Here is a course in
boundary element methods for the absolute
beginners. It assumes some prior basic knowledge of vector calculus (covering topics such as line, surface and volume integrals and the various integral theorems), ordinary and partial differential equations, complex variables, and computer programming. If you would like to create a hyperlink to this webpage or bookmark it, please use the following URL: www.ntu.edu.sg/home/mwtang/bem2011.html. Tell a friend about this site! This course has a webpage at
:
www.facebook.com/bcbem.A related book dealing with boundary element methods for analysing cracks in elastic solids is Hypersingular Integral Equations in Fracture Analysis. Preface Chapters and Fortran source codes The complete course About the author (Also visit WT Ang's author page at Amazon.com) Errata updated on 2 May 2012 Additional articles updated on 23 May 2011 (Item 5: Fortran codes for axisymmetric potential problem) Libraries where one may borrow a hardcopy of this course Citations by research papers, theses and so on Comments and suggestions View statistics of recent visitors click at Recent Visitor Activity for some details of visitors
PREFACE During the last few decades, the boundary element method, also known as the boundary integral equation method or boundary integral method, has gradually evolved to become one of the few widely used numerical techniques for solving boundary value problems in engineering and physical sciences. In implementing the method, only the boundary of the solution domain has to be discretized into elements. In the case of a two-dimensional problem, this is really easy to do. Put closely packed points on the boundary (a curve) and join up two consecutive neighboring points to form straight line elements. In March 1985, when I started my PhD work in the Department of Applied Mathematics at the University of Adelaide, Australia, I was introduced to the method by my supervisor, David L Clements. At that time, the term "boundary element method" was relatively new. It was first used in the 1977 paper "Boundary element methods for potential problems" by CA Brebbia and J Dominguez in the journal Applied Mathematical Modelling (volume 1, page 372). Carlos Brebbia and his co-researchers had undoubtedly played a significant and important role in introducing the method to the engineering research community. Less than 200 journal papers whose titles contained the term "boundary element method" could be found in 1985. In 2006, there were several thousand or perhaps even more such papers. The history of the method could, however, be traced back to an earlier time, well before the 1970s. The mathematics that laid the theoretical foundation for the development of the method could be found in the works of famous mathematicians like Laplace, Gauss, Fredholm, Betti, Muskhelishvili and Mikhlin. In the 1960s, there were attempts at using electronic computers to approximate solutions of potential problems through the use of boundary integral equations, notably the pioneering works of MA Jaswon (in MA Jaswon, "Integral equation methods in potential theory I" Proceedings of the Royal Society of London Series A, volume 275, page 23, 1963) and GT Symm (in GT Symm, "Integral equation methods in potential theory II" Proceedings of the Royal Society of London Series A, volume 275, page 33, 1963). The work of Frank J Rizzo (in FJ Rizzo, "An integral equation approach to boundary value problems of classical elastostatics" Quarterly of Applied Mathematics, volume 25, page 83, 1967) (which earned him a PhD and then much later on won him an ASME Warner Medal in 1993) was regarded by many researchers as the beginning of a novel direct boundary integral method for the numerical solution of elasticity problems. After completing my PhD in mid 1987, I continued to keep myself informed on the development of the boundary integral method and related mathematical works, pick up some new ideas now and then, attend conferences, give talks and seminars, and contribute to boundary element research with applications to problems in engineering and physical sciences. Some specific research areas I had worked on using the boundary integral method include linear fracture mechanics (accurate computation of stress intensity factors using special Green's functions), analyses of nonhomogeneous media (such as functionally graded materials), diffusion with specification of mass, modeling of photonic crystal fibers, integral formulation of imperfect interfaces, and bioheat transfer. Occasionally, I undertook the task of introducing the method to beginners, mainly advanced undergraduate and research students who were working on projects under my supervision. To do this, I had produced various notes over a period of time. The chapters posted below were written based on these notes. In writing the chapters, I assume that the readers have some prior basic knowledge of vector calculus (covering topics such as line, surface and volume integrals and the various integral theorems), ordinary and partial differential equations, complex variables, and computer programming. FORTRAN 77 codes for the numerical procedures discussed are listed in the chapters. Some justifications (if any is needed at all) for using good old FORTRAN 77 would be as follows. Firstly, in spite its seniority, it still remains a powerful "number crunching tool". Secondly, its codes are relatively easy to decipher and would be of some use even to readers who are attempting to implement the numerical procedures using newer software tools (such as C++ and MATLAB). Thirdly, free FORTRAN 77 compilers (e.g. FTN77 from Salford Software and GNU Fortran) may be downloaded from the internet. The constant encouragement and support of my dear wife, Young-Soon, had greatly motivated me to start and finish the writing of the chapters. Ean-Hin Ooi, Lukito Jayaputra and Bao-Ing Yun, students at Nanyang Technological University, had read several chapters (to learn the boundary element method). I would like to thank them and Alessandro Vaccari (Italy), Joris Vankerschaver (USA) and Jackson R. Jones for pointing out some typographical errors. Comments, suggestions and queries are always welcome. You may write me at the following e-mail address: wtang at pmail.ntu.edu.sg. Whye Teong Ang 13 August 2007 31 March 2008 4 July 2008 17 July 2010 CHAPTERS AND FORTRAN SOURCE CODES A description of each of the chapters in the course is provided below. Only Chapters 1 and 5 are available in PDF for reading and low resolution printing. Fortran source codes listed in all the chapters may be downloaded below (in WinZip files). As far as I could ascertain, the codes appeared to work well for the numerical examples I had tested. Nevertheless, some of these codes may have to be refined for use under certain extreme situations. If you choose to use them, please bear in mind that the risk involved is yours to bear alone!
Download Chapters 1 and 5 (corrections listed in the errata are incorporated into this document) The complete course Chapter 1: Two-dimensional Laplace's Equation Chapter 2: Discontinuous Linear Elements Chapter 3: Two-dimensional Helmholtz Type Equation Chapter 4: Two-dimensional Diffusion Equation Chapter 5: Green's Functions for Potential Problems Chapter 6: Three-dimensional Problems Errata Back to Top Chapter 1: Two-dimensional Laplace's Equation Essential for a better of understanding of all other chapters. This chapter introduces the boundary element method through solving a relatively simple boundary value problem governed by the two-dimensional Laplace's equation. A derivation of the boundary integral equation needed for solving the boundary value problem is given. The boundary integral equation is discretized using constant elements. That is, the boundary of the solution domain is approximated using straight line segments and the required solution and its normal derivative on the boundary are assumed to be constants over each line segment. Download Chapters 1 and 5 (corrections listed in the errata are incorporated into this document) FORTRAN source codes in Chapter 1 WinZip file (2D Laplace's equation, constant elements) Chapter 2: Discontinuous Linear Elements Best read after going through Chapter 1. In Chapter 1, the
boundary integral equation for the two-dimensional Laplace's equation
is discretized using constant
elements. This chapter shows how
the approximations over the line
segments can be improved through the
use of discontinuous linear elements.
It may be necessary to use higher
order elements when there is a need to
obtain more accurate results using
fewer elements.
Chapter 6:
Three-dimensional ProblemsFORTRAN source codes in Chapter 2 WinZip file (2D Laplace's equation, discontinuous linear elements) Back to List of Chapters Chapter 3: Two-dimensional Helmholtz Type Equation Best read after going through Chapter 1. Sections 3.2 and 3.3 may be read independently of each other. In the first part of this chapter, the analysis in Chapter 1 is extended to the two-dimensional homogeneous Helmholtz equation. The fundamental solution of the Helmholtz equation is given by a special function in the form of a zeroth order Bessel function of the second kind. With this fundamental solution, a boundary integral solution is obtained and discretized in order to solve boundary value problems governed by the Helmholtz equation. A generalized version of the Helmholtz equation with variable coefficients is considered in the second part of this chapter. For the generalized equation, the fundamental solution may be difficult (if not impossible) to obtain in analytical form. Hence, it may not be possible to derive a boundary integral solution for developing a boundary element procedure. An approach that may be used to overcome this difficulty is to "borrow" the fundamental solution of the Laplace's equation (in Chapter 1) to obtain an integral solution for the generalized Helmholtz equation. The integral solution contains not only the usual boundary integral but also a double integral over the entire solution domain, however. The so called dual-reciprocity method may be applied to convert the double integral approximately into a line integral. This gives rise to the dual-reciprocity boundary element method which allows the boundary element approach to be used for solving a wider range of engineering problems. FORTRAN source codes in Chapter 3 WinZip file (2D Helmholtz equation, 2D generalized Helmholtz equation, constant elements, dual-reciprocity method) Back to List of Chapters Chapter 4: Two-dimensional Diffusion Equation Best read after going through Chapters 1, 2 and Section 3.3 of Chapter 3. This chapter shows how the dual-reciprocity boundary element approach in Chapter 3 can be extended to solve numerically initial-boundary value problems governed by the two-dimensional diffusion equation. The fundamental solution for the Laplace's equation is applied to obtain an integro-differential formulation for the diffusion equation. For a more accurate spatial approximation, discontinuous linear elements are used to discretize the boundary integral in the integro-differential formulation. This gives rise to a system of linear algebraic-differential equations containing unknown functions of time. The time derivatives of the unknown functions are approximated using a finite-difference formula. A time-stepping dual-reciprocity boundary element method is thus derived for the numerical solution of the diffusion equation. FORTRAN source codes in Chapter 4 WinZip file (2D diffusion equation, discontinuous linear elements, time-stepping scheme, dual-reciprocity method) Back to List of Chapters Chapter 5: Green's Functions for Potential Problems Best read after going through Chapter 1. In this chapter, the basic idea of using Green's functions in boundary element formulations is explained in the context of two-dimensional potential problems. Special Green's functions for a half plane, an infinitely long strip and the region exterior to a circle, which satisfy certain boundary conditions, are given with examples of applications. Download Chapters 1 and 5 (corrections listed in the errata are incorporated into this document) FORTRAN source codes in Chapter 5 WinZip file (2D Laplace equation, constant elements, Green's functions) Read about the mathematician originally behind the idea. Back to List of Chapters Best read after going through Chapters 1 and 3. This chapter shows how the analyses and boundary element procedures in Chapters 1 and 3 for Laplace's and Helmholtz type equations can be extended to include three-dimensional problems. FORTRAN source codes
in Chapter 6 WinZip file (3D
Laplace's equation, 3D Helmholtz equation, 3D
generalized Helmholtz equation, constant
elements, triangular elements, dual-reciprocity
method)
Back to List of Chapters ERRATA I am posting here an errata (last updated on 2 May 2012) for the book "A Beginner's Course in Boundary Element Methods" published by Universal Publishers. (I hope that this list of corrections would not turn out to be a long one.) If you discover any mistake (even a very small one) in the book or if you wish to be informed of any update in the errata (assuming that you possess a copy of the book), please let me know. You may write me at the following e-mail address: wtang at pmail.ntu.edu.sg.
Back to List of Chapters Back to Top ADDITIONAL ARTICLES From time to time, additional articles covering more topics on boundary element methods will be posted here. **A potential problem with a non-classical boundary condition**. Recently, when he was a visiting researcher at the National University of Singapore, Wu Hao approached me for help on how to develop a BEM procedure for a potential problem with a non-classical boundary condition. The problem is as described in this**document**. The subroutines DLELAP1 and DLELAP2 in Chapter 2 can be used (with some post-processing needed after calling DLELAP1) to solve the problem. The complete programme for solving the particular test problem in the above mentioned document is given**here**. 22 July 2010
- A boundary integral method for the anti-plane harmonic vibration of an electrically loaded ceramic slab: application to piezoelectric transformers. This short article gives an example of how the boundary integral method in Section 3.2 of Chapter 3 can be applied to solve a vibration problem involving an electrically loaded ceramic slab. 14 July 2010
**The Complex Variable Boundary Element Method**. This document explains how a boundary element procedure for solving boundary value problems governed by the two-dimensional Laplace's equation can be derived from Cauchy integral formulae in the theory of complex functions. Fortran codes for the boundary element procedure may be downloaded**here**. 5 July 2010**(revised on 14 September 2010)**
- Cauchy Principal and Hadamard
Finite-part Integrals. This document looks at
specific forms of Cauchy principal and Hadamard
finite-part integrals. Starting from the boundary
integral equation for the two-dimensional Laplace's
equation, it also shows how a simple crack problem
can be formulated in terms of an integral equation
containing a Hadamard finite-part integral. 8 May 2009
- Boundary
Integral Equation for Axisymmetric
Potential Problem. This note shows how
the boundary integral equation for the
three-dimensional Laplace's equation can be used
to derive a boundary integral equation for an
axisymmetric potential problem. The axisymmetric
boundary integral equation involves integration
over a curve instead of a three-dimensional
surface. 26
February 2009
**(revised on 3 November 2010-added text and corrections highlighted in yellow)**[**Dr Pablo Suarez**, currently at Delaware State University, wrote to inform me that he had adapted the subroutines in CPF, CELAP1 and CELAP2 given in Chapter 1 for the 2D Laplace's equation to discretize and solve the boundary integral equation for the axisymmetric potential problem. The adapted subroutines (called AXICPF, AXICELAP1 and AXICELAP2) together with the supporting subroutine SOLVER and an example program for a particular test problem are in this**WinZip file**. The adapted subroutines are made available here with the consent of Dr Suarez. 23 May 2011]
- 2D Potential Problems
with Periodic Boundary Conditions. This note explains
how the codes in Chapter 1 may be modified to
solve a particular 2D potential problem with
periodic boundary conditions. The subroutine
CELAP1 for setting up the system of linear
algebraic equations for the problem is modified.
The modified subroutine is renamed CELAPPER and
listed in the file CELAPPER.txt. The main program
for the problem is in EXPRD.txt. Other subprograms
needed to run the program are in Chapter 1.
9 January 2009 [ Mr
Kushlesh Kumar, an undergraduate student
in the Department of Chemical Engineering at
Indian Institute of Technology Bombay, wrote
to inform me that he had modified in a
similar manner the subroutine DLELAP1 in Chapter
2 (on discontinuous linear elements) to solve
the same problem involving periodic boundary
conditions. The modified subroutine
renamed as DLELAPER is listed in the file DLELAPER.txt.
The main program for the problem is listed in EXPRD2.txt. Other
subprogams needed to run the main program can be
found in Chapters 1 and 2. 4 March 2010]
Back to Top LIBRARIES As far as I know, a hardcopy of this course may be borrowed from libraries of the institutes listed below. If you know of any other library that keeps a hardcopy of this course, please kindly let me know. You may write me at the following e-mail address: wtang at pmail.ntu.edu.sg.
- Nanyang Technological University (Singapore) (Library
2, Call no. TA347.B69A581)
- National Taiwan Ocean University (Taiwan) (more details here)
- Memorial University of Newfoundland (Canada) (more details here)
- Texas A&M University (USA) (more details here) copy lost
- Oregon State University (USA) (more details here)
- Technische Universitat Chemnitz (Germany) (more details here)
- Tamkang University (Taiwan) (more details here)
- Universiti
Tunku Abdul Rahman (Malaysia) (more
details here)
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*Numerical modelling of processes governing selective laser sintering*, PhD Thesis, Technische Universitat Darmstat, Germany.
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*Introduction to Numerical Electrostatics Using MATLAB*, Wiley & Sons. - A. Dikaiou (2013),
*Biomedical applications of a hybrid fluorescence molecular tomography/magnetic resonance imaging (FMT/MRI) system for small animal imaging*, DSc Thesis, ETH ZURICH, Switzerland.
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approaches for ecohydraulic studies: 3D, 2D,1D and
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*Ecohydraulic: An Integrated Approach*edited by I. Maddock, A. Harby, P. Kemp and P. Wood, Wiley and Sons.
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- M. C. Tenwick (2012), Error Estimates for Numerical Solutions of One- and Two-dimensional Integral Equations, PhD thesis, University of Leeds.
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*Speciális félvezető eszközök szimulációja szukcesszív hálózatredukciós módszerrel*, PhD Thesis, Budapest University of Technology and Economics, Hungary.
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*Numerical Modeling Of Red Blood Cells Dynamics Using The Level Set Method**,*PhD Thesis, Grenoble University, France.
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*A Study on Scattering of Acoustic Sources by a Circular Cylinder using the Fast Multipole Boundary Element Method*, MSc Thesis, KAIST, Korea.
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from a Partially Submerged Vertical Plate,
PhD Thesis, Universidad Carlos III de Madrid (Spain).
**Link to thesis**.
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Non-lineaires Entre Vagues et Structures Immergees,
Appliquee a la Simulation de Systemes Houlomoteurs,
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Paris-Est, France.
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interaction with a submerged slightly inclined porous
plate with a partially reflecting sidewall, Proceedings of the
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**Link to paper**.
- K. V. Malyshev (2011), Polarization of nanoparticles
in tunelling microscope (Поляризация
наночастиц в туннельном микроскопe),
*Science and Education*, Issue No. 10, October 2011.**Link to article**(in Russian).
- B. J. Galluzzo (2011), A Finite-difference based Approach to
Solving Subsurface Fluid Flow Equation in
Heterogeneous Media, Doctoral
dissertation, University of Iowa, USA.
**Link to thesis**.
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(2011), Quadrant searching: A new
technique to reduce the search space of the
inverse problem of EIT using BEM to the direct
problem,
*International Journal of Modern Physics C***22**, 825-839.
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solution of two-dimensional mixed problems with
variable coefficients by the boundary domain integral
and integro-differential equation methods,
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F.-W. Sun (2011), Negative Goos–Hänchen shift on
a concave dielectric interface, Optics Letters 36, 624-626.
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- M. Dehghan and A. Ghesmati (2010), Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method, Engineering Analysis with Boundary Elements 34, 51-59.
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Governed by a Brutal Damage Law using a Coupled
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- K. M. O'Donnell (2009), Field Ionization Detection for Atomic
Microscopy, PhD Thesis, University of
Newcastle, Australia.
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MEMs-based Electron Optical Systems for Electron
Beam Direct-write Lithography, Master Thesis
(in Chinese), National Taiwan University.
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- O. H. Menin (2009), Método dos Elementos de Contorno para Tomografia de Impedância Elétrica, Master Thesis (in Portuguese), University of Sao Paulo, Brazil. Link to thesis.
- E. H. Ooi (2009), Studies
of Ocular Heat Transfer using the Boundary Element
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- D. Mirzaei and M, Dehghan (2009), Boundary element solution of the two-dimensional sine-Gordon equation using continuous linear elements, Engineering Analysis with Boundary Elements 33, 12-24.
- E. H. Ooi, W. T. Ang and E. Y. K. Ng (2009), A boundary element model for investigating the effects of eye tumor on the temperature distribution inside the human eye, Computers in Biology and Medicine 39, 667-677.
Back to Top COMMENTS AND SUGGESTIONS All comments and suggestions are welcome. You may write me at the following e-mail address: wtang at pmail.ntu.edu.sg.-------------------------------------------------------------------------------------------------------------------------- Kirana Kumara P, India (2 May 2012) wrote: ... ... I would like to add that you have done an excellent work by writing the book on BEM. The book explains the salient features (both theory and implementation details) of BEM in a very straight forward way, which helps a reader to quickly grasp the fundamentals and to start coding very soon. Your book has helped me a lot to understand the basics of BEM quickly and easily. -------------------------------------------------------------------------------------------------------------------------- Stephan Helma (17 November 2010) wrote: Thanks for your wonderful book! It
helps to master the steep learning curve which
is associated with the BEM...-------------------------------------------------------------------------------------------------------------------------- Jerome
Huck, France
(6
August 2010) wrote: I ordered your book on BEM
a few days ago. I just received it this morning.
Before (that), I downloaded your sample (chapters)
(chapters 1 and 5). It is a brilliant and clear-cut
book for an introduction on BEM, an excellent balance
between theory and numerical experiments ...
--------------------------------------------------------------------------------------------------------------------------
Kushlesh
Kumar, India
(4
March 2010) wrote:
I have successfully
converted the code for constant elements into
discontinuous linear elements (for the problem with
periodic boundary conditions). I am attaching the codes
and also the validations for the code using the example
solved by you, so that you can share it with others. I
have modified my code so that it matches with the
conventions used by you as much as possible, for ease of
readability and uniformity. Feel free to modify them, if
you feel necessary. Again, thank you very much for your
wonderful material on BEM.Reply: Thanks for sending the codes. I have posted them here. -------------------------------------------------------------------------------------------------------------------------- Dr Joseph Ladish, USA (9 December 2009) wrote: I would greatly appreciate it if you would send me the Fortran 77 codes referred to in your book, "A Beginner's Course in Boundary Element Methods." Thank you for writing such an excellent book, and thank you for making the codes available in electronic format. Reply: Yes, right here in Winzip files (chapter by chapter). -------------------------------------------------------------------------------------------------------------------------- Laurent Masse-Navette, EU (22 July 2009) wrote: I have the pleasure to have bought your book. I must say this is an excellent book and I have been able to get into an implementation of the problem I am trying to solve with BEM in a matter of hours. I have implemented the algorithms you describe in chapter II in C++ in order to implement a fast resistor extraction tool for arbitrary metal shapes for a VLSI CAD application .... -------------------------------------------------------------------------------------------------------------------------- Eduardo V. S. Pouzada, Brazil (15 April 2009) wrote: I bought your book "A Beginner's Course in BEM" a few months ago. In the preface you mention the possibility of sending the codes by electronics means. Could you please send them to my e-mail or perhaps indicate some URL from where it could be downloaded? Reply: Yes, right here in Winzip files (chapter by chapter). ---------------------------------------------------------------------------------------------------------------------------- Joris Vankerschaver, USA (11 March 2009) wrote: I am a postdoc in dynamical systems and I found your book extremely useful in implementing the BEM method. I appreciated the interplay between theory and numerical experiments, which helped me in getting rid of the bugs in my own code ... ------------------------------------------------------------------------------------------------------- Junhan Cho, South Korea (9 January 2009) wrote: Today, I purchased your book entitled "A beginner's course in boundary element methods", since I found it very useful for my research (polymer science). However, I have a question regarding using BEM with periodic boundary conditions. Could you possibly give me a sample program (in fortran 77) dealing with Laplace equation with a periodic boundary condition? ---------- Note: A note explaining how the codes in Chapter 1 can be modified to solve a particular 2D potential problem with periodic conditions and the modified codes are posted here. ---------------------------------------------------------------------------------------------------------------------------- Firdaus Prabowo, Singapore (28 August 2008) wrote: I'm currently studying your book (I download the first chapter from your website) as a basis for fluid dynamics simulation in my PhD research... Thank you very much. I think your book is good, since it has a straightforward way to attack problems. ------------------------------------------------------------------------------------------------------- Alessandro Vaccari, Italy (4 July 2008) wrote: Very nice, clearly written, well exposed book. ---------- Note: I would like to thank Alessandro Vaccari for e-mailing me to report a minor typographical error on page 84 of the book. ---------------------------------------------------------------------------------------------------------------------------- Li Yajun, China (13 May 2008) wrote: I am a graduate student of China Univ of Petroleum and I major in Petroleum Engineering. It's a great honor for me to read your new book "A Beginner's Course in Boundary Element Methods". After a period of study, I gain a lot of knowledge about the basic principle and theory of BEM. To me, it is very useful. ---------------------------------------------------------------------------------------------------------------------------------------------------------- Naveen Gopinathrao, United Kingdom (1 April 2008) wrote: First impression of the book was a bit scary, I thought it will be very mathematical and difficult to understand. But it's a very carefully designed book and very easy to pick up the subject. Numerical implementation with an example at the end of each chapter makes it easy to get the methodology (steps). Thank you very much for the book. ---------------------------------------------------------------------------------------------------------------------------------------------------------- Gerd Baumann, Germany (31 March 2008) wrote: Thank you very much for this nice and down to the point written book. I got it today and was fascinated about the approach right from the beginning. I wrote a small program in Mathematica 6.0 to carry out some calculations discussed in your book ... ---------- Note: After a bit of Google search, I found that Gerd Baumann is the author of several books on the use of the software Mathematica for studying various problems in physics (details at Amazon USA). W.T . Ang (31 March 2008) ---------------------------------------------------------------------------------------------------------------------------------------------------------- Godwin Kakuba, The Netherlands (12 February 2008) wrote: I am .. a PhD student at Eindhoven University of Technology .. I have read the sections of your book .. from your website and it is great. .... ---------------------------------------------------------------------------------------------------------------------------------------------------------- Pablo Suarez, USA (15 October 2007) wrote: I hope you are doing well... A few months back I e-mailed you concerning your now published book. I would like to thank you for writing such a wonderful text that truly aids me in trying to understand the Boundary Element Method. I now own a copy of the book and I always use it as a standard reference for my numerical work. I do thank you for answering my questions in the past and for providing code when needed. Overall I am quite pleased with your Book. (Earlier posting) ---------- Note: Dr Suarez has recently published a book on BEM. W. T. Ang (30 January 2011).---------------------------------------------------------------------------------------------------------------------------------------------------------- Georg Wenig, Germany (4 September 2007) wrote: Just recently I started learning about boundary elements from your great beginner's course. This course represents an ideal smooth introduction into the topic. It turns the complete beginner into a working practitioner within (a) very short time, especially with the help of the source code you provide ... ---------- Further note: Georg Wenig went on to point out that the formulae in Section 1.6 may need some modifications if the point coincides with one of the endpoints ("border points") of the element over which the integration is carried out. I had overlooked the need for the modifications mainly because was not required to be one of those endpoints in the boundary element procedures described in the course. Nevertheless, if I revise the course in the near future, I will add a brief note on this point ! W.T . Ang (5 September 2007) ---------------------------------------------------------------------------------------------------------------------------------------------------------- Yoav Levy, Israel (7 August 2007) wrote: I started to read your book on the net ... However, it's too complicated to be read online. Is there any way to buy this book in order to read it from hardcopy? ---------- Reply: Yes! You may now purchase a hardcopy of the book from Universal Publishers or Amazon (at USA, UK, Canada, France, Germany or Japan) or Barnes and Noble. Or do a Google book search to find a suitable bookstore or bookseller. W.T . Ang (13 August 2007) Note: The query above was received just at about the time the course was published by Universal Publishers. Prior to that, the chapters were posted here in non-printable PDF documents. ---------------------------------------------------------------------------------------------------------------------------- Ray Hari Manalan, Florida, USA (5 August 2007) wrote: I am refering to your chapters to get some basic insight to BEM. Thanks. --------------------------------------------------------------------------------------------------------------------------- Pablo Suarez, USA (25 June 2007) wrote: I am a graduate student in RPI (Renssalaer Polytechnic Institute) in Troy, New York. I read your chapter 1 of the book you are writing and (it) is terrific (that) it is exactly what I needed ... (More) --------------------------------------------------------------------------------------------------------------------------- Ronald W. J. Koers, The Netherlands (6 June 2007) wrote: I have read your introduction to the boundary element with great enthusiasm and would like to do simulation with your code. It is the best introduction I have found. Would it be possible to get access to an electronic version of the fortran source code listed in your chapters? ---------- Reply: Yes! Starting around mid August 2007, I have made the codes available here for downloading. But if you do use them, please bear in mind the risk is yours alone to bear! W.T . Ang (13 August 2007) -------------------------------------------------------------------------------------------------------------------------- Phang Piau, Malaysia (6 June 2007) wrote: I am interested in numerical methods and would like to learn the BEM. I browsed through a few books but found that it was quite hard for me to understand them. When I started reading your chapters, it brought about a strong determination in me for learning that method ... The mathematical notation is very clear ... -------------------------------------------------------------------------------------------------------------------------- James Hilton, Ireland (1 June 2007) wrote: Just a quick note to say your notes on the BEM are excellent. I'm trying to couple BEM to the level set method for charged fluids and after reading a couple of books and getting thoroughly confused I found and followed your notes. After reading them I understood the principles and the system I needed to solve straight away. Thank you and please keep up the good work! ---------- Note: Here's an acknowledgment from James Hilton and his co-author A. van der Net in their research paper entitled "Dynamics of charged hemispherical soap bubbles". --------------------------------------------------------------------------------------------------------------------------- Back to List of Chapters Back to Top |