Method of Volume Coordinates — from Tetrahedral to Hexahedral Elements

In the development history of finite element method, an element model with high performance possesses important significance. The three-dimension (3D) hexahedral isoparametric elements are widely used in scientific and engineering computations. However, their accuracy may drop obviously in an irregular mesh division. In order to improve the robustness of these elements, many researchers have made great efforts, and these jobs have never been stopped. On the other hand, for two-dimension (2D) problems, the area coordinate method has been successfully genrealized from triangular to quadrilateral elements. Compared with the models constructed by isoparametric coordinates, those quadrilateral elements by the area coordinate method are less sensitive to mesh distortion. Following some successful applications of the area coordinate method for quadrilateral elements in 2D problems, a new volume coordinate method for hexahedral elements in 3D problems is systematically established in this paper. (i) the shape parameters of a convex hexahedron are defined, and the related characteristic conditions under which a hexahedron degenerates into other special polyhedron are discussed in details; (ii) the volume coordinates for hexahedral elements are defined; (iii) the transformation relations between the volume coordinate and the Cartesian or isoparametric coordinates are presented; (iv) several important differential formulas for hexahedral volume coordinates are given. This new system has several notable advantages: firstly, the coordinate transformation between volume coordinate and global coordinate is linear; secondly, boundary conditions are easy to express and be satisfied; thirdly, the stiffness matrix of the element constructed by the volume coordinate method can be easily formulated with exact numerical integration. It provides a new tool for developing high performance hexahedral element models.