Truly meshless local Petrov-Galerkin (MLPG) solutions of traction & displacement BIEs

The numerical implementation of the truly Meshless Local Petrov-Galerkin (MLPG) type weak-forms of the displacement and traction boundary integral equations is presented, for solids undergoing small deformations. In the accompanying part 1 of this paper, the general MLPG/BIE weak-forms were presented [Atluri, Han and Shen (2003)]. The MLPG weak forms provide the most general basis for the numerical solution of the non-hyper-singular displacement and traction BIEs [given in Han, and Atluri (2003)], which are simply derived by using the gradients of the displacements of the fundamental solutions [Okada, Rajiyah, and Atluri (1989a,b)]. By employing the various types of test functions, in the MLPG-type weak-forms of the non-hyper-singular dB1E and tBIE over the local sub-boundary surfaces, several types of MLPG/BIEs are formulated, while also using several types of non-element meshless interpolations for trial functions over the surface of the solid. Specifically, three types of MLPG/BIEs are formulated in that paper, i.e. MLPG/BIE1, MLPG/BIE2, and MLPG/BIE6, as per the consistent categorizations of the MLPG domain methods [Atluri and Shen (2002a)]. As the accompanying part II, this paper is devoted to MLPG/BIE6. In particular, the moving least squares (MLS) method has been extended for the approximation on three dimensional surfaces, which makes it possible for the MLPG/BIE methods to be truly meshless. Numerical examples, including crack problems, are presented to demonstrate that the present methods are very promising, especially for solving the elastic problems in which the singularities in displacements, strains, and stresses, are of primary concern.