On the shape of tetrahedra from bisection

Abstract

We present a procedure for bisecting a tetrahedron T successively into an infinite sequence of tetrahedral meshes ¡Tü , ZTX , ST1, ... , which has the following properties: (1) Each mesh !Tn is conforming.(2) There are a finite number of classes of similar tetrahedra in all the ¿Tn , n > 0. (3) For any tetrahedron T? in ¿T" , n(T") > cxn(T), where n is a tetrahedron shape measure and cx is a constant. (4) <5(T?) < c2(l/2)"/3«5(T), where <J(T') denotes the diameter of tetrahedron T and c2 is a constant.Estimates of cx and c2 are provided. Properties (2) and (3) extend similar results of Stynes and Adler, and of Rosenberg and Stenger, respectively, for the 2-D case. The diameter bound in property (4) is better than one given by Kearfott.

On generalized bisection of 𝑛-simplices

“…Sharper bounds and a number of additional results regarding conformal triangular meshes and similarity classes are known for the two-dimensional case of triangles [1,15,17,18] and for some special three-dimensional simplices [5,11,16].…”

Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods

“…½ Ù is smooth (27) In order to illustrate the efficiency of the smoothness indicator, we consider the following problem.…”

“…Determine the elements that cross a discontinuity using the algorithm (27); In elements where the solution is smooth, use (14) and (15) to compute an anisotropic metric field; In elements where the solution is discontinuous, use the reconstructed gradients ÖÛ solution of (20) to compute the normal direction Ò to the discontinuity. Then, build up an anisotropic metric (14) and (15)).…”

“…Sliver tetrahedra (poorly-shaped tetrahedra not bounded by any short mesh edge in transformed space) may exist even if the edge length criteria is met, so an additional criteria is needed to determine and eliminate sliver tetrahedra. One of the standard non-dimensional shape measure, the cubic of mean ratio [27] in the transformed space, is used for this purpose. Let É be the associated transformation matrix of the tetrahedron 4 , the cubic of mean ratio in transformed space, ¼ , is:…”

On a Bisection Algorithm That Produces Conforming Locally Refined Simplicial Meshes

“…Later, in the 80-th, mainly due to efforts of M. C. Rivara, bisection-type algorithms became popular also in the FEM community for mesh refinement/adaptation purposes [12,13,14,15]. Several variants of the algorithm suitable for standard FEMs were also proposed, analysed and numerically tested in [1,2,3,8,10,11] (see also references therein). It has been commonly noticed that inspite of a general simplicity of this type of bisection algorithms, it turns to be hard to provide mesh conformity and simultaneously to prove relevant mesh regularity results [4,20], especially in the case of local mesh refinements and in higher dimensions.…”