An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics

Smoothed finite element method (SFEM) using quadrilateral elements was recently proposed by Liu et al. [A smoothed finite element method for mechanics problems, Comput. Mech. 39 (2007) 859–877; Free and forced vibration analysis using the smoothed finite element method (SFEM), J. Sound Vib. 301 (2007) 803–820; Theoretical aspects of the smoothed finite element method (SFEM), Int. J. Numer. Methods Eng. (2006), in press] to improve the accuracy and convergence rate of the existing standard four-node finite element method (FEM). In this paper the SFEM is further extended to a more general case, n-sided polygonal smoothed finite elements (nSFEM), in which the problem domain can be discretized by a set of polygons, each with an arbitrary number of sides. Stability condition is examined for this type of new elements and some criteria are provided to avoid the presence of spurious zero-energy modes. Approach to constructing nSFEM shape functions are also suggested with emphasis on a novel and simple averaging method. Selective integration scheme is recommended to overcome volumetric locking for nearly incompressible materials. Several numerical examples are investigated and the present results are in good agreement with exact solutions or FEM results. It is found that the present method gives very accurate stresses and desirable convergence rate as compared with FEM. In addition problem domain can be discretized in a very flexible manner as demonstrated in the examples.