Hexahedral volume coordinate method (HVCM) and improvements on 3D Wilson hexahedral element

A new volume coordinate method for developing 3D hexahedral elements, called hexahedral volume coordinate method (HVCM), is systematically established in this paper: (i) several characteristic parameters of a hexahedron are defined and the degeneration conditions under which a hexahedron degenerates into other special polyhedrons are given; (ii) the volume coordinates (L1,L2,L3,L4,L5,L6)(L1,L2,L3,L4,L5,L6) of any point within a convex hexahedron are defined; (iii) transformation relations between the volume coordinates and the Cartesian or isoparametric coordinates are presented; (iv) the differential formulas for volume coordinates in hexahedral elements are given. This new coordinate system can keep not only the advantages of local natural coordinate system, but also a linear relation with the Cartesian coordinate system. Then, for checking the validity of the new HVCM, it is used to formulate three new incompatible eight-node hexahedral elements, HVCC8, HVCC8-ES and HVCC8-EM, by a similar procedure of famous Wilson’s incompatible mode. Numerical results show that the present elements exhibit much better performance than that of conventional isoparametric elements in most distorted mesh cases, especially for MacNeal’s thin beam problem. It demonstrates that the new HVCM is a powerful tool for constructing high-performance hexahedral finite element models.