Numerical Stress Computation Around Cavity Using Boundary Element Method for Elastic Material

ABSTRACT: Precise estimation of field variable values around a cavity is vital for excavation operation safety. The singular nature of stress around the cavity makes the computations challenging for Finite Element Method (FEM) software. Although the Boundary Element Method (BEM) offers a semi-analytical formulation, it gives rise to a set of new challenges in boundary condition enforcement, allocating the traction variables at corners, integration, solving, and post-processing. In this study, the problem of a small cavity inside a three-dimensional elastic domain has been explored using continuous and semi-discontinuous elements. The results of the BEM code were compared with an analytical solution and industrial software (Abaqus). The comparison of stress values indicates that with much fewer elements, BEM is still capable of providing proper results with a few percent of errors. For remote coordinates, error values are less than one percent; for locations too close to the cavity, with a finer mesh, error values are reduced to one percent with the same simple integration scheme. Additionally, for such simulation, semi-discontinuous elements result in higher error values even at remote locations and low deviation ratios. 1. INTRODUCTION Stress concentration near a cavity and its higher values compared to the remote stress values at the boundary is a well-established subject. This stress intensity could impact the stability of rock during an excavation operation. A precise estimation of stress values using analytical solutions or numerical software can reduce the chance of rock failure. Yet analytical solutions are a proper basis for code and software error evaluation, they just address a set of specific problems and geometries. Boundary Element Method (BEM) is a numerical tool for engineers to approximate stress and displacement values by scattering elements over boundary surfaces. In this study, direct BEM formulation has been used which includes singular integrands and discontinuous boundary-imposing challenges (Aliabadi, 2002; Crouch et al., 1983; Gao Hall, 1994; Leng, 2007). Among many methods for dealing with discontinuous Boundary Conditions (BC), non-conforming elements have been extensively utilized. Semi-Discontinuous Element (SDE) is one of the methods that simplify the BC imposing by introducing new nodes to differentiate degrees of freedom (Aliabadi, 2002; Dyka Mi Mitra Parreira, 1988; Yan & Lin, 1994). A simpler method, referred to as Unique Traction (UT) in this work, neglects all challenges. Both UT and SDE have been utilized in this study.