The Boundary Element Method in Engineering: A Complete Course

348 Pages·1992·11.95 MB·English

PROPERTY
OF
PEDRO J. HERRERA FRANCO

The Boundary Element
Method in Engineering

A complete course

Department of Mechanical Engineering
University of Nottingham

McGRAW-HILL BOOK COMPANY

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Libcary of Congress Catsloging-in-Publication Data

Becker A. A.
The boundary element method i engineering: a complete course/
A. A. Bockor
p. em.
Includes bibliographical references and index.
ISBN 0.07-107839-4
1. Boundary element metheds. 1. Title
TA347.B69R43 | 1992
6207.007°S1535-nde20 91-26214
cr

BOUNDARY ELEMENT METHOD IN ENGINEERING.

Copyright © 1992 MoGraw-Hill International (UK) Limited
All rights served. No part of this publication may be reproduced, stored
in retrieval system, or transmitted, in any form or by any means, electronic,
‘mechanical, photocopying, recording, or otherwise, without the prior permission
of the copyrightholdkr.

17345 CUP 95432

‘Typeset by P:R Typsesetters Ltd, Salisbury, Wiltshire
and printed and bound in Great Britainat the University Press, Cambridge
PROPERTY
OF
PEDRO J. HERRERA FRANCO

In memory of my father
CONTENTS,

NOTATION ix
PREFACE i
CHAPTER ONE INTRODUCTION 1
‘Numerical Methods ia Continuum Mechanies
(2 An Overview of BE Development 6
13. BE Compared to FE 8
(4 About This Book: 1
CS) Summary 14
CHAPTER TWO MATHEMATICAL BACKGROUND. Is
21 Veetors and Tensore Is
22 Tensor Notation »
23° About Integral Equations a
24. Green's Theurems 2
25 Stresses and Tractlons 2%
2.6 The Fundamental Solution n
2.7 Reciprocal Work Theorem (Detti) a
28 Integral Identity for Displacements u
29 Numerical Integration Using Gaussian Quadrature 32
2.10 Equation Solving Using Gaussian Elirrination 34
2.10 Etliptic Integrals 3T
242 Summary 9

CHAPTER THREE TWO-DIMENSIONAL POTENTIAL

PROBLEMS 4i
3.1 Analytical Formulation at
2 Numerical Implementation 45
3.3 Two-Dimensional Potential Examples 1

34° Summary o
CHAPTER FOUR TWO-DIMENSIONAL ELAS, OSTATIC

PROBLEMS @
41° Analytical Foumulation 6
4.2, Numerical Implementation nu
43° Treatment of Edges and Corners 8s
4.4 ‘Tsvo-Dimensional Elastostatic Examples RT
43° Summary 90
CHAPTER FIVE TUREE-DIMENSIONAL POTENTIAL AND:
ELASTOSTATIC PROBLEMS oe
5.1. Three-Dimensional Potentiat Formutation 94
$2. Three-Dimensional Etastostatic Formulation ”
3 ion ‘ to4
$4 Three-Dimensional Elasiostatie Examples ug
5.5 Summary 124
CHAPTER SIX AXISYMMETRIC POTENTIAL AND
ELASTOSTATIC PROBLEMS. 125
6.1 Axisymmetric Potential Formulation 126
63 Axisymmetric Elastostatie Examples 137
64 Summary 12
CHAPTER SEVEN THERMOELASTIC PROBLEMS M4
71 The Body Force Term as
72. Throe-Dimensional Thermoelastic Kernels 147
73. Two-Dimensional Thermoelastic Kernels 148
74 Axisymmetric Thesmoclastic Kemels 150
75 . Numerical Implementation (53
1.6 Thermoelastic Examples 154
77° Summary 160
CHAPTER EIGHT MULTI-DOMAIN AND CONTACT
PROBLEMS: 161
8.1 The Contact Variables 162
8.2 Coupling the Contact Variables 165
83° The Iterative Procedure 167
84 Contact Examples 71
85 Summary 19
CHAPTER NINE FRACTURE MECHANICS PROBLEMS: 181
9.1 Linear Blastic Fracture Mechanies 192
9.2 Caleulation of the Stress Intensity Factor 186
19

93° Singularity Elements
9.4. Fracture Examples 192
9.5 Summary 197
CHAPTER TEN REVIEW OF NON-LINEAR PROBLEMS 199
10.1 Theoretical Background 199
10.2 Flastoplastic Differential Equations 203
10.3 Review of BE Elastoplastic Formulation 208
104 BE Approaches to Blastoplastcity 207
30.5 Numerical tmpicrentation 2
10.6 erative and Incremental Schemes a3
407 Suesmary 214
CHAPTER ELEVEN FURTHER APPLICATIONS 26
11.1 Coupling BE and FE Techniques 216
11.2 Cemtrifugal Loading 223.

Conditions BO
11.4 Infinite Boundary Elemente 234
11'5 Time-Dependent Potential Probl
11.6 Summary 24s

CHAPTER TWELVE BE COMPUTER PROGRAM 246
(2.1 Main Features of BEACON 248
122 Structure of BEACON 8
123 Integration Schetoes in BEACON, 29
12.4 Description of the Subroutines 252
12.8 Examples of Applications 256
126 Summary 2m
Appendix A: Gaussian Quattrature Integration Points m
Appendix B: Elliptic Integrals Series Coeficients 16
‘Appendix C: Differentials of the Thrco-Dimensional Shape

Functions am
Appendix: Axisymmetric ‘Traction Kernels 2
Appendix E: Users’ Manual for BEACON 281
Appeniix F: List of the Main Variables in BEACON 288
Appendix G: Pall Listing of BEACON 204
REFCRENCES cry
AUTTIOR INDEX 332
SUBJECT INDEX 334
NOTATION

Some of the key variables used in this book are listed below. Where
"

hols may th

a Crack length

A ‘Area. of the solution domain in two-dimensional and axisym-
ic prob

TA] Matrix containing the kernels multiplying either [@] or [1]

[At] Solution enatrix mutiplying the unknown vector [x]

[8] Matrix containing the keracls muleiplyingeither [39/2] ot £6]

UB* ‘Modified form of BTalier application ofthe houndary conditions

By Veclot containing the body foree integrals

[el Vector containing known quantities on the right-hand side of

the equations
Putt) Thitd-order tensor multiplying the tractions

4 Unit vector it one of the Cartesian directions

E Young's modulus

Etm,m/2) Elliptic incegral of the second kind

if Rody foree vector

R Force vector

G Total number of Gaussian quadrature points

G Strain energy release rate

& Galerkin vector

J(E)——_Tacobian of transformation in two-dimensional and axisyrometcio
problems

Ji 42) Jacobian of transformation in three-dittensional problems

k Conductivity

K(m,/2) Elliptic integral of the first kind
K, Fracture toughness (critical value of the stress intensity factor)
K, (2,2) _ First potential kemel multiplying the potential function
K,(,Q) Second potential kernel muliplying the potential gradients
Ky Ku, Ky Stress intensity factor for fracture modes I, 11 and IIT
Loti.) Linear shape function used in three-dimensiqnal problems
La Maximum distance between any (wo point. on the boundary
i} Laplace transform operaior

w ‘Tangeutial vector iu two-dimensional and axixynunettie problems
Modulus of the elliptic integrals

eiy,m; Tangential vectors in three-dimensional problems

Mi(p,Q) — Thetmoelastic kernel moltiplying the temperature

" Unit outward normal at the boundary

N.() Quadratic shape functions in two-dimensional snd axisymmetric
problems

N.lén&) Quadeatic shape functions in three-dimensional problems

Ni(p,Q) — Thermoctastic kernels multiplying the temperature gradients

> Load point inside the solution domain

P
4 Field point inside the solution domain
2 eld point on the boundary
Fg Oquty —Vatiahle evindrical coordinates of the feld point Q
70). Distance between points p and Q
Ry G,Z, Fixed cylindrical coordinales of the load point p
8 Surface in theee-dimensional problems
Scaling fastor
Sufo.Q) Third-order tensor maubiplying the displacement vector

s; Deviatotie stress component (used in elastoplasticty)
w ‘Traction vector

T Temperature function

F,(e,0) Traction kerael

4; Displacement veewor

* Normal component of displacement

u Sutin energy

Lyle.) Displacement kernel
ey03, 83 Setofthzee orthogonal axes used in three
v Volume in three-dimensional problems
Yiy2.g) Blastoplastic kernels used in the initial stress approach
‘Weight function in ordinary Gaussian quadrature

imensional problems

Mpc ‘Weight function in logarithmic Gaussian quadcature
Was(er4) Elastoplastic kernels used io the initial strain approach.
by Vector containing the unt nowns

%pi¥or2g Variable Cartesian coordinates of the field point @

%,.¥,.Z, Fixed Cartesian coordinates of the load point p
* Angle

« Coefficient of thermal expansion

te Thermal diffusivity

r Surface or houndary in two-dimensional and axisymmetric

problems
NOTATION ai

24 /én___Normal potential gradiont or temperature gradient
3, Kronecker dette funtion

x Interference oe elearanee fa the normal direction
& Amount of tangeotil slip in feietional shpping

oa Equivalent stram (used in elastoplasticte)

oe Cartesian strain components

be Radial strain in axisymmetric problems

a Shear strain jn axisymmetric problems

Axial strain in axisymmetric problems

te oop strain in axisymmetric problems
a Lame’s constant
a Shear modulus
# Cocficient of fction
Poissons zat
g ‘Local intrinsie coordinate in two-dimensional and a
problems
2%, Local intrinsic coordinates in three-dimensional problems
4 jaussian coordina(é in ordinaty Gaussian quadrature
ee Gaussian coordinate in logarithmic Gaussian quadrature
Eid, Local intrinsic coordinates in three-dimensional problems
e Density
& Equivalent stress af a point
oq Equivalent Von Mises stress (used in elastoplastcity)
ey ‘Cantesian stress components
om Hiydrestatic stress component (used in elastoplasticiy)
a ‘Normal stress component
om Redial strees in axisymmetric problems
on Shear stress in axisymmetric problems
* Tangential steess components
ou Axial stress in axisymmetric problems
om Hoop stress in axisymmetric problems
e Potential function of temperature
&, Angular velocity aboot the 2 axis

vi Laplacian operator
PREFACE

‘The Boundary Element (BE) method should be well established by now as
an accurate and powerful numerical fechnique in continuum mechanics. Yet,

prising rae DFO nei
specialize in numerical stress analysis techniques either have no idea what
BE stands for or have heard of it, picked up a book on it, then hurriedly
cout of every page. At the same lime, the Finite Element (FE) method has
attracted a huge following in industry; many engineers use it almost daily
and are quite happy with its performance in practical problems. Although

ws bs matheratiea! mon:

the FE formulation 1s inherently easier Te understand than tke BE formu
lation, it does, however, sues from some drawbacks which perhaps are nat
immediately apparent uetil you start comparing it with the BE method

The interesting fact about FE users is that proportionally few of thera
can claim to fully understand the theary and mathematical background, but
most of them are quite confident that they somehow can pick up’ this
information if they so desire. Mathematics appears in every branch of
engineering; it is not the content but the style of presentation that really
matters. The main obstacle hindering the fast progress of the BE technique
is that it has acquired the reputation (among enginocrs) of having very
unfamiliar and complex mathematics. This book is specifically writlen to
address this problem,

This book is a self-contained text on the BE method. The style of
presentation and range of applications should make the book a complete
cours: on BE techniques aimed spocifically wt engineers and engincering
undergrastuates, Care has been taken (o ensure [hal (he mathematical content
of the book is kept to & minimum and presented clearly (mosuly avoiding
tonsor notation). A diteet BE formulation (hased on physical quantities of,
displacements and strossos)is adopted throughout the book. Tho main feature
of this hook is that al the msthematical background and theories are
contained in only One chapter, which readers con either refer to occasionally
or omit altogether if they so wish, without affecting their understanding of
(he formulation. Another feature is that the BE wumerical itiplementation