A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach

The Galerkin finite element method (GFEM) owes its popularity to the local nature of nodal basis functions, i.e., the nodal basis function, when viewed globally, is non-zero only over a patch of elements connecting the node in question to its immediately neighboring nodes. The boundary element method (BEM), on the other hand, reduces the dimensionality of the problem by one, through involving the trial functions and their derivatives, only in the integrals over the global boundary of the domain; whereas, the GFEM involves the integration of the “energy” corresponding to the trial function over a patch of elements immediately surrounding the node. The GFEM leads to banded, sparse and symmetric matrices; the BEM based on the global boundary integral equation (GBIE) leads to full and unsymmetrical matrices. Because of the seemingly insurmountable difficulties associated with the automatic generation of element-meshes in GFEM, especially for 3-D problems, there has been a considerable interest in element free Galerkin methods (EFGM) in recent literature. However, the EFGMs still involve domain integrals over shadow elements and lead to difficulties in enforcing essential boundary conditions and in treating nonlinear problems.