Application of meshless local Petrov-Galerkin (MLPG) method to elastodynamic problems in continuously nonhomogeneous solids

A new computational method for solving transient elastodynamic initial-boundary value problems in continuously non-homogeneous solids, based on the meshless local Petrov-Galerkin (MLPG) method, is proposed in the present paper. The moving least squares (MLS) is used for interpolation and the modified fundamental solution as the test function. The local Petrov-Galerkin method for unsymmetric weak form in such a way is transformed to the local boundary integral equations (LBIE). The analyzed domain is divided into small subdomains, in which a weak solution is assumed to exist. Nodal points are randomly spread in the analyzed domain and each one is surrounded by a circle centered at the collocation point. The boundary-domain integral formulation with a static fundamental solution in Laplace transform domain is used to obtain the weak solution for subdomains. On the boundary of the subdomain, both the displacement and the traction vectors are unknown generally. If a modified fundamental solution vanishing on the boundary of the subdomain is employed, the traction vector is eliminated from the local boundary integral equations for all interior nodal points. The spatial variation of the displacements is approximated by the MLS scheme.