Adaptive refinement of unstructured finite-element meshes
Finite Elements in Analysis and Design
Lepp-bisection algorithms, applications and mathematical properties
Applied Numerical Mathematics
Multithread parallelization of Lepp-bisection algorithms
Applied Numerical Mathematics
Technical note: Longest-edge algorithms for size-optimal refinement of triangulations
Computer-Aided Design
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The bisection method is the consecutive bisection of a triangle by the median of the longest side. In this paper we prove a subexponential asymptotic upper bound for the number of similarity classes of triangles generated on a mesh obtained by iterative bisection, which previously was known only to be finite. The relevant parameter is γ/σ, where γ is the biggest and σ is the smallest angle of the triangle. We get this result by introducing a taxonomy of triangles that precisely captures the behaviour of the bisection method. We also prove that the number of directions on the plane given by the sides of the triangles generated is finite. Additionally, we give purely geometrical and intuitive proofs of classical results for the bisection method.