Self-regular BEM approaches and error estimation for linear elastic fracture mechanics in two dimensions

The purpose of the current work is to facilitate the use of self-regular boundary element method (BEM) formulations for fracture mechanics problems by resolving issues related to numerical accuracy. The self-regular traction-BEM formulation using the relaxed continuity approach, derived by Dr. Thomas A. Cruse and research associates, is employed. Dr. Cruse served initially as dissertation advisor, becoming co-advisor after his appointment as emeritus faculty.; As a preliminary step, self-regular formulations originally derived for elastostatics were extended to potential theory. The effects of tangential derivatives and mesh grading on the discretization error were highlighted.; The first issue studied for the self-regular traction-BIE involves collocation of this integral formulation at the near-crack-contour region. An analytical study demonstrates that this BIE reproduces correctly the singular behavior of near-crack-contour interior stresses. Also, physical and non-physical singularities are identified in a unique way, and a new near-crack-contour BIE formulation is developed using an extraction technique.; Another issue for the self-regular traction-BIE is the satisfaction of continuity requirements by variables in the discretized boundary. New discretized formulations using variational approaches are proposed and implemented in this work. The first approach enforces uniqueness of the displacement derivatives in global coordinates at corner nodes. The second and more effective approach enforces inter-element continuity of displacement derivatives at nodes in smooth boundary regions.; For the purpose of comparing formulations, two new error estimators were derived and implemented for collocation BEM formulations involving both potential and elastostatics problems. The first estimator is based on a gradient recovery procedure, applied for the first time to BEM. The second estimator is based on evaluation of the integral equation at external points of a closed domain. Also, an adaptive r-refinement procedure based on non-linear optimization procedures was derived and implemented as an application using error estimators.