Preliminaries

• Akpofure E. Taigbenu
Chapter

Abstract

We may not be properly poised to talk about the Green element method (GEM) without making reference to the boundary element method (BEM) since both methods are founded on the same singular integral theory. The theoretical developments in boundary element (BE) circles can be traced to the eighteenth century when related theories of ideal fluid flow (potential flow) and of integral transforms were established, but it was not until the 1960’s that flurries of research activities on BEM intensified because of its acclaimed advantage of being a boundary-only method. Because the earlier boundary element applications were directed at elliptic boundary-value problems, the free-space Green’s function of the differential operator, which is Inr in 2-D. spatial dimensions and 1/r in 3-D. spatial dimensions, is amenable to Green’s identity which transforms the differential equation into an integral one that can essentially be implemented on the boundary (along a line for 2-D. and on a surface for 3-D.) of the computational domain. In the classical approach, a system of discrete equations is obtained from the integral equation by subdividing the boundary into segments over which distributions are prescribed for the primary variable and the flux (Jaswon [1], Jaswon and Ponter [2], Symm [3], Liggett [4], Liu and Liggett [5], Fairweather et al. [6], Rizzo [7], and others). These research thrusts were considered successful when viewed against the background that contemporary methods, like the finite element method (FEM) solved similar problems by discretizing the entire computational domain. This led to claims by most investigators that BEM was more superior than existing computational methods in terms of accuracy and computational efficiency. It is not in doubt that the integral replication of the differential equation provided by the boundary element theory evolves quite naturally, making use of the response function (free-space Green’s function) to a unit instantaneous input. It is only at the computational stage that the distribution of the dependent variable is approximated by some interpolating polynomial function. It is, thus, expected that for problems where the unit response function can be obtained, BEM achieves secondorder accuracy. The claim that BEM is superior to FEM in computational efficiency is not, in most cases, arrived at after actual CPU comparisons are carried, but it is alluded to on the basis of the boundary-only character of the method.

Keywords

Source Node Boundary Element Boundary Element Method Domain Integration Green Element
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