Mathematical Basis of G Spaces

This paper represents some basic mathematic theories for Gs spaces of functions that can be used for weakened weak (W-2) formulations, upon which the smoothed finite element methods (S-FEMs) and the smoothed point interpolation methods (S-PIMs) are based for solving mechanics problems. We first introduce and prove properties of Gs spaces, such as the lower boundedness and convergence of the norms, which are in contrast with H-1 spaces. We then prove the equivalence of the Gs norms and its corresponding seminorms. These mathematic theories are important and essential for the establishment of theoretical frame and the development of relevant numerical approaches. Finally, numerical examples are presented by using typical S-FEM models known as the NS-FEM and alpha S-FEM to examine the properties of a smoothed method based on Gs spaces, in comparison with the standard FEM with weak formulation.