Local topological modification of hexahedral meshes. Part I: A set of dual-based operations

Abstract

Abstract. For hexahedral meshes, it is difficult to make topological modifications which preserve conformal property and which are local to the modified elements. This is most easily understood in how changes affect the dual surfaces ("sheets") and lines ("chords"), which have non-local extent in hexahedral meshes. A set of three operations is proposed which represent distinct local changes to the hex mesh and its dual. These operations are shown to compose the larger set of "flipping" operations described by Bern et. al. The relation between these dual-based operations and the insertion of a Boy surface in the dual, described by Bern, is also discussed. . INTRODUCTIONFinite element analysis and other methods for numerical solution of PDE's require the decomposition of space into a "mesh" (a polyhedral complex where each polyhedron has a prescribed topology). Such decompositions are often required to be conformal [1], where elements fill space, constitute a topology, and whose intersections are either empty or consist of lowerdimensional elements of the topology. Two types of elements are most common in FEA, hexahedra and tetrahedra. Hexahedral meshes are preferred for some kinds of analysis because of their accuracy for a given amount of compute cost [2] [3] 4 A general-purpose method for generating allhexahedral meshes of suitable geometric quality for FEA has not yet been demonstrated. We assert that an important part of any reliable method will be the ability to locally modify the connectivity of the hex mesh; we refer to this as "local hex mesh modification". Previous efforts to describe hex mesh modification have been either ad-hoc or have not described the relation to the hex mesh dual. In this paper, a set of dual-based hex mesh modification operations are described. We demonstrate how other known modifications can be composed from our set of operations. The paper concludes with a discussion of the Boy surface and how it relates to these operations.Body-fitted hexahedral mesh generation has been a topic of study for over ten years. Algorithms can be loosely classified into two groups. "Inside-out" algorithms start by filling a solid with a prescribed regular mesh, usually cartesian, then trim the mesh to match the boundary geometry and topology [4][5]. These meshes often have poor quality near the boundary which, for some applications, is precisely where good quality is most needed. These algorithms also tend to produce more hex elements than is strictly necessary. Inside-out algorithms are not widely used for these reasons. "Outside-in" or advancing-front algorithms start with a prescribed quadrilateral boundary mesh, and seek to fill the interior with hex elements. Methods have been investigated for doing this geometrically [6] or based on connectivity [7], with some of the connectivity-based methods making some boundary modifications [8] or losing some advancing front qualities [9][10] in order to improve chances of closing the mesh. Where mesh closure was reliably achieved [9], mesh quality typically...

Unconstrained plastering—Hexahedral mesh generation via advancing‐front geometry decomposition

“…After all-hexahedral elements are created during unconstrained plastering, element quality should be optimized. In 3D, the ability to locally modify hexahedral topology is limited [68][69][70]. However, at a minimum, smoothing [63] should be performed to improve the quality of the mesh.…”

Fun sheet matching: towards automatic block decomposition for hexahedral meshes

“…Unlike tetrahedral meshes, hexahedral meshes have an inherent layered structure, which makes both local modifications [21][22][23][24][25] and automatic generation of all-hexahedral meshes difficult. The layered structure in hexahedral meshes is the primary reason for robustness issues with previous attempts at all-hexahedral meshing which rely on a pre-meshed boundary.…”

Hexahedral Mesh Matching: Converting non‐conforming hexahedral‐to‐hexahedral interfaces into conforming interfaces

“…Thus, insertion of self-touching and self-intersecting sheets should be avoided. In addition, neither pillowing nor dicing can insert several theoretical sheets, such as sheets containing triple intersections and atomic pillow sheets [32], which are rarely useful for anything other than academic research. However, for completeness, in this section we introduce a new operator, called sheet inflation, which extends the functionality spectrum by allowing the insertion of both arbitrary self-intersecting and arbitrary self-touching sheets.…”