An accurate solution to the meshless local Petrov-Galerkin formulation in elastodynamics

A meshless local Petrov-Galerkin (MLPG) method for solving elastodynamic problems is developed and numerically implemented. The proposed MLPG approach is based on a locally symmetric weak form and shape functions from the moving least squares (MLS) approximation. This approach is truly meshless, as it does not involve a finite element mesh, either to interpolate the solution variables, or to integrate the energy. However, complex vibrating-modes or -frequencies may arise from asymmetric mass and stiffness matrices formulated by the MLPG method. Unlike the commonly used finite difference methods such as the Newmark method, the accurate approach in this study is the time-discontinuous Galerkin (TDG) method for solving second-order ordinary differential equations in the time domain. Numerical results indicate that the TDG method provides very stable and accurate results in the sense that the crucial modes are accurately integrated and the spurious modes are successfully filtered out.