Boundary Element Methods for Engineers and Scientists: An Introductory Course with Advanced Topics

491 Pages·2003·10.795 MB·English

by  Prof. Dr.-Ing. habil. Lothar Gaul, Dr.-Ing. Martin Kögl, Dr.-Ing. Marcus Wagner (auth.)

L. Gaul, M. Kogl, M. Wagner
Boundary Element Methods for Engineers and Scientists
Springer-Verlag Berlin Heidelberg GmbH
ONLINE LIBRARY
Engineering
http://www.springer.de/engine/
Lothar Gaul · Martin Kogl • Marcus Wagner
Boundary Element Methods
for Engineers and Scientists
An Introductory Course with Advanced Topics
With 135 Figures
Springer
Prof. Dr.-Ing. habil. Lothar Gaul
Full Professor and Head Institute A of Mechanics, University of Stuttgart. Former
Dean Process Engineering and Engineering Cybernetics, University of Stuttgart.
Former Dean Mechanical Engineering, University of the Federal Armed Forces
University Hamburg.
e-mail: [email protected]
Dr.-Ing. Martin Kogl
Currently working as post-doctoral researcher at the Department of Structures and
Foundation Engineering, University of Sao Paulo, Brazil. Former research coworker
at the Institute A of Mechanics, University of Stuttgart.
e-mail: [email protected]
Dr.-Ing. Marcus Wagner
Currently research engineer at BMW Group, Munich. Former post-doctoral fellow,
Division of Mechanics and Computation, Stanford University, Stanford, CA, USA.
Former research assistant at the Institute A of Mechanics, University of Stuttgart.
e-mail: [email protected]
Universitat Stuttgart
Institut A fur Mechanik
Pfaffenwaldring 9
70550 Stuttgart
Germany
ISBN 978-3-642-05589-8 ISBN 978-3-662-05136-8 (eBook)
DOI 10.1007/978-3-662-05136-8
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Preface
Over the past decades, the Boundary Element Method has emerged as a ver
satile and powerful tool for the solution of engineering problems, presenting in
many cases an alternative to the more widely used Finite Element Method.
As with any numerical method, the engineer or scientist who applies it to
a practical problem needs to be acquainted with, and understand, its basic
principles to be able to apply it correctly and be aware of its limitations.
It is with this intention that we have endeavoured to write this book: to
give the student or practitioner an easy-to-understand introductory course
to the method so as to enable him or her to apply it judiciously. As the title
suggests, this book not only serves as an introductory course, but also cov
ers some advanced topics that we consider important for the researcher who
needs to be up-to-date with new developments.
This book is the result of our teaching experiences with the Boundary
Element Method, along with research and consulting activities carried out in
the field. Its roots lie in a graduate course on the Boundary Element Method
given by the authors at the university of Stuttgart. The experiences gained
from teaching and the remarks and questions of the students have contributed
to shaping the 'Introductory course' (Chapters 1-8) to the needs of the stu
dents without assuming a background in numerical methods in general or
the Boundary Element Method in particular. This part can be used both for
self-study or as a basis for a graduate university course on the method. The
remaining chapters cover more advanced topics and are aimed at researchers
who need to learn morc about extensions and alternative approaches to the
'classical' method. We sincerely hope that we have succeeded in the task
to find an approach that is suitable for a first introduction to the method,
since we believe that the first contact is of great importance in awaking the
students' interest and to motivate them to delve deeper into the subject.
Before embarking on the Boundary Element Method, we have included
two chapters on the mathematical basis and continuum physics; this way the
reader will not have to consult other textbooks to be able to follow ours. We
assume that the reader is familiar with the basic concepts of vector algebra
and calculus as taught in any undergraduate course on engineering mathe
matics - the chapter on mathematics is intended only to introduce some con
cepts that might not be covered in those courses. The chapter on continuum
VI Preface
physics attempts to describe in a concise way the basic physical principles
that lead to the differential equations on which we base our numerical solu
tion procedures. While standard textbooks on numerical methods commonly
omit this subject, we feel that an understanding of the physical principles is
of great importance for successfully solving engineering problems, not only
for the correct choice of the model and the pre-processing, but also for the
interpretation and critical judgement of the results.
While the classical Boundary Element Method, described in the first part
of this book, is the most widely known and universally employed approach,
the last two decades have seen the emergence of a number of complementary
and concurrent developments in Boundary Element research, dealing with
problems where the classical method presents difficulties or fails completely.
We have therefore included two of these additional methods in the book; we
believe their study is of great importance for the practitioner or researcher
who needs or wants to extend his or her knowledge about the method: the
Dual Reciprocity Method and the Hybrid Boundary Element Methods.
In the first two chapters of Part II, we give an introductory course on the
Dual Reciprocity Method and describe in detail the solution of the equations
of motion. Later, we will apply the method to more advanced problems. The
Dual Reciprocity Method is an extremely versatile tool for the transformation
of general source terms to the boundary, a problem that occurs in many
Boundary Element applications (coupled field problems, dynamic analysis,
inhomogeneities, non-linearities, to name but a few). It is not a separate
Boundary Element Method in itself but rather an extension to the classical
method which can be used in the many cases in which we have to deal with
sources that cannot be transformed to the boundary by other means.
In Part III, we present a new development in the field of symmetric for
mulations, the Hybrid Boundary Element Methods. These methods are based
upon variational principles rather than upon weighted residual formulations.
The name 'hybrid' describes the fact that we use a Boundary Element-type
approximation with fundamental solutions for the domain field, and a Finite
Element-type approximation on the boundary. We will describe two formu
lations of the method for elastodynamics and acoustics, and also give an
introduction to fluid-structure interaction.
The insight and experience that have contributed to this book could only
be gained by an intensive cooperation in the framework of a research group on
the Boundary Element Method. The authors are indebted and would like to
thank their colleagues and ex-colleagues Dr. Christian Fiedler, Dr. Wolfgang
Wenzel, Dr. Friedrich Moser, and Dipl.-Ing. Matthias Fischer at the Institute
A of Mechanics at the University of Stuttgart for their cooperation and the
fruitful discussions on the subject. We would also like to thank our Brazil
ian colleagues Prof. Ney Dumont of PUC Rio de Janeiro and Prof. Marcos
Noronha of the University of Sao Paulo (USP), with whom an intensive ex
change of researchers and ideas has taken place. The authors appreciate the
Preface VII
continuous advice of Professor Patrick Selvadurai, McGill University Mon
treal, during his stay as recipient of the Alexander von Humboldt Senior
Scientist Award at the Institute A of Mechanics. We are as well indebted to
his kind wife Mrs. Sally Selvadurai for her thorough proofreading and editing
of the manuscript. We shall not forget our students who have made valuable
contributions through a number of excellent theses.
This research would not have been possible without the excellent pro
motion by the Deutsche Forschungsgemeinschaft (DFG -- German Research
Foundation) and the Deutscher Akademischer Austausch Dienst (DAAD -
German Academic Exchange Service), who have financed a number of our
research projects on the Boundary Element Method.
Stuttgart, Lothar Gaul
November 2002 Martin K ogl
Marcus Wagner
Contents
Part I. The Direct Boundary Element Method
1. Introduction.............................................. 3
1.1 Numerical Solution of Engineering Problems. . . . . . . . . . . . . . . 3
1.2 Historic Development of Boundary Element Method ... . . . . . 6
1.3 The Method of Weighted Residuals . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Collocation Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Method of Moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Galerkin's :\lethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.4 Collocation by Subregions. . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.5 Least Squares. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10
1.3.6 Summary........................................ 11
1.4 Boundary Elements vs. Finite Elements .. . . . . . . . . . . . . . . . .. 11
1.4.1 General Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12
1.4.2 Comparison of FE and BE Formulations ............ 13
1.5 Boundary Integral Method for 1-D Differential Equation. . . .. 16
1.6 General Boundary Element Approach. . . . . . . . . . . . . . . . . . . .. 20
2. Mathematical Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
2.1 Notation.............................................. 23
2.1.1 Vector Notation and Matrix Representation of Vectors 23
2.1.2 Index Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25
2.2 Theory of Distributions ................................. 26
2.2.1 Definition and Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27
2.2.2 Dirac Distribution and Heaviside Function. . . . . . . . . .. 27
2.3 Integral Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29
2.4 Time-Harmonic ;vIotion ................................. 30
2.5 Introduction to the Calculus of Variations. . . . . . . . . . . . . . . .. 31
3. Continuum Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35
3.1 Introduction........................................... 35
3.1.1 Continuous l\ledia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35
3.1.2 Variables and Equations of State. . . . . . . . . . . . . . . . . .. 36
3.1.3 Crystal Structure of a Continuous Medium. . . . . . . . .. 37
3.2 Elastodynamics........................................ 38
X Contents
3.2.1 Kinematics...................................... 39
3.2.2 Kinetics......................................... 43
3.2.3 Conservation of Mass - The Continuity Equation. . . .. 44
3.2.4 Conservation of Linear Momentum ................. 46
3.2.5 Conservation of Angular Momentum. . . . . . . . . . . . . . .. 47
3.2.6 Conservation of Energy ........................... 48
3.2.7 Constitutive Equation ............................ 50
3.2.8 Field Equations and Boundary Conditions. . . . . . . . . .. 53
3.2.9 Elastodynamic Wave Equations. . . . . . . . . . . . . . . . . . .. 55
3.2.10 Waves in Anisotropic Materials .................... 56
3.3 Heat Conduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60
3.3.1 First Law of Thermodynamics. . . . . . . . . . . . . . . . . . . .. 60
3.3.2 Second Law of Thermodynamics ................... 61
3.3.3 Field Equations of Heat Conduction . . . . . . . . . . . . . . .. 63
3.3.4 Boundary and Initial Conditions ................... 64
3.4 Electrodynamics........................................ 65
3.4.1 Maxwell's Equations in Vacuum. . . . . . . . . . . . . . . . . . .. 65
3.4.2 Electromagnetic Wave Equations. . . . . . . . . . . . . . . . . .. 67
3.4.3 Electrostatic Field in Macroscopic Media. . . . . . . . . . .. 68
3.4.4 Field Equation for Dielectrics . . . . . . . . . . . . . . . . . . . . .. 68
3.4.5 Electric Jump Conditions at Interfaces. . . . . . . . . . . . .. 70
3.4.6 Electric Boundary Conditions for an Ideal Conductor. 70
3.5 Thermoelasticity....................................... 71
3.5.1 First Law of Thermodynamics. . . . . . . . . . . . . . . . . . . .. 71
3.5.2 Second Law of Thermodynamics ................... 72
3.5.3 Constitutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . .. 73
3.5.4 Field Equations of Thermoelasticity ................ 76
3.5.5 Simplified Thermoelastic Theories. . . . . . . . . . . . . . . . .. 77
3.6 Acoustics.............................................. 79
3.6.1 The Acoustic Wave Equation. . . . . . . . . . . . . . . . . . . . .. 80
3.6.2 Constitutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . .. 81
3.6.3 The Velocity Potential . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86
3.6.4 Boundary Conditions in Acoustics. . . . . . . . . . . . . . . . .. 86
3.6.5 Radiation Condition in Infinite Domains ............ 87
3.7 Piezoelectricity......................................... 89
3.7.1 Polarisation in Piezoelectrics. . . . . . . . . . . . . . . . . . . . . .. 90
3.7.2 Energy Balance and Thermodynamic Potentials. . . . .. 91
3.7.3 Constitutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . .. 92
3.7.4 Field Equations of Piezoelectricity. . . . . . . . . . . . . . . . .. 93
4. Boundary Element Method for Potential Problems ....... 95
4.1 Introduction........................................... 95
4.2 BE Formulation of Laplace's Equation . . . . . . . . . . . . . . . . . . .. 96
4.2.1 Inverse Formulation of Differential Equation. . . . . . . .. 96
4.2.2 Green's Representation Formula. . . . . . . . . . . . . . . . . . .. 98