An error estimation measure in the direct BEM formulation by the external Somigliana's identity

In this paper we consider a physical interpretation of Somigliana's identities as the basis of the formulation of the direct Boundary Element Method as well as its error estimation measure. While Somigliana's internal identity gives the solution to a given boundary value problem for a domain D⁽⁺⁾, the external identity of Somigliana gives zero stress and displacement for the domain D^(-) complementary to D⁽⁺⁾ in the infinite region. The BEM error equally affects the accuracy of both identities and error estimation for the complementary domain D^(-) is adopted for that for the original domain D⁽⁺⁾. We propose the domain integral for the strain energy in the complementary domain D^(-) as an error measure which, in turn, is converted to a boundary integral over the boundary of D^(-). The individual contribution to this boundary error integral comes from each boundary element and it serves as an error measure for each boundary element, which can be used in an adaptive re-meshing strategy. Numerical examples strongly support our error measure and adaptive re-meshing strategy.