ISSN:
0029-5981

Keywords:
Engineering
;
Engineering General

Source:
Wiley InterScience Backfile Collection 1832-2000

Topics:
Mathematics
,
Technology

Notes:
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 Vi internal and Vb boundary vertices and Si internal and Sb boundary sides, then \documentclass{article}\pagestyle{empty}\begin{document}$$\begin{array}{l} T = \frac{1}{3}\left({S_b + 2S_i} \right) = V_b + 2V_i + 2H - 2\quad{\rm (for\,triangular\,elements)}\\ T = \frac{1}{4}\left({S_b + 2S_i} \right) = \frac{1}{2}\left({V_b + 2V_i} \right) + H - 1\quad{\rm (for\,quadrilateral\,elements)} \\ \end{array} $$\end{document} 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 \documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{l} T = \frac{1}{6}\left({F_b + 2Fi} \right){\rm\quad (for\,cuboid\,elements)} \\ T = \frac{1}{4}\left({F_b + 2Fi} \right){\rm\quad (for\,tetrahedral\,elements)} \\ \end{array} $$\end{document} where there are Fb boundary and Fi 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 Ei internal and Eb boundary edges: \documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{l} T = \frac{1}{8}E_b + \frac{1}{2}\left({E_i - V_i + H - h - 1} \right){\rm\quad (for\,cuboid\,elements)} \\ T = \frac{1}{3}E_b + E_i - V_i + H - H - 1{\rm\quad (for\,tetrahedral\,elements)} \\ \end{array} $$\end{document} where there are H through holes and h cavities.

Additional Material:
3 Ill.

Type of Medium:
Electronic Resource

Permalink