Rules governing the numbers of nodes and elements in a finite element mesh

Abstract

In any mesh, rules exist that interrelate the number of internal and external sides, vertices, etc. and the total number of elements. These are given explicitly for plane meshes of triangles and quadrilaterals, and for solid meshes of tetrahedra and cuboidal elements. The method is quite general and discovers all such independent rules that exist. Thus, for a plane mesh of T elements having V_i internal and V_b boundary vertices and S_i internal and S_b boundary sides, then where H is the number of internal boundaries (holes) there might be. For solid meshes, these two‐dimensional equations relating elements to sides generalize to where there are F_b boundary and F_i internal faces. Unfortunately, there is no direct generalization of the two‐dimensional equations relating vertices and elements: it is only possible to do this by including the E_i internal and E_b boundary edges: where there are H through holes and h cavities.