On global and local mesh refinements by a generalized conforming bisection algorithm

Abstract

MSC: 65M50 65N30 65N50Keywords: Zlámal's minimum angle condition Finite element method Nested triangulations Conforming longest-edge bisection algorithm High aspect ratio elements a b s t r a c tWe examine a generalized conforming bisection (GCB-)algorithm which allows both global and local nested refinements of the triangulations without generating hanging nodes. It is based on the notion of a mesh density function which prescribes where and how much to refine the mesh. Some regularity properties of generated sequences of refined triangulations are proved. Several numerical tests demonstrate the efficiency of the proposed bisection algorithm. It is also shown how to modify the GCB-algorithm in order to generate anisotropic meshes with high aspect ratios.

The minimum angle condition for d-simplices

“…Assume first that the condition in Definition 2.3 is satisfied, so that there exists a constant C > 0 as in (7). Using (8), we get that…”

A mathematical proof of how fast the diameters of a triangle mesh tend to zero after repeated trisection

“…It is already well established that the assumption of regularity over the meshes [3], i.e., the bounded ratio between the outer and inner diameters, leads to the convergence of standard finite element methods. As a consequence of the convergence of the diameters to zero, the bisection method has been proven to be useful in FEM for approximating solutions of differential equations [20,24,25].…”

“…and others. In this sense, the generation of robust, reliable local mesh refinements for the production of meshes for finite element or finite difference methods is a significant area of study, together with the geometric and topological properties of the triangle or tetrahedral partitions [3,5,15,19,28].…”

A Weak Compatibility Condition for Newest Vertex Bisection in Any Dimension

“…Possible implementation of methods OT0 and SRN2 Data: List of active faces F, List of verticesV, Mesh T 1 F = V = ∅ 2 Choose initial T ∈ T 3 Order according to announced refinement edge E(T ) 4 V = V(T ) 5 add all non-boundary F ∈ T d−1 (T ) with E(T ) ∈ F at the beginning of F 6 add the other non-boundary F ∈ T d−1 (T ) at the end of F with the flag noRefEdge 7 mark T as treated 8 while ∃ untreated T ∈ T (⇔ F = ∅) do 9Get untreated neighbor T and treated element T of first F ∈ F10 v = T 0 (T ) \ T 0 (F ) v = T 0 (T ) \ T 0 (F ) 11if v ∈ V then /* do nothing */ noRefEdge set then /* In this case v is the first or last vertex of T */14 insert v after last or before first vertex of T , depending on the announced refinement edge15 else non-boundary F ∈ T d−1 (T ) do 22 if F ∈ F then /* possibly check for strong compatibility */ E(T ) ⊂ F then 26 add F at beginning of F 27 else 28 add F at the end of F…”