Engineering Analysis with Boundary Elements 29 (2005) 268–302 www.elsevier.com/locate/enganabound Heritage and early history of the boundary element method Alexander H.-D. Chenga,*, Daisy T. Chengb aDepartment of Civil Engineering University of Mississippi, University, MS, 38677, USA bJohn D. Williams Library, University of Mississippi, University, MS 38677, USA Received 10 December 2003; revised 7 December 2004; accepted 8 December 2004 Available online 12 February 2005 Abstract This article explores the rich heritage of the boundary element method (BEM) by examining its mathematical foundation from the potential theory, boundary value problems, Green’s functions, Green’s identities, to Fredholm integral equations. The 18th to 20th century mathematicians, whose contributions were key to the theoretical development, are honored with short biographies. The origin of the numerical implementation of boundary integral equations can be traced to the 1960s, when the electronic computers had become available. The full emergence of the numerical technique known as the boundary element method occurred in the late 1970s. This article reviews the early history of the boundary element method up to the late 1970s. q 2005 Elsevier Ltd. All rights reserved. Keywords: Boundary element method; Green’s functions; Green’s identities; Boundary integral equation method; Integral equation; History 1. Introduction industrial settings, mesh preparation is the most labor intensive and the most costly portion in numerical After three decades of development, the boundary modeling, particularly for the FEM [9] Without the need element method (BEM) has found a firm footing in the of dealing with the interior mesh, the BEM is more cost arena of numerical methods for partial differential effective in mesh preparation. For problems involving equations. Comparing to the more popular numerical moving boundaries, the adjustment of the mesh is much methods, such as the Finite Element Method (FEM) and easier with the BEM; hence it is again the preferred tool. the Finite Difference Method (FDM), which can be With these advantages, the BEM is indeed an essential part classified as the domain methods, the BEM distinguish in the repertoire of the modern day computational tools. itself as a boundary method, meaning that the numerical In order to gain an objective assessment of the success of discretization is conducted at reduced spatial dimension. For the BEM, as compared to other numerical methods, a search example, for problems in three spatial dimensions, the is conducted using the Web of ScienceSM, an online discretization is performed on the bounding surface only; bibliographic database. Based on the keyword search, the and in two spatial dimensions, the discretization is on the total number of journal publications found in the Science boundary contour only. This reduced dimension leads to Citation Index Expanded 195 was compiled for several smaller linear systems, less computer memory require- numerical methods. The detail of the search technique is ments, and more efficient computation. This effect is most described in Appendix. The result, as summarized in pronounced when the domain is unbounded. Unbounded Table 1, clearly indicates that the finite element method domain needs to be truncated and approximated in domain (FEM) is the most popular with more than 66,000 entries. methods. The BEM, on the other hand, automatically The finite difference method (FDM) is a distant second with models the behavior at infinity without the need of more than 19,000 entries, less than one third of the FEM. deploying a mesh to approximate it. In the modern day The BEM ranks third with more than 10,000 entries, more than one half of the FDM. All other methods, such as the * Corresponding author. Tel.: C1 662 915 5362; fax: C1 662 915 5523. finite volume method (FVM) and the collocation method E-mail address: [email protected] (A.H.-D. Cheng). (CM), trail far behind. Based on this bibliographic search, 0955-7997/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2004.12.001 A.H.-D. Cheng, D.T. Cheng / Engineering Analysis with Boundary Elements 29 (2005) 268–302 269 Table 1 to tell, there is an indication that the number of annual BEM Bibliographic database search based on the Web of Science publications is reaching a steady state at about 700–800 Numerical Search phrase in topic field No. of entries papers per year. For comparison, this number for the FEM is method about 5000 articles per year, and for the FDM, it is about FEM ‘Finite element’ or ‘finite elements’ 66,237 1400. FDM ‘Finite difference’ or ‘finite differences’ 19,531 As the BEM is on its way to maturity, it is of interest to BEM ‘Boundary element’ or ‘boundary 10,126 visit its history. Although there exist certain efforts toward elements’ or ‘boundary integral’ the writing of the history of the FEM [84,127] and the FDM FVM ‘Finite volume method’ or ‘finite volume 1695 methods’ [131,193], relatively little has been done for the BEM. The CM ‘Collocation method’ or ‘collocation 1615 present article is aimed at taking a first step toward the methods’ construction of a history for the BEM. Refer to Appendix A for search criteria. (Search date: May 3, 2004). Before reviewing its modern development, we shall first explore the rich heritage of the BEM, particularly its we can conclude that the popularity and versatility of BEM mathematical foundation from the 18th century to the early falls behind the two major methods, FEM and FDM. 20th. The historical development of the potential theory, However, BEM’s leading role as a specialized and Green’s function, and integral equations are reviewed. To alternative method to these two, as compared to all other interest the beginners of the field, biographical sketches numerical methods for partial differential equations, is celebrating the pioneers, whose contributions were key to unchallenged. the mathematical foundation of the BEM, are provided. The Fig. 1 presents the histogram of the number of journal coverage continues into the first half of the 20th century, papers published annually, containing BEM as a keyword. It when early numerical efforts were attempted even before shows that the growth of BEM literature roughly follows the the electronic computers were invented. S-curve pattern predicted by the theory of technology Numerical methods cannot truly prosper until the diffusion [75]. Based on the data, we observe that after the invention and then the wide availability of the electronic ‘invention of the technology’ in the late 1960s and early computers in the early 1960s. It is of little surprise that both 1970s, the number of published literature was very small; the FEM and the BEM started around that time. For the but it was on an exponential growth rate, until it reached an BEM, multiple efforts started around 1962. A turning point inflection point around 1991. After that time, the annual that launched a series of connected efforts, which soon publication continued to grow, but at a decreasing rate. A developed into a movement, can be traced to 1967. In the sign of a technology reaching its maturity is marked by the 1970s, the BEM was still a novice numerical technique, but leveling off of its production. Although it might be too early saw an exponential growth. By the end of it, textbooks were Fig. 1. Number of journal articles published by the year on the subject of BEM, based on the Web of Science search. Refer to Appendix for the search criteria. (Search date: May 3, 2004). 270 A.H.-D. Cheng, D.T. Cheng / Engineering Analysis with Boundary Elements 29 (2005) 268–302 written and conferences were organized on BEM. This gradient of temperature distribution article reviews the early development up to the late 1970s, q ZKkVT leaving the latter development to future writers. (1) Before starting, we should clarify the use of the term where q is the heat flux vector, k is the thermal conductivity, ‘boundary element method’ in this article. In the narrowest and T is the temperature. The steady state heat energy view, one can argue that BEM refers to the numerical conservation requires that at any point in space the technique based on the method of weighted residuals, divergence of the flux equals to zero: mirroring the finite element formulation, except that the $ Z weighing function used is the fundamental solution of V q 0 (2) governing equation in order to eliminate the need of domain Combining (1) and (2) and assuming that k is a constant, we discretization [19,21]. Or, one can view BEM as the obtain the Laplace equation numerical implementation of boundary integral equations 2 based on Green’s formula, in which the piecewise element V T Z 0 (3) concept of the FEM is utilized for the discretization [108]. For groundwater flow, similar procedure produces Even more broadly, BEM has been used as a generic term for a variety of numerical methods that use a boundary or V2h Z 0 (4) boundary-like discretization. These can include the general where h is the piezometric head. It is of interest to mention numerical implementation of boundary integral equations, that the notation V used in the above came form William known as the boundary integral equation method (BIEM) Rowan Hamilton (1805–1865). The symbol V, known as [54], whether elements are used in the discretization or not; ‘nabla’, is a Hebrew stringed instrument that has a triangular or the method known as the indirect method that distributes shape [73]. singular solutions on the solution boundary; or the method The above theories are based on physical quantities. A of fundamental solutions in which the fundamental solutions second way that the Laplace equation arises is through the are distributed outside the domain in discrete or continuous mathematical concept of finding a ‘potential’ that has no fashion with or without integral equation formulation; or direct physical meaning.