In this work, a non-symmetric variational approach is derived to enforce C 1;a continuity at interelement nodes for the self-regular traction-BIE. This variational approach uses only Lagrangian C 0 elements. Two separate algorithms are derived. The first one enforces C 1;a continuity at smooth inter-element nodes, and the second enforces continuity of displacement derivatives in global coordinates at corner nodes, where C 1;a continuity cannot be enforced. The variational formulation for the traction-BIE is implemented in this work for two elastostatics problems with various discretizations and polynomial interpolants. Local and global measures of the discretization error are obtained by means of an error estimator recently derived by the authors. Comparisons are also made with the displacement-BIE, which does not require C 1;a continuity for the displacement. The lack of smoothness of the displacement derivatives at the inter-element nodes is shown to be an important source of both local and global error for the traction-BIE formulation, especially for quadratic elements. The accuracy of the boundary solution obtained from the traction-BIE improves significantly when C 1;a continuity is enforced where possible, i.e., at the smooth interelement nodes only.
IntroductionRegularization of the traction-BIE eliminates the need for special numerical integration procedures for singular and hyper-singular integrals, but stricter continuity requirements must be satisfied by the density for the boundary solution to exist. The self-regular formulation, developed by Cruse and co-workers [1, 2, 3], based on conforming C 0 elements, uses a relaxed continuity approach and does not satisfy these continuity requirements, even though reasonably accurate numerical results are obtained. In this work, a variational formulation is presented for the selfregular traction-BIE, enforcing the necessary inter-element continuity conditions as subsidiary constraining equations.Somigliana's Stress Identity can be regularized by adding and subtracting appropriate terms [1]. This process is equivalent to imposing on the problem a simple linear solution for the displacement, or, equivalently, imposing a constant stress state [4]. The regularized SSI can be written with regularization based on a collocation boundary point [5]. When the limit is taken as the collocation point tends to a smooth point of the boundary and the resulting stress-BIE is multiplied with the surface normal at P, a self-regular traction-BIE is obtained [2].For problems in which the stress tensor is unique, the traction BIE gives a unique value for tractions at all smooth boundary points. A condition for the existence of a limit to the boundary of the traction BIE requires that the primary density (the displacement) must be Hölder continuous of the type C 1;a at the collocation points [6,7].The discretization of the self-regular traction-BIE into C 0 elements violates the C 1;a continuity requirement for the displacement at the interface between elements. Three approaches can...