Meshless Local Petrov-Galerkin (MLPG) approaches for solving the weakly-singular traction & displacement boundary integral equations

The general Meshless Local Petrov-Galerkin (MLPG) type weak-forms of the displacement & traction boundary integral equations are presented, for solids undergoing small deformations. These 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 dBIE 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. Three specific types of MLPG/BIEs are formulated in the present study, according to three different types of test functions assumed over a local sub-boundary surface, as: a) the weight function in the MLS, for formulating the MLPG/BIE1; b) a Dirac delta function for formulating the collocation method (MLPG/BIE2); c) the trial function itself, for formulating the MLPG/BIE6. As a special case, the MLPG/BIE6 leads to symmetric systems of equations, and are presented in default in the present study. Numerical examples, presented in the accompanying part II of this paper, show that the present methods are very promising, especially for solving the elastic problems in which the singularities in displacemens, strains, and stresses, are of primary concern.