Highly accurate smoothed finite element methods based on simplified eight-noded hexahedron elements

Abstract

Compared with the tetrahedron elements, hexahedron elements are preferred for their high accuracy. However, coordinate mapping required in the hexahedron elements of FEM formulation costs huge running time, leading to poor performance. Besides, the high quality of Jacobian matrix and mesh is required, which affects the accuracy of the strain results greatly. In order to solve these problems, we propose a novel simplified integration technique based on the smoothed finite element method (S-FEM) for the eight-noded hexahedron elements, where coordinate mapping is not demanded. The proposed new S-FEM-H8 models include simplified NS-FEM-H8 (using node-based smoothing domains) and simplified FS-FEM-H8 (using face-based smoothing domains). In the work, we divide a quadrilateral surface segment of a smoothing domain into two triangular sub-segments, so that the strain-displacement matrix can be calculated using a simple summation in the S-FEM theory instead of the integration in FEM. Then we conduct the Gauss integration scheme in each triangular surface sub-segment in order to avoid the coordinate mapping required in quadrilateral surface segments. The rest solving algorithm is the same as the standard S-FEM. Intensive numerical examples demonstrate that the simplified S-FEM-H8 possess the following features: (1) The strain energy of simplified NS-FEM-H8 is an upper bound of the exact solutions; (2) The simplified NS-FEM-H8 can overcome the volume locking problems for incompressible materials; (3) The method of dividing boundary surface into two triangular surfaces in smoothing domain keeps nearly the same accuracy as the standard S-FEM-H8.