A coupling of FEM and weakly singular SGBEM for analysis of a three-dimensional infinite medium

Abstract

To offer an efficient numerical technique for analysis of a three-dimensional infinite medium that contain both a line of singularity introduced by cracks and a localized nonlinear region introduced by high intensity loads. It is well-known that nonlinearities present within the domain render the modeling by methods of boundary integral equations computationally inefficient while treatment of a medium that is unbounded and/or contains the discontinuity surface requires special numerical treatments and can lead to a substantial computational cost. In this investigation, we establish a coupling procedure by exploiting advantageous features of both a standard finite element method (FEM) and a symmetric Galerkin boundary element method (SGBEM). The infinite medium is first decomposed into two subdomains; the first one that is finite, localized, and may contain a nonlinear region is modeled by the FEM while the other that is unbounded and may contain the discontinuity surface is treated by the SGBEM. Use of boundary integral equations to treat an infinite region instead of the FEM offers two advantages; one corresponding to the mathematical ease of the governing equations and the numerical discretization effort (i.e. a set of governing equations involving only integrals over the boundary of the domain). Another key feature of the current technique is the use of a pair of weak-form integral equations for the displacement and for the traction to establish the formulation for the SGBEM. Such a pair of integral equations are weakly singular in the sense that they contain only kernels of order O(1/r) and, as a consequence, they only require continuous interpolations in the approximation procedure. To demonstrate accuracy and versatility of the current technique, numerous numerical experiments are performed and numerical solutions for selected cases are reported and discussed.