For a point set P on the plane, a four element subset S ⊂ P is called a 4-hole of P if the convex hull of S is a quadrilateral and contains no point of P in its interior. Let R be a point set on the plane. We say that a point set B covers all the 4-holes of R if any 4-hole of R contains an element of B in its interior. We show that if |R| ≥ 2|B|+5 then B cannot cover all the 4-holes of R. A sim...

This article presents special quadrilateral quadratic refinement elements, which provide geometry and field continuity across Tjunctions where two elements are connected to one side of a larger quadrilateral. The main idea in element refinement is to place one or more nodes outside the element area and to modify element shape functions in order to maintain continuity at refinement edges. Specia...

An automatic adaptive quadrilateral mesh conversion scheme for the generation of adaptive refinement meshes over analytical curved surfaces is proposed. The starting point of the quadrilateral mesh generator is a background triangular mesh of the curved surface. By a carefully controlled process to merge two triangles at a time the triangular mesh can be completely converted to quadrilaterals. ...

\The chance of : : : the quadrilateral formed by joining four points, taken arbitrarily within any assigned boundary, constituting a reentrant or convex quadrilateral, will serve as types of the class of questions in view." |J.J. Sylvester 11] We prove that two fundamental constants of the geometry of the plane are equal. First, if R is an open set in the plane with nite Lesbesgue measure, let ...

Abstract In this paper, we give upper and lower bounds on the number of Steiner points required to onstru t a stri tly onvex quadrilateral mesh for a planar point set. In parti ular, we show that 3⌊ 2 ⌋ internal Steiner points are always suÆ ient for a onvex quadrilateral mesh of n points in the plane. Furthermore, for any given n ≥ 4, there are point sets for whi h ⌈ 2 ⌉ − 1 Steiner points are...

In this work, a new method for mesh simpliication and surface reconstruction speciically designed for the needs of CAD/CAM engineering design and analysis is introduced. The method simpliies the original free-form face model by rst constructing restricted curvature deviation regions, generating a boundary conforming nite element quadrilateral mesh of the regions, and then tting a smooth surface...

This chapter explains techniques for the generation of quadrilateral and hexahedral element meshes. Since structured meshes are discussed in detail in other parts of this volume, we focus on the generation of unstructured meshes, with special attention paid to the 3D case. Quadrilateral or hexahedral element meshes are the meshes of choice for many applications, a fact that can be explained emp...