Meshless Local Petrov-Galerkin (MLPG) mixed finite difference method for solid mechanics

The Finite Difference Method (FDM), within the framework of the Meshless Local Petrov-Galerkin (MLPG) approach, is proposed in this paper for solving solid mechanics problems. A "mixed" interpolation scheme is adopted in the present implementation: the displacements, displacement gradients, and stresses are interpolated independently using identical MLS shape functions. The system of algebraic equations for the problem is obtained by enforcing the momentum balance laws at the nodal points. The divergence of the stress tensor is established through the generalized finite difference method, using the scattered nodal values and a truncated Taylor expansion. The traction boundary conditions are imposed in the stress equations directly, using a local coordinate system. Numerical examples show that the proposed MLPG mixed finite difference method is both accurate and efficient, and stable.