A MLPG (LBIE) method for solving frequency domain elastic problems.

*(English)*Zbl 1064.74170Summary: A new meshless local Petrov-Galerkin (MLPG) method for solving two-dimensional frequency domain elastodynamic problems is proposed. Since the method utilizes, in its weak formulation, either the elastostatic or the frequency domain elastodynamic fundamental solution as test function, it is equivalent to the local boundary integral equation (LBIE) method. Nodal points spread over the analyzed domain are considered, and the moving least squares (MLS) interpolation scheme for the approximation of the interior and boundary variables is employed. Two integral equations suitable for the integral representation of the displacement fields in local sub-domains are used. The first utilizes the frequency domain fundamental solution, comprises only boundary integrals and exploits the elastodynamic companion solution, which is derived in the framework of the present work. The second equation makes use of the simple elastostatic fundamental solution, employs the elastostatic companion solution in order to get rid of tractions on the local boundaries and contains both boundary and volume integrals. On the global boundary, derivatives of the shape functions of the MLS approximation are avoided by considering displacements and tractions as independent variables. Direct numerical techniques for the accurate evaluation of both surface and volume integrals are employed and presented in detail. All the strongly singular integrals are computed directly through highly accurate integration techniques. Three representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.

##### MSC:

74S15 | Boundary element methods applied to problems in solid mechanics |

74H45 | Vibrations in dynamical problems in solid mechanics |

74H15 | Numerical approximation of solutions of dynamical problems in solid mechanics |