Source: http://www.ntu.edu.sg/home/mwtang/bem2011.html
Timestamp: 2019-04-20 09:22:30+00:00

Document:
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.
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.
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.
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, Joris Vankerschaver 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.
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!
W. T. Ang, A Beginner's Course in Boundary Element Methods, Universal Publishers, Boca Raton, USA, 2007.
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.
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.
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.
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.
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.
Read about the mathematician originally behind the idea.
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.
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.
From time to time, additional articles covering more topics on boundary element methods will be posted here.
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.
For a list of citations as compiled by Google Scholar, please click here.
P. Saidi and J. J. Hoyt (2015), Diffusion-controlled growth rate of stepped interfaces, Physical Review E 92, 012404.
R. Dayal (2014), Numerical modelling of processes governing selective laser sintering, PhD Thesis, Technische Universitat Darmstat, Germany.
A. J. Nowak (2014), Boundary element method in heat conduction, Encyclopedia of Thermal Stresses, 415-425, Springer.
L. N. Dworsky (2014), 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.
D. Tonina and K. Jorde (2013), Hydraulic modelling approaches for ecohydraulic studies: 3D, 2D,1D and non-numeric models, In Chapter 3 of Ecohydraulic: An Integrated Approach edited by I. Maddock, A. Harby, P. Kemp and P. Wood, Wiley and Sons.
D. Haley, M. P. Moody and GDW Smith (2013), Level set methods for modelling field evaporation in atom probe, Microscopy and Microanalysis 19, 1709-1717.
S. Sirca and M. Horvat (2013), Computational Methods for Physicists: Compendium for Students, Springer.
Q. H. Qin and H. Wang (2013), Special circular hole elements for thermal analysis in cellular solids with multiple circular holes, International Journal of Computational Methods 10, 1350008 (24 pages).
P. Ghaderi Daneshmand and R. Jafari (2013), A 3D hybrid BE–FE solution to the forward problem of electrical impedance tomography, Engineering Analysis with Boundary Elements 37, 757-764.
M. C. Tenwick (2012), Error Estimates for Numerical Solutions of One- and Two-dimensional Integral Equations, PhD thesis, University of Leeds.
P. László (2012), Speciális félvezető eszközök szimulációja szukcesszív hálózatredukciós módszerrel, PhD Thesis, Budapest University of Technology and Economics, Hungary.
M.A. AL-Jawary, J. Ravnik, L.C. Wrobel and L. Škerget (2012), Boundary element formulations for the numerical solution of two-dimensional diffusion problems with variable coefficients, Computers & Mathematics with Applications 64, 2695–2711.
E. H. Ooi and V. Popov (2012), A simplified approach for imposing the boundary conditions in the local boundary integral equation method, Computational Mechanics. DOI: 10.1007/s0046 6-012-0747-1.
M. A. Al-Jawary and L. C. Wrobel (2012), Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods, International Journal of Computer Mathematics 89, 1488-1503.
Y. Liu, H. J. Li and Y. C. Li (2012), A new analytic solution for wave scattering by a submerged horizontal porous plate with finite thickness, Ocean Engineering 42, 83-92.
M. A. Al-Jawary and L. C. Wrobel (2012), Radial integration boundary integral and integro-differential equation methods for two-dimensional heat conduction problems with variable coefficients, Engineering Analysis with Boundary Elements 36, 685-695.
E. H. Ooi, V. Popov and H. Dogan (2012), Three-dimensional solution for acoustic and transport problems using the radial basis integral equation method, Applied Mathematics and Computation 218, 9470-9488.
A. Laadhari (2011), Numerical Modeling Of Red Blood Cells Dynamics Using The Level Set Method, PhD Thesis, Grenoble University, France.
J. A. Jeun (2011), A Study on Scattering of Acoustic Sources by a Circular Cylinder using the Fast Multipole Boundary Element Method, MSc Thesis, KAIST, Korea.
P. Martinez-Legazpi Aquilo (2011), Corner Waves Downstream from a Partially Submerged Vertical Plate, PhD Thesis, Universidad Carlos III de Madrid (Spain). Link to thesis.
E. Guerber (2011), Modelisation Numerique des Interactions Non-lineaires Entre Vagues et Structures Immergees, Appliquee a la Simulation de Systemes Houlomoteurs, Doctorat de Mecanique des Fluides del l'Universite Paris-Est, France. Link to thesis.
Y. Liu and Y. C. Li (2011), Analysis of wave interaction with a submerged slightly inclined porous plate with a partially reflecting sidewall, Proceedings of the 6th International Conference on Asia and Pacific Coasts, Hong Kong. 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.
O. H. Menin and V. Rolnik (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.
M. A. Al-Jawary and L. C. Wrobel (2011), Numerical solution of two-dimensional mixed problems with variable coefficients by the boundary domain integral and integro-differential equation methods, Engineering Analysis with Boundary Elements 35, 1279-1287.
L.-M. Zhou, C.-L. Zou, Z.-F. Han, G.-C. Guo, and F.-W. Sun (2011), Negative Goos–Hänchen shift on a concave dielectric interface, Optics Letters 36, 624-626.
Y. Liu, S. He and S. Han (2010), Effect of coastline reflection on the performance of a submerged horizontal plate breakwater, at http://www.paper.edu.cn (in Chinese).
V. Janecek and V. S. Nikolayev (2010), Influence of surface forces on the apparent contact angle at partial wetting and in the presence of heat and mass transfer, Proceedings of the 2nd European Conference om Microfluidics-Microfluidics 2010-Toulouse, France, December 8-10. Link to paper.
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.
M. Dehghan and A. Ghesmati (2010), Application of the dual reciprocity boundary integral equation technique to solve the nonlinear Klein-Gordon equation, Computer Physics Communications 181, 1410-1418.
A. L. Guilmin (2009), Study of the Propagation of Defects Governed by a Brutal Damage Law using a Coupled Boundary Element and Level Set Technique, Master Thesis, Ecole Centrale de Nantes, France.
K. M. O'Donnell (2009), Field Ionization Detection for Atomic Microscopy, PhD Thesis, University of Newcastle, Australia. Link to thesis.
H. H. Chen (2009), Analysis and Optimization Design of MEMs-based Electron Optical Systems for Electron Beam Direct-write Lithography, Master Thesis (in Chinese), National Taiwan University. Link to thesis.
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 Method, PhD Thesis, Nanyang Technological University, Singapore. Link to thesis.
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.
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.
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).
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?
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.
Note: Dr Suarez has recently published a book on BEM. W. T. Ang (30 January 2011).
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?
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.
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?
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".

References: V. 
 V. 
 V. 
 V. 

V. 
 V. 
 V.