The boundary element method revisited

Abstract

The collocation boundary element method is derived on the basis of the weighted-residuals statement. Only the static case is addressed, as it already involves all relevant conceptual issues. The present outline brings to discussion some relevant aspects and implementation issues of the method that should belong in any text book. It is shown that, if the boundary element method is consistently formulated, an inherent error term -related to arbitrary rigid-body displacements -is naturally taken into account and has no influence on the resultant matrix equation, with traction force parameters that are always in balance independently of mesh discretization. The constitutive matrices of the method -the single-layer and double-layer potential matrices G and H -present some spectral properties that are per se interesting but that also have applicability consequences. The matrix G is rectangular, if consistently obtained. For and adequately formulated problem, the solution of the resultant matrix equation is always possible (and unique) whether directly or approximated in terms of equivalent nodal forces. The effects of body forces, whenever transformable to boundary actions, may be expressed in terms of the boundary interpolation functions, which renders the final matrix equation more elegant and speeds up calculations in no detriment to accuracy. There is a novel proposition for the interpolation of traction forces along curved boundaries, with results that may be only slightly improved, as compared to the classical procedure, but that simplifies numerical computation and adds to the consistency of the method in terms of patch test assessments. The conceptual and numerical developments are illustrated by means of a few examples.

The expedite boundary element method

“…The singular integrals can be always evaluated mathematically in correspondence to simple mechanical meanings [1]. A conceptual assessment of eqn (10) is given in Reference [8].…”

“…They are actually theorems that must be proven from some mechanical axioms. Some of them have already been dealt with in References [6][7][8][9][10]. In this paper, one attempts to keep full consistency of the equations while being not too formal.…”

“…The numbers of degrees of freedom for traction forces n t and displacements n d are not necessarily the same, since one may need more than one traction-force parameter to represent tractions that are not single valued at the boundary surface, generally at nodes where adjacent boundary segments present different outward normals [8]. Then, it results that n t ≥ n d , as t in eqns (6) and (7) are tractionforce attributes on boundary segments, whereas u in in eqn (3) are displacement attributes at nodal points.…”

“…The numerical model is implemented in Fortran with double precision. The scheme shown in Figure 1 has a total of 124 nodes [8]. The problem is modeled using linear, quadratic and cubic elements with differently refined meshes, for the total numbers of nodes shown in Table 1.…”

“…which leads to the modified boundary element method (MBEM) [8]. In this equation, |J| (at ) is the value of the Jacobian of the global (x, y, z) to natural (ξ, η) coordinate transformation at the nodal point and the term |J| (at ) /|J| features a term in the denominator that cancels the Jacobian term of the infinitesimal boundary segment dΓ = |J|dξdη in the numerator of two integral expressions introduced in eqns (10) and (14).…”