.1

17

76

8 'X

100

0.17

4217

X10

0

-0.1

7226

0 X

10-2

0.

6978

72 X

100

-0.8

4901

9 X

10-1

0.

6544

35 X

10-2

2 122

-0.8

7329

6 X

100

-0.8

8362

1 X

100

-0.4

11

69

8 X

100

0.16

2872

X10

1

-0.4

23

92

6 X

100

0.26

2760

X10

0

0.11

6119

X10

1

0.78

2916

X10

0

0.26

5874

X10

0

0.27

9530

X10

0 ,

U1

-....I

CHAPTER 4 APPLICATIONS IN ELASTICITY AND ELASTO-PLASTICITY

4-1 INTRODUCTION

The previous chapter discussed how formulae (3-7-32)

and (3-7-33) can be used to transform the domain integrals

(3-7-1) and (3-7-2) into boundary integrals and examined

the accuracy of these formulae. The present chapter

shows some applications of these concepts to solve elasto-

statics and elasto-plasticity problems.

It is not difficult to apply the previously deduced

formulae to elastostatics provided that the body forces

are known. After evaluating the Fourier coefficients for

the body force functions analytically, the transformation

formulae (3-7-32) and (3-7-33) can be used directly. The

accuracy of the solutions depends on the accuracy of the

Fourier series representations and usually the relative

errors of the solutions are better than those of the

Fourier series. This is because the order of the coeff-

~ ~ ~ ~ icients of (Pj)n' (Pj)nm or (uj)n' (uj)nm shown in

Tables 3-7-1 and 3-7-2, is one or two orders higher than

the order of Fourier coefficients of b j , when n (or m)

becomes infinite. One can see that they approach zero

faster than the Fourier coefficients as n (or m) becomes

infinite. For the continuous body force functions, only

a few terms of Fourier expansions need to be considered,

hence the CPU time required when using transformation

formulae is less than the CPU required when using the

original domain integrals, because with these formulae

159

the integrations are only performed on the boundary,

i.e. they are one dimension less than domain integrals.

In addition, one does not need to divide the domain into

cells. In the cases of discontinuous body force functions,

however, more terms of Fourier series must be taken to

obtain sufficiently accurate solutions.

A short review of plasticity will now be presented

and then the boundary integral equations for elasto­

plasticity will be deduced.

Applications of boundary integral equations to

plasticity have been developed only since 1971 starting

with a paper by Swedlow and Cruse [53] and then followed

by a few other references [54-55]. Recent works

[56-59] especially have shown how the boundary element

formulations for three-, two-dimensional and semi­

infinite plasticity problems can be successfully developed

and those formulations are used in the present work.

In the area of plasticity problems, there are three

kinds of formulations in boundary element methods:

initial strain, initial stress and fictitious traction

and body force approaches. All of them require some

types of domain integrals to be computed. In principle,

each of these domain integrals can be transformed into

boundary ones with a similar technique as previously

shown, but here one will consider only the fictitious

traction formulation.

The domain integral terms in plasticity are due to

not only the actual body forces, but also tQe plastic

strains or stresses. These plastic strains also constitute

160

fictitious traction and body force components. As one

solves the plastic problems by means of an incremental

procedure, the prescribed loads or displacements applied

to the body or on the boundary are divided into increments,

which are applied step by step. For each step an

iterative technique is used until convergent solutions

are obtained. The plastic strains and stresses are

obtained only at a certain number of points at the end of

each iteration. Because no analytical expressions generally

exist for them, the Fourier coefficients must be calculated

from these finite values numerically. The numerical

method applied here is also described in this chapter.

Unfortunately, it is time consuming to compute the Fourier

coefficients. The main reason is that the Fourier

coefficients must be recalculated at the end of each

step during the iterative procedures. The advantage of

this method is however that it requires only the dis­

cretization of the boundary and there is no need to

subdivide the domain into subdomains (cells) which need

to be numbered in advance.

4-2 AN EXAMPLE OF GRAVITATIONAL LOADING

In this example a simple body force, i.e. gravitational

load, is applied to an elastic body of rectangular cross

section. The problem is assumed to be in plane strain.

The boundary conditions are free surfaces on three

sides and the other is constrained in the x direction.

161

In order to avoid the rigid body motion in y direction,

the middle point of that side is fixed (shown in Fig.

4-2-1). The properties of material are as follows:

E gm, g 10, m 250

For this kind of constant body force function, the

Galerkin tensor can be used, which is related to the

fundamental solution Uij by the following expression

[60-62) :

There

1 Gij,kk - 2(1-v) G*ik,kj (4-2-1)

G~. consists of two (for 2-D) or three (for 3-D) ~J

vectors each of them corresponding to the infinite space

in which the unit load is applied in the direction i.

For this infinite space the Galerkin tensor can be

represented as follows

Gij 1 ro .. 8nG ~J

(for 3-D) (4-2-2)

Gij 1 1

8nG rZln (-) o .. r ~J

(for plane strain) (4-2-3)

In order to satisfy Eq.(4-2-1) for 2-D plane strain,

a constant term must be added to the fundamental solution

for displacements (3-3-4), i.e.

u~. ~J

1 8n(1-v)G [(3-4v)ln rOij -

o

(7-8v) - r .r . + --2-- o .. J (for plane strain) (4-2-4) ,~ ,J ~J

162

///L_LLL / / / /' /'//// ~ J

! -y

I I I I

I j j j 2. 0'

5

j j j I 2 L

Figure 4-2-1 Geometry, body force and boundary conditions

163

substituting the Galerkin tensor into domain

integrals:

1. f * b j drl 1 Uij

Q

f (G~. kk -1

G\ k') b. dQ (4-2-5) 1J, 2(1-\1) 1 , J J

Q

one can transform the above domain integral into boundary

integrals for the case of a constant body force b i • The

complete formulation can be found as follows [12]:

where

and

f Uij(s,x)bj(x)dQ(x)

Q

(for 3-D)

(for plane strain)

These domain integrals can be computed with formulae

(3-7-32) and (3-7-33) also integrating only on the

boundary. The meshes and internal points are shown in

Figure 4-2-2 and the results are presented in Table

4-2-1. For the constant body force, the Fourier expansion

results present the same accuracy as those shown in

Tables 3-9-1. In addition, comparing the results with

the formulation using the Galerkin tensor, they are

exactly the same in the numerical sense.

164

• INl'ERNAL POINTS

x MESH POINTS

A • C A • C -A • C

B B I· MESH I MESH II MESH III

Figure 4-2-2 Meshes and internal points

Tab

le

4-2

-1

mes

h p

oin

t

A

I B

C

A

II

B

C

A

III

B

I C

I

Th

e D

isp

lacem

en

ts

and

S

tresses at

Inte

rnal

Po

ints

u u

a a

x y

xx

xy

0.23

473

X10

-2

-0.4

1677

X10

-7

0.38

350

X10

4 -0

.772

48 X

10-3

0.30

738

X10

-2

-0.6

0943

X10

-7

0.25

279

X10

4 0.

3474

2 X

10-2

0.23

008

X10

-2

-0.3

8354

X10

-3

0.37

826

X10

4 0.

4859

1 X

102

0.23

393

X10

-2

-0.5

8844

X10

-8

0.38

273

X10

4 -0

.475

88 X

10-3

0.30

655

X10

;;2

-0.8

3701

X10

-8

0.25

266

X10

4 0.

2641

7 X

10-3

0.22

939

X10

-2

-0.3

8152

X10

-3

0.37

651

X10

4 0.

5086

9 X

102

0.23

375

X10

-2

-0.1

98

72

X10

-7

0.38

227

X10

4 -0

.567

44 X

10-4

I 0.3

0627

X10

-2

-0.2

7570

X10

-7

0.25

238

X10

4 -0

.484

94 X

10-3

i 0.

2291

9 X

10-2

-0

.38

09

6 X

10-3

0.

3760

1 X

104

0.51

754

X10

2

a yy

-0.5

0304

X10

2

-0.1

5619

X10

2

-0.5

1258

X10

2

-0.5

2201

X

102

-0.2

8905

X10

2

-0.3

3713

X10

2

-0.5

2203

X10

2

-0.2

9514

X10

2

-0.3

2237

X10

2

ozz

0.11

354

X10

4

0.75

368

X10

3

0.11

194

X10

4

0.11

325

X10

4

0.74

931

X10

3

0.11

194

X10

4

0.11

311

X10

4

0.74

829

X10

3

0.11

183

X10

4

I I

0)

C11

166

4-3 AN EXAMPLE WITH A MORE GENERAL TYPE OF DISTRIBUTED

* LOADING

This section presents an example with body force

functions in the form of

b 1f. (1fX) 1 2a Sl.n 2a

(4-3-1)

It is not difficult in this case to find out the

analytical solutions for stresses, strains and displacements

which satisfy equilibrium equation (3-2-3), the relation-

ship between stresses and strains (3-2-8) and the

definition of strains (3-2-5), are given by

°11 cos G:)

°22 v cos G:) (4-3-2)

°12 °21 sin G:)

1-v 2 cos G:) £11 -E-

£22 0 (4-3-3)

1 sin G:) £12 £21 2G

u l 1-v 2 2a . (1fX) -- - Slon-

E 1f 2a (4-3-4)

2a cos (~X) u 2 - TIG .. a

* In this section the coordinate Xl is replaced by x for simplicity

167

The domain of the problem under consideration is

defined as n: [0 ~ x ~ 5, 0 < y ~ 4), a = 10 and

the mesh points, internal points are shown in Fig. 4-3-1.

The boundary conditions are prescribed as

on r 1 , r3

and (4-3-5)

on r 2 , r 4

the values of ui and Pi at nodes on the boundary are

evaluated by Eqs. (4-3-4) and (4-3-2) together with the

following equation:

p. = o .. n. ~ J~ J

where nj (j = 1,2) are the components of the outward

normal on the boundary r.

In this case, body forces can be expressed as

Fourier series with only one term, if one chooses the

domain for the harmonic expansion nf : [-2a < x ~ 2a).

Using formulae (3-7-32) and (3-7-33),one can expect to

obtain with n = 1 results as accurate as those shown

in Tables 3-9-2 and 3-9-3. Later on one will adopt the

notation I 2a for the solutions obtained with this

expansion.

One can also select the domain of harmonic expansion

nf : [-a ~ x ~ a) to see what influence this has on the

results. The Fourier series for the body forces b i are

now in the form of

168

• intemal points .37

xmesh points -36 ,

• 29 - 30 '31 -32 -33 4 r2

-:35 .~

f- ~4

r1

5

Figure 4-3-1 Geometry, mesh and internal points

169

1 N (_l)n . (nn ) a I -1---- sJ.n a x

n=l '4 - n 2

1 1 a + 2a I (-l)~n cos(:n x )

n=l n 2 - '4

(4-3-6)

After setting E = 2 x 10 6 and v = 0.3, one can

divide the boundary into elements as shown in Figure

4-3-1. The solutions are listed in Tables 4-3-1 and

4-3-2, in which displacements and stresses at the internal

points are presented. The relative errors of the solutions

are also shown in these tables. One can see that the

solutions, even for N = 1, are quite accurate by

comparison with solutions I 2a -and solutions I 2a are very

accurate comparing with the analytical solutions.

4-4 RELATIONSHIP BETWEEN PLASTIC STRESSES AND PLASTIC

STRAINS

Up to now, the materials under consideration have

been assumed as elastic, which means:

A. After unloading, the configuration of the body for

the original unstrained configuration is recovered.

B. The deformations of the body depend only on the final

stresses, not on the previous load history or strain

path.

In plasticity, these two conditions are not satisfied,

residual strains are expected to occur when the loads are

removed and the final deformations depend not only on the

final stresses, but also on the path stress history from

the beginning of yielding.

Tab

le 4

-3-1

D

isp

lace

men

ts a

t In

tern

al

Po

ints

in

Exa

mpl

e 4-

3-1

All

th

e nu

rrbe

rs o

f d

isp

laca

nen

ts m

ust

be

mu

l tip

ied

by

10

-6 i

n t

his

tab

le.

I2a

N=1

N=2

u1

u2

u1

u2

u1

u2

u1

29

0.37

388

-8.1

85

3

0.46

578

-8.1

81

5

0.34

858

-8.1

89

3

0.37

206

-1. 1

1 %

-0

.24

%

23.1

9 %

-0

.29

%

-7.8

0 %

-0

.19

%

-1.5

9 %

30

0.74

288

-7.9

76

0

0.82

419

-7.9

73

5

0.72

148

-7.9

78

7

0.74

046

-0.9

1

%

-0.2

3 %

9

.94

%

-0

.26

%

-3.7

6 %

-0

.19

%

-1.2

3 %

31

1.09

90

-7.6

28

6

1.11

766

-7.6

29

5

·1.0

78

8

-7.6

29

5

1.09

66

-0.8

6 %

-0

.23

%

6.1

4

%

-0.2

2 %

-2

.68

% -

0.2

2 %

-1

.07

%

32

1.43

66

-7.1

50

2

1.51

80

-7.1

55

0

1.41

53

-7.1

49

8

1.43

43

-0.8

1

%

-0.2

4

%

4.81

%

-0

.17

%

-2.2

8 %

-0

.24

%

-0.9

7 %

33

1. 7

506

-6.5

47

7

1.84

37

-6.5

55

0

1.72

65

-6.5

47

1

1.74

75

-0.7

2 %

-0

.28

%

4.5

6

%

-0.1

7

%

-2.0

9 %

-0

.29

%

-0.9

0 %

34

1.10

29

-7.6

33

5

1.16

19

-7.6

36

2

1.08

82

-7.6

36

2

1.10

08

-0.5

0 %

-0

.16

%

4.81

%

-0

.13

%

-1.8

3 %

-0

.13

%

-0.6

9 %

35

1.10

00

-7.6

29

8

1.17

28

-7.6

31

4

1. 0

812

-7.6

31

4

1.09

77

-0.7

7 %

-0

.21

%

5

.80

%

-0.1

9 %

-2

.46

%

-0.1

9 %

-0

.97

%

36

1.09

99

-7.6

29

5

1.17

30

-7.6

29

7

1.08

09

-7.6

29

7

1.09

76

-0.7

8 %

-0

.22

%

5.8

2 %

-0

.21

%

-2

.48

%

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Tab

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-U

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172

The problem of formulating physical relations

describing the actual behaviour of a material during

plastic flow is a very complex one. In order to simplify

it, one considers the behaviour of a specimen stressed

in simple tension shown in Figure 4-4-1. If one expresses

the total strain as composed of elastic strain £e and

plastic strain £P, one has

£

e ° where £ = E .

(4-4-1)

When the stress has not reached the yield point y:

a - y < 0

only the pure elastic behaviour is obtained for the

loading.

Once a exceeds Y, however, one has the following

relationship from Figure 4-4-1:

After rearranging the previous equation, one obtains

Y + (4-4-2)

1

If the stress a is less than 00' the behaviour is

elastic. For general stress state, the yield condition

at a point is assumed to satisfy some criteria. One of

them commonly used is the von Mises yield criterion

which can be written as

a

ao-_

y

173

/1 / I

I I / I

I I / I

~~----------~--------~---------- &

Figure 4-4-1 Uniaxial stress-strain diagram showing elastic and plastic strains

174

F(o .. ,k) = 13J2 - a ~J 0

o (4-4-3)

where J 2 is the second invariant of the stress deviator

tensor:

s .. ~J

(4-4-4)

In Eq. (4-4-3), one can define I.3J"2 as an equivalent

s~ress ~, so that the von Mises yield criterion can be

expressed as

a e

a o (4-4-5)

which is used in the examples shown in this chapter.

In order to consider strain hardening, one applies

the load step by step in small increments after yielding.

Each increment loading step creates increments of strain

and stress tensor components, i.e. U, E .. } and {t1O .. } • . ~J ~J

Next one considers the relationship between the

plastic strain increments and stresses. Assume that a

loading path is found to reach a certain level and to

have then Elj plastic strains. Then, the load is increased

by a small increment, the additional plastic strains

produced are ~Elj and the total strain Eij are expressed

as follows:

(4-4-6)

where

175

£7. are total elastic strains including the ~J

current load increment.

The definition of modified total strain is

£! . ~J

£ •. - £~. ~J ~J

(4-4-7)

Considering £7. satisfy the Hooke's law (3-2-8), ~J

one can rewrite Eq. (4-4-7) as

£! . ~J

The equivalent strain is defined by

(4-4-8)

(4-4-9)

where e!. is the modified strain deviator tensor: ~J

(4-4-10)

Therefore, plastic strain increments are computed

by the following expression derived from the well-known

Prandtl-Reuss equations [63,64],

(4-4-11)

where ~£p is the equivalent plastic strain increment: e

I (4-4-12)

176

which can be calculated through the uniaxial stress-

strain curve. Thus one has

3G e:et - 0e 3G + H' (4-4-13)

where H' is the slope of uniaxial tensile curve given

by

H' do e

de: P

one can find H' from (4-4-2) after considering (4-4-5)

as

H' (4-4-14)

One has now all the relationships needed. For a

step and during the iterative procedure, the convergence

of ~e:P can be checked at the nodes. The recent procedures e

can be described as follows:

A. After the stresses are obtained from the boundary

element formulation, e:ij can be computed from (4-4-8).

B. The equivalent strain e:et is calculated with Eq.(4-4-9).

C. Next, from Eq. (4-4-13) the equivalent plastic strain

increment ~e:P can be obtained. It must be greater e

than zero, otherwise set it equal to zero.

D. Finally, the plastic strain increment ~e:lj can be

found from Eq. (4-4-11).

Repeat the above procedures until ~e:P are convergent e at every node.

177

4.5 THE GOVERNING EQUATIONS FOR ELASTO-PLASTICITY

In this section the basic differential equations and

boundary integral equations for elasto-plasticity are

presented. Here only some formulations are given without

detailed derivations, which can be found in reference

[12] •

First one define the strain increments as:

~(l1u .. + l1u .. ) ~,J J,~

(4-5-1)

where l1£ij , l1£~j and l1£lj are total, elastic and plastic

strain increments respectively.

The equilibrium equation in incremental form can

be rewritten as

l10.. . + l1 b J. ~J,~ o

and on the boundary one has

in n (4-5-2)

on r (4-5-3)

The Hooke's law (3-2-7) is still valid for the

elastic part:

l10ij = 2~(l1£ij - l1£lj) + A(l1£kk - l1£kk)&ij

(4-5-4)

The above Eq.(4-5-4) can be rewritten as:

l10 .. ~J

(4-5-5)

where l101 j are the components of the initial stresses,

which satisfy relationship:

t:.of? • 1.)

178

(4-5-6)

In the above equations t:.of? and t:.£f? can be 1.) 1.)

considered as "initial stress" and "initial strain"

terms.

Eq.(4-5-5) is valid for 3-D and plane strain problems,

but for plane stre$V must be replaced by v = vft1+v)in the

expression of A in (3-2-9).

The t:.£~k in Eq. (4-5-4) is as follows [12]

(4-5-7)

(for plane stress)

substituting Eq. (4-5-5) into Eqs. (4-5-2) and (4-5-3)

and considering Eq. (4-5-1), one obtains

G • + __ G__ • uUj,kk 1-2v UUk,kj 2G t:.£f? . - b.

1.),1. )

and (4-5-8)

+ G(t:.u .. + t:.u .. In. 1.,) ),1.)

(4-5-9)

Eq. (4-5-8) is the incremental form of the Navier

equation and Eq. (4-5-9) is incremental form of the traction

boundary condition.

179

The above expressions can alternatively be written

in the following form:

G A + _G_ 6 Lluj,kk 1-2\1 Uk,kj - 6'0.

J (4-5-10)

and

6Pi (4-5-11)

where 6b j and 6Pi are fictitious body forces and fictitious

tractions defined by

and

lib J. - 6oi? . 1J,1

(4-5-12)

(4-5-13)

After the concepts of fictitious body forces and

fictitious tractions are introduced in expressions (4-5-12)

and (4-5-13), one can find that there are no substantial

differences between the Navier equations (4-5-10) and

(3-2-10) and traction conditions (4-5-11) and (3-2-11)

except for the following:

A. Now in the incremental form.

B. The fictitious tractions and body forces instead

of previous actual ones.

All the boundary integral equations, fundamental

solutions and boundary implementations mentioned in

Chapter 3 are still valid for plasticity in the incremental

180

form with the concept of "fictitious":

c .. L'lU J. 1J I pij L'lU j df + I Uij L'lb j dQ f Q

(4-5-14)

Using the concepts of initial strains or initial

stresses instead of using fictitious force, one produces

the following boundary integral equations:

c .. llu. I u~. llpj df - I pij L'lu. df + 1J J 1J J

f f

+ I u~. lib. d~ + I ajki llE~k dQ 1J J

(4-5-15)

Q ~

c .. llu. I u~. lip. df- I pij llu. df + 1J J 1J J J

f f

+ I u* L'lb. d~ + I E'\. L'la~k dQ J J 1

(4-5-16)

Q ~

where Eijk and aijk are the same as Eqs. (3-3-6) and

(3-3-7) .

By comparing with the elasticity formulation, one

can see that there is another term in the domain integral,

which is due to the plasticity phenomenon. This term is

usually computed using subdomain integrations. However,

using the transformation technique previously discussed,

one can transform these terms into boundary integrals.

As one can not have the analytical expressions of plastic

stress or strain increments , during the iterative

181

procedures, one obtains the values of 601j and 6Elj

at a certain number of points only. The locations of

these points are chosen in advance. Hence, the Fourier

coefficients must be evaluated numerically and this will

be discussed in the next section.

The boundary element methods to solve the boundary

integral equation (4-5-14), (4-5-15) or (4-5-16) are

completely similar to those mentioned before, so one does

not need to repeat them in this chapter.

4-6 NUMERICAL ANALYSIS USING FINITE FOURIER SERIES

Let us suppose again that b(x) is of the period 2n

as in Section 2-5, but that its values are known only at

a discrete set of equally spaced points in this period

interval [65,66), i.e. at the 2N+l points

in/N i -N, -N+l, .•. , O,l, ••• ,N

According to the Fourier theorem, Fourier series converges

to the average of the right- and left-hand limits of b(x)

at each point where b(x) is discontinous, so one sets

b(-n) bIn) ~[lim b(x) + lim b(x)) (4-6-1)

After that, one has 2N independent points, which may be

expected to determine the coefficients of 2N terms in

the finite Fourier series:

b(x) kl cos nx + n

N-l I

n=l k 2 sin nx n (4-6-2)

182

Let us denote the ith coordinate as

in/N i -N+l, -N+2, ... ,D,1, ... ,N

so that the 2N independent values b i

The finite set of functions

(4-6-3)

b(x i ) are prescribed.

{I, cOS(nx), sin (mx) I n=l, ... ,N m=l, ... ,N-l}

is an orthogonal set under summation over the set of

points (4-6-3):

N t: n = m 1, ... , N-l L sin nx. sin mx.

i= -N+l 1 ]-# n m

N L sin nX i cos mx. D

i= -N+l 1

N

{,: n = m 1, ... ,N-l

L cos nx. cos mx. n = m D,N i -N+l 1 ~

n ,m

(4-6-4)

Now an approximation is assumed in the form of

b(x) =

N ko + L

n=l kl cos nx +

n

N-l L sin nx

n=l

(4-6-5)

and the least-square criterion is adopted as follows:

183

N N L [b(x i )- ko L kl cos nXi

i= -N+l n=l n

N-l L k 2 sin nXi )2 min

n=l n (4-6-6)

This leads to the following conditions:

N N L [b(xi ) - k L kl cos nx.

i= -N+l a n=l n ~

N-l L k 2 sin nxil 0

n=l n

(4-6-7)

N N L cos pxi[b(xi ) - k L kl cos nXi

i= -N+l a n=l n

N-l L k 2 sin nxil 0

n=l n

P l, ••• ,N

N N L sin pxi[b(xi ) - k L kl cos nx.

i= -N+l a n=l n ~

N-l L k 2 sin nxil 0

n=l n p l, ••• ,N-l

Considering Eqs. (4-6-4) and (4-6-7) , one obtains

1 N

k 2N L b(xi ) a i= -N+l

1 N

kl N L b (xi) cos nx. n = l, ••• , N-l n i= -N+l ~

(4-6-8)

1 N

kl 2N L b (xi) cos Nx. N i= -N+l ~

184

1 N -N L b(xl.') sin nxl.'

n= -N+1 n = 1, ..• ,N-1

A derivation for two-dimensional functions is

completely analogous to the one shown above for a

one-dimensional function and one has

where

4 N ko + L L

JI.=l n=l

k~ Nm

1 N

k ~ = 0 nM

4NM L n= -N+1 m

4 N M L L L

JI.=l n=l m=l

(4-6-9)

n=l, ... ,N m 1, ••• ,M

n = 1, ••• ,N-1

JI. = 1,3

1 N M NM L L b(x , x ) fJl. (x , xm)

n= -N+1 m= -m+1 n m nm n

1 N M 2NM L L

n= -N+1 m= -M+1 b(x n

JI. 1,3

1 N M 2NM L L b(xn

n= -N+1 m= -M+1 xm) fJl. (x , xm) Nm n

JI. 1,2

1 N M 4NM L L b(xn n= -N+1 m= -M+1

(4-6-10)

1

185

Using formulations (4-6-10), one can calculate the

coefficients of finite Fourier series from several

discrete values of the function and express the function

approximately as a Fourier series. Hence, in elasto-

plasticity, after every step in the iteration, one can

obtain the plastic strains and plastic stresses at certain

points inside the domain and on the boundary, so that

essentially, one can compute the domain integral terms

in Eqs. (4-5-14), (4-5-15) and (4-5-16) with the trans-

formation technique mentioned before.

If the period of the functions bi(x) is not equal to

2n, the transformation can be carried out as shown in

Eq. (2-5-6) of Section 2-5. Similarly the corresponding

finite Fourier expansion and the formulation of Fourier

coefficients can be obtained.

4-7 APPLICATION TO ELASTO-PLASTIC PROBLEMS

As mentioned in Section 4-5, the boundary integral

equation for elastoplastic problems can be equivalent

using (4-5-14), (4-5-15) or (4-5-16). There are two

kinds of domain integrals in each of those formulations.

One kind is due to the body forces b i and can be trans­

formed into boundary integrals with the formula (3-7-32),

which has already been discussed before. Another is due

to the plastic stress increments 6a~. or the plastic 1)

strain increments 6£lj. They can also be computed

with the same technique. For example, by means of initial

strain one can compute the last domain integral in Eq.

(4-5-15) with formula (3-7-33) because aijk

186

However when one calculates the stresses at internal

points, there is a domain integral in the initial strain

formulation, i.e. [12]

(4-7-1)

In order to compute it using boundary integrals, a

derivation analogous to the one shown in Eqs. (3-7-32) to

Cl·/-3l j.s u:1~d. The space derivatives of (3-7-33) with

respect to the spatial coordinate, yields a new trans-

formation formula.

Thu:'> in principle, using any of the formulations,

(i.e. initial strain, initial stress or fictitious body

forces and tractions), the domain integrals involved in

the boundary integral equations can be taken to the boundary.

In the present section, only the fictitious body force

and traction formulation (4-5-14) is discussed mainly

because of its simplicity, i.e.

A. After replacing b i and Pi by bi and Pi' the

formulation is the same as that of elasticity.

B. In the expression of fictitious body forces (4-5-12)

there are derivatives of the plastic strains ~£~ •• , ~],~

which are difficult to obtain numerically, but when

the finite Fourier series of the plastic strains

in the form of (4-6-9) is used, it is easy to evaluate

analytically these space derivatives.

After solving Eq. (4-5-14) , one obtains the displace-

ments and tractions at all nodes on the boundary. Then

the internal stresses can be computed by using Eqs.(4-5-1)

and (4-5-4) as follows [12]:

187

60ij J Uijk 6Pk df - J pijk 6uk df +

f f (4-7-2)

+ J Uijk 6bk dn - Cijk~ a

6Ek~ n

EXAMPLE 4-7-1

The simplesiexample is a rectangular specimen under

tensile force. The geometry of the specimen, mesh

points and internal points are shown in Fig. 4-7-1.

The material properties are

A. The body is composed of an elastic perfectly plastic

material (shown in Fig. 4-7-2 A), i.e. El = O.

B. The material has hardening properties (shown in

Fig. 4-7-2 B), i.e. EI

and, in addition, E = l~, v = 0.33. Yield point 00

1.3*10~.

The final displacement is 6 = 0.015.

The solutions are shown in Fig. 4-7-3.

EXAMPLE 4-7-2

In this example, a solid plane strain specimen is

indented by a rigid flat punch (shown in Fig.4-7-4).

The properties of material are the same as those in

Example 4-7-1 (Fig. 4-7-2 B). The final displacement of

the rigid punch is 6 = 0.02. The internal points, at

which the plastic strains are computed and the Fourier

coefficients are evaluate, can be seen in Fig. 4-7-4. The

dimensions are as follows:

188

• intemal p:>ints

x Iresh points

.- . I 1 I I I

..----T--'- --

I I 2 I I

~---t- - - -+ - ___ l-

I I I I 1 1 1/

3

ct. Figure 4-7-1 Geometry of the specimen in Example 4-7-1

a

y

189

a

y

~L-____________________ ,-£ ____________________

Figure 4-7-2 Uniaxial stress-strain diagram in Examples4-7-1 and 4-7-2.

6 .... ., 0

B 0 0 M r-~ ~ -" a g " ., I"il

Z9L

M M

,,; " "

cr,

~1 I I I I

\~

190

..... I

r­I ... Cl) ..... 0.

~ ><

I"il

'" o s:: o .... .... ::I ..... o ., Cl) .r: E-o

M I ..... I ...

o

00

o

.... o

N .,

o

B 13.5

h 8.5

b 6

191

Accurate results are shown in Fig. 4-7-5. By

comparison with the results presented in reference [121,

the tractions and plastic strains at the point A (in Fig.

4-7-4), at which the solution is discontinuous, are

less than 5% different.

192

x mesh points

• internal points

. I 1

I

H-++~ H--tt-H-i I~ I ._-LLj ____ . ~~: _____ b __ ~ ___ ·_-_____ B ____________ ~

Figure 4-7-4 Plane strain punch problem

h

193

>I,Q

I

\ N I

I"-I .... .... _0

0

\ Q) .... 0. e ., I< iii

..... 0

~

\ 0 .r! .oJ ::J .... 0 M

VI -0

\ Q) 0

.<::

\ E-t

III

\ I

l"-I ....

\ . \ Q) J.< ::J

'", 0>

.r! r.. N

0

0

.", .

'" "" ."" '" _0 . 0

'" ~9L

"';1 .,; o

CHAPTER 5 PROGRAMMING

In this chapter, a general description of the programm­

ing is presented. These programs were developed for

implementating the transformation formulae derived in

chapters two and three. They can be applied to solve

potential, elasticity and elasto-plasticity problems.

All these programs are coded in FORTRAN 77 and have

data structures suitable for most computers and micro­

computers. For simplicity of preparation of the input­

data file a special pre-processor is used [51].

5.1 POTENTIAL PROBLEMS

The program BEPOLI, for solving the potential problems

allows for general distributed source functions using a

linear continuous element. It consists of the following

subroutines.

1. Subroutine INPUT

The function of this subroutine is to read initial data

from input-data file.

The input data can be divided into the following

types:

A. General control numbers, such as the number of

boundary elements, the number of nodes on the boundary f of

the problem under consideration, the number of

internal points.

B. Basic numbers needed for the transformation formula,

such as number of boundary elements on f b , along

which the transformation formula is integrated, the

195

periods of source function in xl and x 2 directions,

the number of terms in finite Fourier expression Nand

M.

C. Geometry description, such as the coordinates of

nodes on boundary rand r b , the coordinates of

internal points, and the connecting node numbers

for every boundary element.

D. The coefficients of the Fourier series of the source

function.

E. Prescribed boundary conditions.

2. Subroutine MATC

The values of C are calculated for each node according

to the formulation (2-2-10).

3. Subroutine BFPO

The integrals due to the source function are computed

at every node required using transformation formula.

In this subroutine, a suitable number of integration

points is determined for each boundary element according

to the distance between the middle point of this element

and the source point. A judgement is made: if the source

point is located on the boundary element over which the

integrals are carried out, the transformation (2-7-3)

is used automatically, otherwise the original Gauss quad­

rature is used.

4. Subroutine MATHG

According to the formulation (2-3-5), one computes

the elements of matrices Hand G through all the boundary

196

and assembles them into the global matrices Hand G. Then,

after substituting the prescribed boundary conditions and

adding the integrals due to the source function, a set

of linear algebraic equations is created by rearranging

the original matrices.

5. Subroutine SOLV

The function of this subroutine is to solve the Eet

of linear equations with pivoting.

6. Subroutine INTER

This subroutine computes the potentials at the

internal points. The difference between this program and

that for Laplace equation is only the last integral term

of Eq. (2-2-9), the values of which have been computed

after calling subroutine BFPO.

7. Subroutine OUTPUT

This subroutine arranges the output data including

the values of potential and flux at the boundary nodes

and the values of potential at the internal points.

A main program controls the above subroutines.

5-2 ELASTICITY PROBLEMS

The program BEELLI uses linear continuous element to

solve two-dimensional elasticity problems with general

body forces. This program consists of the following

subroutines.

197

1. Subroutine INPUT

Besides reading the initial data the same as that

in potential problems, this subroutine reads some basic

data from input-data file, such as properties of material

(Young's modulus, Poisson's ratio) and the type of problem,

i.e. plane strain or plane stress.

2. Subroutine MATe

In this subroutine, the matrices c(s), the components

of which are expressed in Eqs. (3-8-11), are computed at wbere

the pointsAintegrals (3-7-1) or (3-7-2) are required

during the computations.

3. Subroutine BFEL

In this subroutine, the integrals (3-7-1) are computed

at all the mesh points on the boundary r and the internal

points in the domain with the transformation formula

(3-7-32). The integrals (3-7-2) are computed at all the

internal points with the formula (3-7-33).

If the source point s is located on the element over

which the integrals are carried out, the transformation

(2-7-3) and the finite-part integral (3-8-6) are used.

4. Subroutine MATHG

This subroutine computes the matrices Hand G element

by element and assembles them into the global matrices H

and G.

Then, after substituting the prescribed boundary

conditions into the matrix equation and rearranging the

equation, one obtains a set of linear algebraic equations.

198

The values of integral (3-7-1) are added to the

corresponding locations on the right-hand side of the

linear equations.

5. Subroutine SOLV

This subroutine is the same as that used in program

BEPOLI for solving the set of equations.

6. Subroutine INTER

The functions of this subroutine are to calculate

the displacements and stresses with formulations (3-5-4)

and (3-5-5). The statements are almost the same as for

the case of the program without body forces except that

one now adds the last integrals of Eqs. (3-5-4) and

(3-5-5) to the solutions. The values of these integrals

have already been computed when calling the subroutine

BFEL.

7. Subroutine OUTPUT

The function of this subroutine is arranging the

ouput-data file, which produces the displacements and

tractions at the boundary nodes and the displacements,

stresses and displacements at the internal points.

A main program calls the subroutines listed above.

5-3 ELASTO-PLASTICITY PROBLEMS

The program BEEPLI uses the fictitious tractions

and fic-titious body forces applied to study elasto­

plastic problems.

The program BEEPLI is based on the program BEELLI and

is extended to cover iterative and incremen~techniques.

199

In addition, a new subroutine is necessary in order to

obtain the Fourier coefficients numerically. The

differences between this program and the program BEELLI

are as follows.

1. Subroutine FOCO

The functions of this subroutine are:

A. After obtaining the plastic strains the Fourier

coefficients of plastic strain increments are

calculated with Eqs.(4-6-10).

B. Then, the Fourier coefficients of fictitious traction

increments ~p. and fictitious body force increments ~

~b. are combined with Eqs. (4-5-12) and (4-5-13). ~

2. Subroutine ITERA

From step A to step D, described in the last part

of section 4-4, is carried out for certain increment of

boundary conditions.

After executing subroutine ITERA, there are two

possible options.

A. If the equivalent plastic strain increment ~E~ converges,

(i.e. the differences of Eet between two adjacent

iterative steps are less than the prescribed error

at every node), the program proceeds to the next

incremental step.

B. If the solutions do not converge, the values of ~EP e

and ~Elj obtained during the last iterative step are

recorded and a new iterative procedure is started.

200

3. In addition to the above, the actual tractions are

calculated with Eq. (4-5-13) after each iterative step.

4. In the main program, the increment loops are added.

After obtaining convergent solutions, one can then add

the increment to the prescribed boundary conditions and

continue with the iterative procedures.

CHAPTER 6 GENERAL DISCUSSION AND CONCLUSIONS

The present work describes a way to transform domain

integrals into boundary integrals in boundary element

formulations. The transformation is presented for both

potential and elasticity problems with detailed treatments

of how they can be implemented numerically. The results

tested term by term demonstrate the accuracy of this

method. The transformation formulae are proved to be

reliable using the numerical methods and the programs

developed in this work. For the case of arbitrary source

or Body force functions, the solutions obtained by the

transformation technique are sufficiently accurate

provided that enough number of terms are taken in the

Fourier series. The examples presented demonstrate

that this type of transformation technique is feasible

for applications to Poisson's or elasticity type problems

including cases with arbitrary source or body force

functions. The numerical solutions are accurate for

engineering problems when a relatively low number of terms

of Fourier series is taken. Therefore the transforma­

tion technique is also efficient in computer time.

The obvious advantage of the approch is that only

the discretizatio~ of the boundary is necessary and

hence its efficiency is mainly due to the equations

having one degree less of dimensionality.

The approach can be extended to the non-linear or

time-dependent problems. In these cases the Fourier

coefficients need to be calculated numerically which

202

presents the following problems:

A. In order to obtain accurate Fourier expansion

numerically, a sufficiently large number of integral

points must be taken and both operations, i.e.

computing the solutions at the internal points and

finding the Fourier coefficients require a consider­

able amount of CPU time.

B. After each step of iteration, the Fourier coeffic­

ients must be recalculated. This additional

computation makes the technique less efficient than

the original domain integral method.

In spite of the above, the transformation technique

presents a general and alternative way to solve non­

linear problems without having to discretize the domain

cells.

Throughout this work, the boundary element integrals

can be computed numerically following two different

approaches: a. using direct numerical integrations to

compute the boundary integrals presented in the trans­

formation formulae, which contain the harmonic functions

for Fourier series. b. using polynomial interpolation

functions one can express the harmonic functions over

each element in terms of nodal values, with which the

boundary integrals can be computed using a series of

influence coefficients and nodal values of the harmonic

functions.

The former approach is used throughout this work.

The advantage of it is great accuracy even using coarse

boundary elements. The latter approach is more efficient

than the former; particularly in non-linear problems

203

the integrals have to be computed at each itera~ion.

In this case, the computations for the transformation

formulae will consume almost the same time as taking

only one term in the Fourier expansion. It can be done

as further work including the error analysis of results.

Throughout this work, harmonic functions are chosen

to expand the source or body force functions. One of

the advantages of this set of functions is that it is

easy to find out the solutions which satisfy the original

differential operators of the problems. Another is that

arbitrary functions can be expanded as a convergent

Fourier series in most practical engineering problems.

Despite these obvious advantages, there exist

certain disadvantages for this technique. One of them is

that convergence may not be fast enough in some particular

cases, such as for discontinuous functions. Another

disadvantage is that it is necessary to find out the

Fourier coefficients required in the transformation.

One way of increasing the efficiency of the method is

to compute the Fourier coefficients analytically, which

can only be done in the case of having analytical

expresions for the source or body force functions. Other­

wise, a numerical approach must be used. This usually

leads to loss of accuracy and requires considerably

more CPU time

A suggestion for further work will be to investigate

the numerical behaviour and computer efficiency of complete

functions other than Fourier series. This work may intro­

duce new difficulties but it is worth pursuing.

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Lecture Notes in Engineering

Edited by CA Brebbia and SA Orszag

Vol. 1: J. C. F. Telles, The Boundary Element Method Applied to Inelastic Problems IX, 243 pages. 1983.

Vol. 2: Bernard Amadei, Rock Anisotropy and the Theory of Stress Measurements XVIII, 479 pages. 1983.

Vol. 3: Computational Aspects of Penetration Mechanics Proceedings of the Army Research Office Workshop on Computational Aspects of Penetration Mechanics held at the Ballistic Research Laboratory at Aberdeen Proving Ground, Maryland, 27-29 April, 1982 Edited by J. Chandra and J. E. Flaherty VII, 221 pages. 1983.

Vol. 4: W.S. Venturini Boundary Element Method in Geomechanics VIII, 246 pages. 1983.

Vol. 5: Madassar Manzoor Heat Flow Through Extended Surface Heat Exchangers VII, 286 pages. 1984.

Vol. 6: Myron B. Allen III Collocation Techniques for Modeling Compositional Flows in Oil Reservoirs VI, 210 pages. 1984.

Vol. 7: Derek B.lngham, Mark A. Kelmanson Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems IV, 173 pages. 1984.

Vol. 8: Linda M. Abriola Multiphase Migration of Organic Compounds in a Porous Medium A Mathematical Model VIII, 232 pages. 1984.

Vol. 9: Theodore V. Hromadka II The Complex Variable Boundary Element Method XI, 243 pages. 1984.

Vol. 10: C. A. Brebbia, H. Tottenham, G. B. Warburton, J. M. Wilson, R. R. Wilson Vibrations of Engineering Structures VI, 300 pages. 1985.

Vol. 11: M. B. Beck Water Quality Management: A Review of the Development and Application of Mathematical Models VIII, 108 pages. 1985.

Vol. 12: G. Walker, J. R. Senft Free Piston Stirling Engines XIV, 286 pages. 1985.

Vol. 13: Nonlinear Dynamics of Transcritical Flows Proceedings of a DFVLR International Colloquium, Bonn, Germany, March 26, 1984 VI, 203 pages. 1985.

Vol. 14: A. A. Bakr The Boundary Integral Equation Method in Axisymmetric Stress Analysis Problems XI, 213 pages. 1986.

Vol. 15: I. Kinnmark The Shallow Water Wave Equation: Formulation, Analysis and Application XXIII, 187 pages, 1986.

Vol. 16: G. J. Creus Viscoelasticity - Basic Theory and Applications to Concrete Structures VII, 161 pages. 1986.

Vol. 17: S. M. Baxter C. L. Morfey Angular Distribution Analysis in Acoustics VII, 202 pages. 1986.

Vol. 18: N. C. Markatos, D. G. Tatchell, M. Cross, N. Rhodes Numerical Simulation of Fluid Flow and Heat/Mass Tranfer Processes VIII, 482 pages. 1986.

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Vol. 20: S. M. Niku Finite Element Analysis of Hyperbolic Cooling Towers VIII, 216 pages. 1986.