An effective sub-domain smoothed Galerkin method for free and forced vibration analysis

Free and forced vibration analysis is significant in structural mechanics and engineering. A sub-domain smoothed Galerkin method is developed for free and forced vibration analyses of 2D solids. This method integrates the advantages of mesh-free Galerkin method and finite element method (FEM). The arbitrarily shaped sub-domains are predefined in the problems domain represented by mesh-free nodes. In each sub-domain, based on mesh-free Galerkin weak formulation, the local discrete equation can be obtained by using moving kriging interpolation, which is similar to the discrete equation of a high-order element in FEM. The strain smoothing technique can be subsequently applied to the nodal integration of sub-domain by dividing the sub-domain into several smoothing cells. Moreover, condensation of degrees of freedom can also be introduced into discrete equations by transfer equations of inner nodes to equations of boundary nodes based on sub-domains. The global dynamic equations of the present method are obtained based on the scheme of FEM by assembling all local discrete equations of sub-domains and are solved by using the standard implicit Newmark’s time integration scheme. Numerical examples proved that the sub-domain smoothed Galerkin method is a robust technique to simulation the elastic dynamic problems, which has high computational efficiency and good accuracy.