The simplex subdivision of a complex region: a positive and negative finite element superposition principle

In the case of very complex geometric morphology in three-dimensional (3-d) calculation regions, the subdivision of finite element meshes tends to be very difficult or even impossible. This problem was especially dominant in the element subdivision of rock blocks cut by arbitrary fracture networks. This study proposed a directed simplex subdivision method of 2- and 3-d regions which can readily subdivide a complex region and automatically adapt to arbitrary variations of region boundaries. The directed simplex was employed as finite elements for the first time. When the nodes of 2-d simplex (triangle) were connected anticlockwise and the nodes of 3-d simplex (tetrahedron) were linked by the right hand screw rule, the finite element is treated as a positive element. Otherwise, it is treated as a negative element. However, some generated elements presented superposition, which was unallowable in the normal finite element method (FEM). Therefore, the superposition principle of positive and negative finite elements was proposed. Such superposition method can transform a normal FEM mesh into a positive–negative FEM mesh with maintaining its completeness and compatibility. Finally, 1-d, 2-d, and 3-d cases were used as examples to verify the validity of the proposed method. The completeness and compatibility of the positive–negative FEM and the normal FEM are consistent. The accuracy of these two methods differs due to the difference of nodal connection relationships, node numbers, and element sizes. However, as the mesh is refined, the calculated results tend to be consistent.